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On Cacti and Crystals

  • Arkady BerensteinEmail author
  • Jacob Greenstein
  • Jian-Rong Li
Chapter
Part of the Progress in Mathematics book series (PM, volume 330)

Abstract

In the present work we study actions of various groups generated by involutions on the category \(\mathscr O^{int}_q({\mathfrak {g}})\) of integrable highest weight \(U_q({\mathfrak {g}})\)-modules and their crystal bases for any symmetrizable Kac–Moody algebra \({\mathfrak {g}}\). The most notable of them are the cactus group and (yet conjectural) Weyl group action on any highest weight integrable module and its lower and upper crystal bases. Surprisingly, some generators of cactus groups are anti-involutions of the Gelfand–Kirillov model for \(\mathscr O^{int}_q({\mathfrak {g}})\) closely related to the remarkable quantum twists discovered by Kimura and Oya (Int Math Res Notices, 2019).

MSC:

17B37 17B10 (primary); 20F36 18D10 20F55 (secondary) 

1 Introduction

In the present work we study the action of various groups generated by involutions on the category \(\mathscr O^{int}_q({\mathfrak {g}})\) of integrable highest weight \(U_q({\mathfrak {g}})\)-modules for any symmetrizable Kac–Moody algebra \({\mathfrak {g}}\) (the necessary notation is introduced in Sect. 2).

Let \(V\in \mathscr O^{int}_q({\mathfrak {g}})\). We claim that for every node i of the Dynkin diagram I of \({\mathfrak {g}}\) there exists a unique linear operator \(\sigma ^i_V\) on V  such that
$$\displaystyle \begin{aligned}\sigma^i_V(E_i^{(k)}(u))=E_i^{(l-k)}(u) \end{aligned} $$
(1.1)
for all l ≥ k ≥ 0 and for all \(u\in \ker F_i\cap \ker (K_i-q_i^{-l})\). Clearly, \((\sigma ^i_V)^2= \operatorname {\mathrm {id}}_V\). Denote by W(V ) the subgroup of Open image in new window generated by the \(\sigma ^{i}_V\), i ∈ I.

Theorem 1.1

For any non-zero module \(V\in \mathscr O^{int}_q({\mathfrak {g}})\) , the assignments
$$\displaystyle \begin{aligned} \sigma^i_V\mapsto \begin{cases}1,& i\in J(V)\\ s_i,& \mathit{\text{otherwise}} \end{cases} \end{aligned}$$

where J(V ) = {i  I  :  Fi(V ) = {0}}, define a homomorphism ψV from W(V ) to the Weyl group W of \({\mathfrak {g}}\).

We prove Theorem 1.1 in Sect. 3.3 by showing that the image of ψV can be described in terms of a natural action of W on a certain set of extremal vectors in V . In particular, ψV is surjective if and only if J(V ) = ∅. Moreover, we show that \(\sigma ^i_V= \operatorname {\mathrm {id}}_V\) if and only if i ∈ J(V ). This suggests the following

Conjecture 1.2

The homomorphism ψV is injective for any \(V\in \mathscr O^{int}_q({\mathfrak {g}})\).

Clearly, it is equivalent to \((\sigma ^i\sigma ^j)^{m_{ij}}= \operatorname {\mathrm {id}}_V\), i ≠ j ∈ I for appropriate choices of mij. We proved it for mij = 2 and we have ample evidence that this conjecture holds for mij = 3. We also verified it for all modules in which weight spaces of non-zero weight are one-dimensional (see Theorem 7.2). This class of modules includes all miniscule and quasi-miniscule ones. Conjecture 1.2 combined with Theorem 1.1 implies that W acts naturally and faithfully on objects in \(\mathscr O^{int}_q({\mathfrak {g}})\), which is quite surprising. Informally speaking, this conjecture asserts that Kashiwara’s action of the Weyl group on crystal bases lifts to an action on the corresponding module (see Remark 5.7).

Remark 1.3

The definition (1.1) of \(\sigma ^i_V\) makes sense for any integrable \(U({\mathfrak {g}})\)-module where \({\mathfrak {g}}\) is a semisimple or a (not necessarily symmetrizable) Kac–Moody Lie algebra. The “classical” Theorem 1.1 holds verbatim. Moreover, Conjecture 1.2 implies its classical version for all (even not symmetrizable) Kac–Moody algebras.

It turns out that we can extend the group W(V ) by adding involutions σJ for any non-empty J ⊂ I such that the subgroup WJ = 〈si  :  i ∈ J〉 is finite; we denote the set of all such J by \(\mathscr J\). Note that \(\{i\}\in \mathscr J\) for all i ∈ I and in particular \(\mathscr J\) is non-empty.

Proposition 1.4 (Proposition 4.14)

For any \(V\in \mathscr O^{int}_q({\mathfrak {g}})\) , \(J\in \mathscr J\) there exists a unique Open image in new window -linear map \(\sigma ^J=\sigma ^J_V:V\to V\) such that
  1. (a)

    σJ(v) = vJ for any \(v\in \bigcap _{i\in J} \ker E_i\) where vJ is a distinguished element in \(\bigcap \limits _{i\in J}\ker F_i\cap U_q({\mathfrak {g}}^J)v\) defined in Proposition 4.14(a) ;

     
  2. (b)

    \(\sigma ^J(F_j(v))=E_{j^\star }(\sigma ^J(v))\), \(\sigma ^J(E_j(v))=F_{j^\star }(\sigma ^J(v))\) for all j  J, v  V  where  : J  J is the involution on J induced by the longest element \(w_\circ ^J\) of WJ via \(s_{j^\star }=w_\circ ^J s_j w_\circ ^J\), j  J (see Sect.2.1).

     

Moreover, for any morphism f : V  V in \(\mathscr O^{int}_q({\mathfrak {g}})\) the following diagram commutes

By definition, σJ = σi if J = {i}. The following is the main result of this paper.

Theorem 1.5

Let \(V\in \mathscr O^{int}_q({\mathfrak {g}})\) . Then for any \(J\in \mathscr J\) we have in Open image in new window
  1. (a)

    σJ ∘ σJ = 1;

     
  2. (b)

    If J = J J′′ where J and J′′ are orthogonal (that is, J J′′ = ∅ and \(s_{j^{\prime }}s_{j^{\prime \prime }}=s_{j^{\prime \prime }}s_{j^{\prime }}\) for all j J, j′′ J′′), then \(\sigma ^J=\sigma ^{J^{\prime }}\circ \sigma ^{J^{\prime \prime }}\); in particular, \(\sigma ^{J^{\prime }}\circ \sigma ^{J^{\prime \prime }}=\sigma ^{J^{\prime \prime }}\circ \sigma ^{J^{\prime }}\) if \(J^{\prime },J^{\prime \prime }\in \mathscr J\) are orthogonal.

     
  3. (c)

    \(\sigma ^{J}\circ \sigma ^{K}=\sigma ^{K^\star }\circ \sigma ^J\) for any K  J, where  : J  J is as in Proposition 1.4(b) .

     

We prove Theorem 1.5 in Sect. 4.3 using appropriate modifications of Lusztig’s symmetries (which we introduce in Sect. 4.1).

Following (and slightly generalizing) [29] (see also [10]), we denote \( \operatorname {\mathrm {\mathsf {Cact}}}_W\) the group generated by the τJ, \(J\in \mathscr J\) subject to all relations of Theorem 1.5. Indeed, this definition coincides with that in [29, (1.1)] if W is finite because \(\tau _J=\tau _{J^{\prime }}\tau _{J^{\prime \prime }}\) for any J as in Theorem 1.5(b). By definition, the assignments \(\tau _J\mapsto \sigma ^J_V\), \(J\in \mathscr J\) define a representation of \( \operatorname {\mathrm {\mathsf {Cact}}}_W\) on V . In view of (1.2) we obtain the following immediate corollary of Theorem 1.5 (see Sect. 4 for the notation)

Corollary 1.6

The group \( \operatorname {\mathrm {\mathsf {Cact}}}_W\) acts on the category \(\mathscr O^{int}_q({\mathfrak {g}})\) via \(\tau _J\mapsto \sigma ^J_\bullet \), \(J\in \mathscr J\).

The study of cactus groups began with \( \operatorname {\mathrm {\mathsf {Cact}}}_n:= \operatorname {\mathrm {\mathsf {Cact}}}_{S_n}\) which appeared, to name but a few, in [13, 14, 16, 33, 35] in connection with the study of moduli spaces of rational curves with n + 1 marked points and their applications in mathematical physics. It is easy to see that \( \operatorname {\mathrm {\mathsf {Cact}}}_n\) is generated by involutions τi,j = τ{i,…,j−1}, 1 ≤ i < j ≤ n subject to the relations
$$\displaystyle \begin{aligned}\begin{array}{r*{20}l} &\tau_{i,j}\tau_{k,l}=\tau_{k,l}\tau_{i,j},&\qquad & i<j<k<l\\ &\tau_{i,l}\tau_{j,k}=\tau_{i+l-k,i+l-j}\tau_{i,l},&& i\le j<k\le l. \end{array}\end{aligned} $$
Categorical actions of \( \operatorname {\mathrm {\mathsf {Cact}}}_n\) on n-fold tensor products in symmetric coboundary categories (first introduced in [15]) were studied in [20, 34] and also implicitly in [9] where the braided structure on the category \(\mathscr O_q^{int}({\mathfrak {g}})\) was converted into a symmetric coboundary structure for any complex reductive Lie algebra \({\mathfrak {g}}\) (for non-abelian examples of coboundary categories, see the discussion after Theorem 1.8). It would be interesting to compare these actions of \( \operatorname {\mathrm {\mathsf {Cact}}}_n\) with the one given by Corollary 1.6. We expect that they are connected in some cases via the celebrated Howe duality (see, e.g., the forthcoming paper [4]). In view of Corollary 1.6 it is natural to seek other categorical representations of \( \operatorname {\mathrm {\mathsf {Cact}}}_W\) for all Coxeter groups W.

Conjecture 1.2 suggests that our representation of \( \operatorname {\mathrm {\mathsf {Cact}}}_W\) on \(\mathscr O^{int}_q({\mathfrak {g}})\) is not faithful. Namely, in view of the discussion after the conjecture, we expect that the kernel \(\mathsf K_{{\mathfrak {g}}}\) of this representation of \( \operatorname {\mathrm {\mathsf {Cact}}}_W\) contains all elements \((\sigma ^i\sigma ^j)^{m_{ij}}\), i ≠ j ∈ I. For example, if \({\mathfrak {g}}={\mathfrak {sl}}_3\), then τ1,2τ1,3 = τ1,3τ2,3 and so \( \operatorname {\mathrm {\mathsf {Cact}}}_W\) is freely generated by involutions τ1,2 and τ1,3. It is easy to see that \(\tau _{1,2}\notin \mathsf K_{{\mathfrak {sl}}_3}\), while \(\tau _{1,3}\notin \mathsf K_{{\mathfrak {sl}}_3}\) by Remark 7.14. Thus, we expect that \(\mathsf K_{{\mathfrak {sl}}_3}=\{ (\tau _{1,2}\tau _{1,3})^{6n}\,:\, n\in {\mathbb Z}\}\).

Therefore, we can pose the following

Problem 1.7

Find the kernel \(\mathsf K_{{\mathfrak {g}}}\) of the representation of \( \operatorname {\mathrm {\mathsf {Cact}}}_W\) on \(\mathscr O^{int}_q({\mathfrak {g}})\).

To outline an approach to Problem 1.7, denote ΦV, \(V\in \mathscr O^{int}_q({\mathfrak {g}})\) the subgroup of Open image in new window generated by the \(\sigma ^J_V\), \(J\in \mathscr J\). Then clearly \(\mathsf K_{{\mathfrak {g}}}\) is the intersection of kernels of canonical homomorphisms \( \operatorname {\mathrm {\mathsf {Cact}}}_W\to \Phi _V\) over all \(V\in \mathscr O^{int}_q({\mathfrak {g}})\). We show (Proposition 4.18) that \(\Phi _V\cong \Phi _{ \underline V}\) where \( \underline V=\bigoplus \limits _{\lambda \in P^+\,:\, \operatorname {\mathrm {Hom}}_{U_q({\mathfrak {g}})}(V_\lambda ,V)\neq 0} V_\lambda \). In particular, \( \operatorname {\mathrm {\mathsf {Cact}}}_W/\mathsf K_{{\mathfrak {g}}}\) is isomorphic to \(\Phi _{\mathcal C_q({\mathfrak {g}})}\) where \(\mathcal C_q({\mathfrak {g}})=\bigoplus _{\lambda \in P^+} V_\lambda \) is the Gelfand–Kirillov model for \(\mathscr O^{int}_q({\mathfrak {g}})\); in fact, it has a structure of an associative algebra (see Sect. 6). Thus, in view of the above we expect that \( \operatorname {\mathrm {\mathsf {Cact}}}_3/\mathsf K_{{\mathfrak {sl}}_3}\) is isomorphic to the dihedral group of order 12. However, it is likely that \(\Phi _{\mathcal C_q({\mathfrak {g}})}\) is infinite for simple \({\mathfrak {g}}\) different from \({\mathfrak {sl}}_2\) and \({\mathfrak {sl}}_3\).

It turns out that the action of \( \operatorname {\mathrm {\mathsf {Cact}}}_W\) on \(\mathscr O^{int}_q({\mathfrak {g}})\) descends to a permutation representation on any crystal basis of any object V  (see Sect. 2.5 for definitions and notation). Thus, we obtain the following refinement of [19, Theorem 5.19].

Theorem 1.8

Let \(V\in \mathscr O^{int}_q({\mathfrak {g}})\). Then for any lower or upper crystal basis (L, B) of V  at q = 0 the group ΦV preserves L and acts on B by permutations.

We prove Theorem 1.8 in Sect. 5 by means of what we call c-crystal bases, which allow one to treat lower and upper crystal bases uniformly. Taking into account that B is graded by the weight lattice of \({\mathfrak {g}}\), all weights occur in a crystal basis of \(\mathcal C_q({\mathfrak {g}})\) and that W acts faithfully on the weight lattice, we obtain an immediate

Corollary 1.9

The assignments \(\sigma ^J\mapsto w_\circ ^J\), \(J\in \mathscr J\) define a surjective homomorphism \( \operatorname {\mathrm {\mathsf {Cact}}}_W/\mathsf K_{{\mathfrak {g}}}\to W\) which refines the natural epimorphism \( \operatorname {\mathrm {\mathsf {Cact}}}_W\to W\) from [29] .

Analogously to the notion of the pure braid group, one calls the kernel of the natural homomorphism \( \operatorname {\mathrm {\mathsf {Cact}}}_W\to W\) the pure cactus group (this term was used for \( \operatorname {\mathrm {\mathsf {Cact}}}_n\) in e.g., [16, 33, 35]). Thus, Corollary 1.9 asserts that \(\mathsf K_{{\mathfrak {g}}}\) is pure.

The involution \(\sigma ^i_V\) was first defined in [8] for \({\mathfrak {g}}={\mathfrak {gl}}_n\) and simple polynomial representations Vλ and explicitly computed on the corresponding crystal in [28]. In fact, it coincides with the famous Schützenberger involution (see Remark 4.11). Following a suggestion of the first author and [28], an action of \( \operatorname {\mathrm {\mathsf {Cact}}}_n\) on the category of crystal bases was constructed in [20], thus turning it into a symmetric coboundary category.

We expect that to solve Problem 1.7 it suffices to find the kernels of permutation representations of \( \operatorname {\mathrm {\mathsf {Cact}}}_W\) on all B.

Since W(V ) is naturally a subgroup of ΦV, its action on V  induces an action on B by permutations which coincides with Kashiwara’s crystal Weyl group (see Remark 5.7).

In case when \({\mathfrak {g}}\) is reductive we can refine Theorem 1.8 as follows.

Theorem 1.10

Let \({\mathfrak {g}}\) be a reductive Lie algebra and let \(V\in \mathscr O^{int}_q({\mathfrak {g}})\). Then for any crystal basis (L, B) of V  the involution \(\sigma ^i_V\) preserves the corresponding upper global crystal basis BV of V .

An analogous result for \(J\subsetneq I\) is weaker. We prove (Proposition 5.8) that the image of any element of BV under σJ, \(J\in \mathscr J\) is a \(\bar \cdot \)-invariant element of V  where \(\bar \cdot \) is the anti-linear involution fixing BV. However, as explained in Remark 7.18, σJ does not need to preserve BV if \(J\subsetneq I\). For example, if V  is the 27-dimensional simple module V2ρ for \({\mathfrak {g}}={\mathfrak {sl}}_3\), then the \(\sigma ^i_V\), i = 1, 2 do not preserve the canonical basis of V .

An analogue of Theorem 1.10 for a simple V  and its lower global crystal basis was deduced from [30, Proposition 21.1.2] in [20, Theorem 5].

We prove Theorem 1.10 in Sect. 6.5. A central role in our argument is played by the following surprising property of σI on the aforementioned quantum Gelfand–Kirillov model \(\mathcal C_q({\mathfrak {g}})\) of \(\mathscr O^{int}_q({\mathfrak {g}})\).

Theorem 1.11 (Theorem 6.21)

For \({\mathfrak {g}}\) reductive finite dimensional, \(\sigma ^I_{\mathcal C_q({\mathfrak {g}})}\) is an algebra anti-involution on \(\mathcal C_q({\mathfrak {g}})\).

Our proofs of Theorems 1.10 and 1.11 rely in a crucial way on the properties of a remarkable quantum twist defined in [27]. We do not expect an analogous result for \(J\subsetneq I\); for example, for \({\mathfrak {g}}={\mathfrak {sl}}_3\), the σi, i ∈{1, 2} are not algebra anti-automorphisms of \(\mathcal C_q({\mathfrak {g}})\).

In view of Theorem 1.8 we can refine Conjecture 1.2 for every \(V\in \mathscr O^{int}_q({\mathfrak {g}})\) with \( \underline V=\mathcal C_q({\mathfrak {g}})\) as follows. We expect that in the notation of Theorem 1.8 the group ΦV acts on B faithfully. Morally, this means that each element of ΦV is semisimple in Open image in new window .

Similarly to Remark 1.3, our constructions, results, and conjectures make sense if one replaces \(U_q({\mathfrak {g}})\) by \(U({\mathfrak {g}})\) for any (symmetrizable or not) Kac–Moody algebra \({\mathfrak {g}}\). Some results (for example, Theorem 1.8) should be possible to rescue even when W is not crystallographic (and so \({\mathfrak {g}}\) does not exist) with the aid of theory of continuous crystals initiated by A. Joseph in [21].

2 Preliminaries

2.1 Coxeter Groups

Let I be a finite set. Let W be a Coxeter group with Coxeter generators si, i ∈ I subject to the relations \((s_i s_j)^{m_{ij}}=1\) where mii = 1, mij = mji, and \(m_{ij}\in \{0\}\cup \mathbb Z_{\ge 2}\) for i ≠ j ∈ I. Let \(\ell :W\to {\mathbb Z}_{\ge 0}\) be the Coxeter length function, that is, (w) is the minimal length of a presentation of w as a product of the si, i ∈ I. We say that i = (i1, …, ir) ∈ Ir is reduced if \(\ell (s_{i_1}\cdots s_{i_r})=r\) and denote by R(w) the set of reduced words for w, that is, \(R(w)=\{ (i_1,\dots ,i_{\ell (w)})\in I^{\ell (w)}\,:\, w=s_{i_1}\cdots s_{i_{\ell (w)}}\}\).

Given J ⊂ I we denote by WJ the subgroup of W generated by the si, i ∈ J. We will need the following standard fact (see [11, IV.1.8, Théorème 2]).

Lemma 2.1

For any J, J I
  1. (a)

    \(W_J\cap W_{J^{\prime }}=W_{J\cap J^{\prime }}\) ;

     
  2. (b)

    \(W_J\subset W_{J^{\prime }}\) if and only if J  J.

     

Let \(\mathscr J=\{ J\subset I,\,:\, |W_J|<\infty \}\). If \(J\in \mathscr J\), we denote by \(w_\circ ^J\) the unique longest element of WJ; thus, \(\ell (s_j w_\circ ^J)<\ell (w_\circ ^J)\) for all j ∈ J. If \(I\in \mathscr J\), we abbreviate \(w_\circ =w_\circ ^I\). Given \(J\in \mathscr J\) and j ∈ J, there exists a unique j ∈ J such that \(s_{j^\star }=w_\circ ^J s_j w_\circ ^J\); the assignments jj define an involution  : J → J.

Given J ⊂ I, we set J = {i ∈ I ∖ J  :  mij = 2, ∀ j ∈ J} = {i ∈ I ∖ J  :  sisj = sjsi, ∀ j ∈ J}. We say that J, J⊂ I are orthogonal if J ∩ J = ∅ and J⊂ J (whence J ⊂ J).

Define a relation ∼ on I by i ∼ j if i = j or mij > 2. Then the transitive closure of this relation is an equivalence on I which we still denote by ∼. In particular, if i ∼ i, then there exists a sequence (called admissible) (i0, …, id) ∈ Id+1 with i0 = i, id = i and \(m_{i_{r-1},i_r}>2\), 1 ≤ r ≤ d. Define \( \operatorname *{\mathrm {dist}}(i,i^{\prime })\) to be the minimal length of an admissible sequence beginning with i and ending with i. Clearly, this defines a metric on I.

Define a topology on I by declaring that the fundamental neighborhood of each i ∈ I is its equivalence class with respect to ∼. In particular, each open set is closed and vice versa and is a union of equivalence classes. For J ⊂ I we denote by \( \operatorname *{\mathrm {cl}}(J)\) its closure in that topology, that is, the union of equivalence classes of elements of J. Denote (J) the boundary of J, that is, the complement of J in \( \operatorname *{\mathrm {cl}}(J)\). The following is immediate.

Lemma 2.2

Let J  I. Then I = J  J if and only if J is closed in the above topology.

The following is a reformulation of a well-known fact [11, IV.1.9, Proposition 2]

Lemma 2.3

Let J  I be a closed subset. Then W is the internal direct product of WJ and \(W_{J^\perp }\).

Given a group G acting on a set X, denote by KX(G) the kernel of the natural homomorphism of groups \(G\to \operatorname {Bij}(X)\) induced by the action. By definition, the action of G on X is faithful if and only if KX(G) = {1}.

The following is the main result of Sect. 2.1 (which is probably known although we could not find it in the literature).

Theorem 2.4

We have \(\mathsf K_J:=\mathsf K_{W/W_J}(W)=W_{I\setminus \operatorname *{\mathrm {cl}}(I\setminus J)}\) for any J  I. In particular, if I is connected and \(J\subsetneq I\), then W acts faithfully on WWJ.

Proof

The following is immediate.

Lemma 2.5

Let G be a group and H be a subgroup of G. Then KGH(G) = {k  H  :  g−1kg  H, ∀ g  G} is a subgroup of H.

The following Lemmata are apparently well-known. We provide their proof for the reader’s convenience.

Lemma 2.6

Let w  W and let J  I be such that ℓ(sjw) = ℓ(w) − 1 for all j  J. Then WJ is finite and \(w=w_\circ ^J w^{\prime }\) for some w W with \(\ell (w)=\ell (w^{\prime })+\ell (w_\circ ^J)\).

Proof By [11, Ch. IV, Ex. 3], every u ∈ W can be written uniquely as [u]J ⋅J[u] where [u]J ∈ WJ, J[u] ∈JW = {x ∈ W  :  (sjx) > (x), ∀ j ∈ J} and (u) = ([u]J) + (J[u]). The uniqueness of such a presentation implies that [sjw]J = sj[w]J and J[sjw] =J[w] for all j ∈ J and so that (sj[w]J) < ([w]J) for all j ∈ J. This implies that WJ is finite and [w]J is its longest element \(w_\circ ^J\). The assertion follows with w =J[w]. □

Lemma 2.7

For i  I and u  WI∖{i} the following are equivalent.
  1. (a)

    \(u\in W_{\{i\}^\perp }\) (in particular, siu = usi);

     
  2. (b)

    siusi ∈ WI∖{i}.

     
Proof The implication (a) ⇒ (b) is obvious. To prove the opposite implication, note that the assumption in (b) implies that siu = usi for some u∈ WI∖{i}. Then (siu) = (u) + 1 and (usi) = (u) + 1 whence (u) = (u). We prove the assertion
$$\displaystyle \begin{aligned} s_i u=u^{\prime}s_i\implies u=u^{\prime}\in W_{\{i\}^\perp} \end{aligned} $$
(2.1)
by induction on (u) = (u), the case (u) = (u) = 0 being obvious. If (u) = (u) > 0, then there exists j ≠ i ∈ I such that (sju) < (u). Let w = siu. Then (sjw) < (w) and (siw) < (w). Applying Lemma 2.6 to w and J = {i, j} we conclude that \(w=(\underbrace {s_i s_j\cdots }_{m_{ij}})u^{\prime }\) with (w) = mij + (u) and so \(u=(\underbrace {s_js_i\cdots }_{m_{ij}-1})u^{\prime }\) with (u) = mij − 1 + (u). Since u ∈ WI∖{i}, a reduced word for u cannot contain i, yet for any (i1, …, ir) ∈ R(u), \((\underbrace {j, i,\dots }_{m_{ij}-1},i_1,\dots ,i_r)\in R(u)\). Thus, mij = 2 and so j ∈{i}. Then si(sju) = sjsiu = (sju)si. Thus, sju, sju satisfy (2.1) and (sju) < (u). Then the induction hypothesis implies that \(s_j u=s_j u^{\prime }\in W_{\{i\}^\perp }\) and hence \(u=u^{\prime }\in W_{\{i\}^\perp }\). □

Lemma 2.8

Let i, i be connected in I and let (i = i0, i1, …, id = i) ∈ Id+1 be an admissible sequence with \(d= \operatorname *{\mathrm {dist}}(i,i^{\prime })\). Suppose that w  WI∖{i} and \(s_{i_0}\cdots s_{i_d}ws_{i_d}\cdots s_{i_0}\in W_{I\setminus \{i\}}\). Then \(w\in W_{\{i_0,\dots ,i_d\}^\perp }\).

Proof The argument is by induction on d. The case d = 0 (that is, i = i) is established in Lemma 2.7. Suppose that d > 0. Let \(u=s_{i_1}\cdots s_{i_d} w s_{i_d}\cdots s_{i_1}\). By Lemma 2.7, \(u\in W_{\{i\}^\perp }\). Since \(m_{i,i_1}>2\), i1∉{i}. Thus, \(u\in W_{I^{\prime }\setminus \{i_1\}}\) where I = I ∖{i} and \( \operatorname *{\mathrm {dist}}(i_1,i^{\prime })=d-1\). By the induction hypothesis, \(u\in W_{\{i_1,\dots ,i_d\}^\perp }\) and in particular u = w. But then \(w\in W_{\{i\}^\perp }\cap W_{\{i_1,\dots ,i_d\}^\perp }= W_{\{i\}^\perp \cap \{i_1,\dots ,i_d\}^\perp }=W_{\{i_0,\dots ,i_d\}^\perp }\) where we used Lemma 2.1(a) and the observation that J∩ J = (JJ). □

By Lemma 2.5, KJ = {w ∈ W  :  uwu−1 ∈ WJ, ∀ u ∈ W} and is a subgroup of WJ. Suppose that w ∈KJ; in particular, w ∈ WJ. Furthermore, using Lemma 2.7 with u = w and i ∈ I ∖ J, we conclude that \(w\in \bigcap _{i\in I\setminus J} W_{\{i\}^\perp }=W_{(I\setminus J)^\perp }\). Let i∈ (I ∖ J). By definition, there exists i ∈ I ∖ J and an admissible sequence (i0, …, id) with \(d= \operatorname *{\mathrm {dist}}(i,i^{\prime })\), i0 = i and id = i. Since uwu−1 ∈ WJ with \(u=s_{i_0}\cdots s_{i_d}\), it follows from Lemma 2.8 that \(w\in W_{\{ i_0,\dots ,i_d\}^\perp }\subset W_{\{i^{\prime }\}^\perp }\). Thus, \(w\in W_{ (I\setminus J)^\perp \cap \partial (I\setminus J)^\perp }=W_{ ( \operatorname *{\mathrm {cl}}(I\setminus J))^\perp }=W_{ I\setminus \operatorname *{\mathrm {cl}}(I\setminus J)}\). We proved that \(\mathsf K_J\subset W_{J_0}\) where \(J_0=I\setminus \operatorname *{\mathrm {cl}}(I\setminus J)\).

To complete the proof of Theorem 2.4 we need the following.

Lemma 2.9

Let J J which is closed in I. Then \(W_{J^{\prime }}\subset \mathsf K_J\).

Proof Since J is closed, \(W_J=W_{J^{\prime }}\times W_{J\setminus J^{\prime }}\) and \(W=W_{J^{\prime }}\times W_{I\setminus J^{\prime }}\) by Lemma 2.3. Then \(W/W_J=W_{I\setminus J^{\prime }}/W_{J\setminus J^{\prime }}\). Since \(W_{J^{\prime }}\) acts by left multiplication in the first factor, this implies that \(W_{J^{\prime }}\) acts trivially on WWJ. □

Applying Lemma 2.9 with \(J^{\prime }=J_0=I\setminus \operatorname *{\mathrm {cl}}(I\setminus J)\) we conclude that \(W_{J_0}\subset \mathsf K_J\). Thus, \(\mathsf K_J=W_{J_0}\). This completes the proof of Theorem 2.4. □

2.2 Cartan Data and Weyl Group

In this section we mostly follow [23]. Let A = (aij)i,jI be a symmetrizable generalized Cartan matrix, that is aii = 2, i ∈ I, \(-a_{ij}\in {\mathbb Z}_{\ge 0}\) and aij = 0 ⇒ aji = 0, i ≠ j and diaij = djaji for some \(\mathbf d=(d_i)_{i\in I}\in {\mathbb Z}_{>0}^I\). We fix the following data:
  • a finite dimensional complex vector space \({\mathfrak {h}}\);

  • linearly independent subsets {αi}iI of \({\mathfrak {h}}^*\) and \(\{\alpha _i^\vee \}_{i\in I}\) of \({\mathfrak {h}}\);

  • a symmetric non-degenerate bilinear form (⋅, ⋅) on \({\mathfrak {h}}^*\), and

  • a lattice \(P\subset {\mathfrak {h}}^*\) of rank \(\dim {\mathfrak {h}}^*\)

such that
  1. 1∘

    \(\alpha _j(\alpha _i^\vee )=a_{ij}\), i, j ∈ I;

     
  2. 2∘

    \((\alpha _i,\alpha _i)\in 2{\mathbb Z}_{>0}\);

     
  3. 3∘

    \(\lambda (\alpha _i^\vee )=2(\lambda ,\alpha _i)/(\alpha _i,\alpha _i)\) for all \(\lambda \in {\mathfrak {h}}^*\);

     
  4. 4∘

    αi ∈ P for all i ∈ I;

     
  5. 5∘

    \(\lambda (\alpha _i^\vee )\in {\mathbb Z}\) for all λ ∈ P;

     
  6. 6∘

    \((P,P)\subset \frac 1d{\mathbb Z}\) for some \(d\in {\mathbb Z}_{>0}\).

     
These assumptions imply, in particular, that \(\dim {\mathfrak {h}}\ge 2|I|-\operatorname {rank}A\).

Denote by Q (respectively, Q+) the subgroup (respectively, the submonoid) of P generated by the αi. Let \(P^+=\{ \lambda \in P\,:\, \lambda (\alpha _i^\vee )\in {\mathbb Z}_{\ge 0},\,\forall i\in I\}\).

Define \(\omega _i\in {\mathfrak {h}}^*\), i ∈ I, by \(\omega _i(\alpha _j^\vee )=\delta _{i,j}\), j ∈ J and ωi(h) = 0 for all \(h\in \bigcap _{i\in I}\ker \alpha _i\). We will assume that ωi ∈ P, i ∈ I and denote by Pint (respectively, \(P_{int}^+\)) the subgroup (respectively, the submonoid) of P generated by the ωi, i ∈ I. Given any J ⊂ I, denote ρJ =∑jJωj ∈ P; we abbreviate ρI = ρ.

Let W be the Weyl group associated with the matrix A, that is, the Coxeter group with mij = 2 if aij = 0, mij = 3 if aijaji = 1, mij = 4 if aijaji = 2, mij = 6 if aijaji = 3 and mij = 0 if aijaji > 3. It is well-known that W is finite if and only if A is positive definite. It should be noted that in that case αi ∈ Pint for all i ∈ I. The group W acts on \({\mathfrak {h}}\) (respectively, on \({\mathfrak {h}}^*\)) by \(s_ih=h-\alpha _i(h)\alpha _i^\vee \) (respectively, \(s_i\lambda =\lambda -\lambda (\alpha _i^\vee )\alpha _i\)), \(h\in {\mathfrak {h}}\), \(\lambda \in {\mathfrak {h}}^*\) and i ∈ I. Then we have ()(h) = λ(w−1h) for all w ∈ W, \(h\in {\mathfrak {h}}\) and \(\lambda \in {\mathfrak {h}}^*\). Clearly, W(P) = P and P = Pint ⊕ PW where \(P^W=\{ \lambda \in P\,:\, w\lambda =\lambda ,\,\forall \, w\in W\}=\{ \lambda \in P\,:\, \lambda (\alpha _i^\vee )=0,\,\forall \, i\in I\}\).

Given J ⊂ I we define a linear map \(\rho _J^\vee :{\mathfrak {h}}^*\to \mathbb C\) by \(\rho _J^\vee (\alpha _i)=1\), i ∈ J and \(\rho _J^\vee (\lambda )=0\) if (λ, αi) = 0 for all i ∈ J. As before, we abbreviate \(\rho ^\vee _I=\rho ^\vee \). If \(J\in \mathscr J\), then it can be shown that \(\rho _J^\vee (\lambda )\) is equal to \(\frac 12\lambda (\sum _{h\in R_J^\vee } h)\) where \(R_J^\vee =\{ h\in {\mathfrak {h}}\,:\, h\in (\bigcup _{i\in J} W_J\alpha _i^\vee )\cap \sum _{i\in J} {\mathbb Z}_{\ge 0}\alpha _i^\vee \}\) is the set of positive co-roots of WJ. In particular, this implies that \(\rho ^\vee _J(P)\subset \frac 12{\mathbb Z}\).

If \(J\in \mathscr J\), then for each j ∈ J we have \(w_\circ ^J(\alpha _j)=-\alpha _{j^\star }\).

Given λ ∈ P+, denote \(J_\lambda =\{ i\in I\,:\, \lambda (\alpha _i^\vee )=0\}=\{i\in I\,:\, s_i\lambda =\lambda \}\). It is well-known that \(\operatorname {Stab}_W\lambda =W_{J_\lambda }\) for λ ∈ P+.

2.3 Quantum Groups

We associate with the datum \((A, {\mathfrak {h}}, \{\alpha _i\}_{i\in I},\{\alpha _i^\vee \}_{i\in I})\) a complex Lie algebra \({\mathfrak {g}}\) generated by the ei, fi, i ∈ I and \(h\in {\mathfrak {h}}\) subject to the relations
$$\displaystyle \begin{aligned}{}[h,h^{\prime}]=0,\quad [h,e_i]=\alpha_i(h)e_i,\quad [h,f_i]=-\alpha_i(h)f_i,\quad [e_i,f_j]=\delta_{i,j}\alpha_i^\vee,\quad h,h^{\prime}\in {\mathfrak{h}},\,i,j\in I\\ (\operatorname{\mathrm{ad}} e_i)^{1-a_{ij}}(e_j)=0=(\operatorname{\mathrm{ad}} f_i)^{1-a_{ij}}(f_j),\qquad i\neq j. \end{aligned} $$
If A is positive definite, then \({\mathfrak {g}}\) is a reductive finite dimensional Lie algebra. For J ⊂ I we denote by \({\mathfrak {g}}^J\) the subalgebra of \({\mathfrak {g}}\) generated by the ei, fi, i ∈ J and \({\mathfrak {h}}\). It can also be regarded as the Lie algebra corresponding to the datum \((A|{ }_{J\times J},{\mathfrak {h}},\{\alpha _i\}_{i\in J},\{\alpha ^\vee _i\}_{i\in J})\). In particular, if \(J\in \mathscr J\), then \({\mathfrak {g}}^J\) is a reductive finite dimensional Lie algebra.
Let Open image in new window be any field of characteristic zero containing \(q^{\frac 1{2d}}\) which is purely transcendental over \(\mathbb Q\). Given any Open image in new window with v2 ≠ 1 define
$$\displaystyle \begin{aligned} (n)_v=\frac{v^{n}-v^{-n}}{v-v^{-1}},\qquad (n)_v!=\prod_{s=1}^n (s)_v,\qquad \binom{n}{k}_v=\prod_{s=1}^k \frac{(n-s+1)_v}{(s)_v}. \end{aligned}$$
Let \(q_i=q^{\frac 12(\alpha _i,\alpha _i)}\). Henceforth, given any associative algebra \(\mathcal A\) over Open image in new window and \(X_i\in \mathcal A\), i ∈ I denote \(X_i^{(n)}:=X_i^n/(n)_{q_i}!\) We will always use the convention that \(X_i^{(n)}=0\) if n < 0.
Define the Drinfeld-Jimbo quantum group \(U_q({\mathfrak {g}})\) corresponding to \({\mathfrak {g}}\) as the associative algebra over Open image in new window with generators Kλ, \(\lambda \in \frac 12 P\) and Ei, Fi, i ∈ I subject to the relations
$$\displaystyle \begin{gathered} K_\lambda E_i=q^{(\lambda,\alpha_i)}E_i K_\lambda,\,\, K_\lambda F_i=q^{-(\lambda,\alpha_i)}F_i K_\lambda,\,\, [E_i,F_j]=\delta_{ij}\, \frac{ K_{\alpha_i}-K_{-\alpha_i}}{q_i-q_i^{-1}},\quad \lambda\in\tfrac12 P,\, i,j\in I\\ \sum_{r+s=1-a_{ij}} (-1)^r E_i^{(r)}E_j E_i^{(s)}=0=\sum_{r+s=1-a_{ij}} (-1)^r F_i^{(r)}F_j F_i^{(s)}\quad i \neq j \in I. \end{gathered} $$
This is a Hopf algebra with the “balanced” comultiplication
$$\displaystyle \begin{aligned} \Delta(E_i)=E_i\otimes K_{\frac 12\alpha_i}+K_{-\frac 12\alpha_i}\otimes E_i,\,\, \Delta(F_i)=F_i\otimes K_{\frac 12\alpha_i}+K_{-\frac 12\alpha_i}\otimes F_i,\, i\in I, \end{aligned} $$
(2.2)
while Δ(Kλ) = Kλ ⊗ Kλ, \(\lambda \in \frac 12 P\). Denote by \(U_q^+({\mathfrak {g}})\) (respectively, \(U_q^-({\mathfrak {g}})\)) the subalgebra of \(U_q({\mathfrak {g}})\) generated by the Ei (respectively, the Fi), i ∈ I. Then \(U_q^\pm ({\mathfrak {g}})\) is graded by ± Q+ with deg Ei = αi = −deg Fi. Given ν ∈ Q+, denote by \(U_q^\pm ({\mathfrak {g}})(\pm \nu )\) the subspace of homogeneous elements of \(U_q^\pm ({\mathfrak {g}})\) of degree ± ν.

Given J ⊂ I we denote by \(U_q({\mathfrak {g}}^J)\) the subalgebra of \(U_q({\mathfrak {g}})\) generated by the Ej, Fj, j ∈ J and Kλ, \(\lambda \in \frac 12 P\) and set \(U_q^\pm ({\mathfrak {g}}^J)=U_q^\pm ({\mathfrak {g}})\cap U_q({\mathfrak {g}}^J)\).

If \(J\in \mathscr J\), then the algebra \(U_q({\mathfrak {g}}^J)\) admits an automorphism θJ defined by \(\theta _J(E_i)=F_{i^\star }\), \(\theta _J(F_i)=E_{i^\star }\) and \(\theta _J(K_\lambda )=K_{w_\circ ^J\lambda }\), \(\lambda \in \frac 12 P\). If \(I\in \mathscr J\), we abbreviate θI = θ.

2.4 Integrable Modules

We say that a \(U_q({\mathfrak {g}})\)-module M is integrable if \(M=\bigoplus \limits _{\beta \in P}M(\beta )\) where \(M(\beta )=\{ m\in M\,:\, K_\lambda (m)=q^{(\lambda ,\beta )}m,\,\forall \lambda \in \frac 12 P\}\) and the Ei, Fi, i ∈ I act locally nilpotently on M. Given any m ∈ M we can write uniquely
$$\displaystyle \begin{aligned}m=\bigoplus_{\beta\in P} m(\beta) \end{aligned} $$
(2.3)
where m(β) ∈ M(β) and m(β) = 0 for all but finitely many β ∈ P. Denote \( \operatorname *{\mathrm {supp}} m=\{ \beta \in P\,:\, m(\beta )\neq 0\}\). By definition, if m ∈ M(β) and \(u_\pm \in U_q^\pm ({\mathfrak {g}})(\pm \nu )\), ν ∈ ± Q+, then u±(m) ∈ M(β ± ν). We say that m ∈ M(β), β ∈ P is homogeneous of weight β and call M(β) a weight subspace of M.

Definition 2.10

The category \(\mathscr O^{int}_q({\mathfrak {g}})\) is the full subcategory of the category of \(U_q({\mathfrak {g}})\)-modules whose objects are integrable \(U_q({\mathfrak {g}})\)-modules M with the following property: given m ∈ M, there exists N(m) ≥ 0 such that \(U_q^+({\mathfrak {g}})(\nu )(m)=0\) for all ν ∈ Q+ with ρ(ν) ≥ N(m).

Given \(V\in \mathscr O^{int}_q({\mathfrak {g}})\), let \(V_+=\bigcap _{i\in I} \ker E_i\) where the Ei are regarded as linear endomorphisms of V . For any subset S of an object V  in \(\mathscr O^{int}_q({\mathfrak {g}})\) we denote S(β) = S ∩ V (β) and S+ = S ∩ V+.

It is well-known (see, e.g., [30, Theorem 6.2.2]) that \(\mathscr O^{int}_q({\mathfrak {g}})\) is semisimple and its simple objects are simple highest weight modules Vλ, λ ∈ P+ with (Vλ)+ = Vλ(λ) one-dimensional. Furthermore, every \(V\in \mathscr O^{int}_q({\mathfrak {g}})\) is generated by V+ as a \(U_q({\mathfrak {g}})\)-module and V+(λ) ≠ 0 implies that λ ∈ P+. Given λ ∈ P+ and \(V\in \mathscr O^{int}_q({\mathfrak {g}})\) denote \(\mathcal I_\lambda (V)\) the λ-isotypical component of V  as a \(U_q({\mathfrak {g}})\)-module. Thus, every simple submodule (and hence a direct summand) of \(\mathcal I_\lambda (V)\) is isomorphic to Vλ and \(\mathcal I_\lambda (V)_+=V_+(\lambda )\). Furthermore, for any v ∈ V+ we have the following equality of \(U_q({\mathfrak {g}})\)-submodules of V
$$\displaystyle \begin{aligned} U_q({\mathfrak{g}})(v)=\sum_{\lambda\in\operatorname*{\mathrm{supp}} v} U_q({\mathfrak{g}})(v(\lambda)), \end{aligned} $$
(2.4)
where the sum is direct and each summand is simple and isomorphic to Vλ.

It is immediate from the definition that every object \(\mathscr O^{int}_q({\mathfrak {g}})\) can be regarded as an object in \(\mathscr O^{int}_q({\mathfrak {g}}^J)\), J ⊂ I. Denote \(P^+_J=\{ \mu \in P\,:\, \mu (\alpha _j^\vee )\in {\mathbb Z}_{\ge 0},\,\forall \, j\in J\}\). Given \(V\in \mathscr O^{int}_q({\mathfrak {g}})\) and \(\lambda _J\in P^+_J\) denote by \(\mathcal I^J_{\lambda _J}(V)\) the λJ-isotypical component of V  as a \(U_q({\mathfrak {g}}^J)\)-module. Clearly, \(\mathcal I^J_{\lambda _J}(\mathcal I_\lambda (V))=\mathcal I_\lambda (V)\cap \mathcal I^J_{\lambda _J}(V)\) for any \(V\in \mathscr O^{int}_q({\mathfrak {g}})\), λ ∈ P+, \(\lambda _J\in P^+_J\). We denote \(V_+^J=\bigcap _{j\in J} \ker E_j\subset V_+\). Then \(\mathcal I^J_{\lambda _J}(V)\) is generated by \(V_+^J(\lambda _J)\) as a \(U_q({\mathfrak {g}}^J)\)-module.

2.5 Crystal Operators, Lattices and Bases

Here we recall some necessary facts from Kashiwara’s theory of crystal bases. To treat lower and upper crystal operators and lattices uniformly, we find it convenient to interpolate between them using c-crystal operators and lattices (for other generalizations, see, e.g., [12, 18]).

The following fact is standard (for example, see [30, Lemma 16.1.4]).

Lemma 2.11

Let \(V\in \mathscr O^{int}_q({\mathfrak {g}})\) and fix i  I. Then
$$\displaystyle \begin{aligned} V=\bigoplus_{0\le n\le l} F_i^n(\ker E_i\cap \ker(K_{\alpha_i}-q_i^l)) =\bigoplus_{0\le n\le l} E_i^n(\ker F_i\cap \ker(K_{\alpha_i}-q_i^{-l})) \end{aligned}$$
Let \(\mathbb D=\{ (l,k,s)\in \mathbb {\mathbb Z}_{\ge 0}\times {\mathbb Z}_{\ge 0}\times {\mathbb Z}\,:\, k-l\le s\le k\le l\}\). Fix a map \(\mathbf c:\mathbb D\to \mathbb Q(z)^\times \) and denote its value at (l, k, s) by cl,k,s. We use the convention that cl,k,s = 0 whenever \((l,k,s)\in {\mathbb Z}^3\setminus \mathbb D\). Using Lemma 2.11 we can define generalized Kashiwara operators Open image in new window , \(s\in {\mathbb Z}\) by
$$\displaystyle \begin{aligned} \tilde e_{i,s}^{\mathbf c}(F_i^k(u))=\mathbf c_{l,k,s}(q_i) F_i^{k-s}(u), \end{aligned} $$
(2.5)
for every \(u\in \ker E_i\cap \ker (K_{\alpha _i}-q_i^l)\), 0 ≤ k ≤ l. Note that under these assumptions on u, \(\tilde e_{i,s}^{\mathbf c}(F_i^{(k)}(u))\neq 0\) if and only if Open image in new window . Clearly, like lower or upper Kashiwara operators, the generalized ones commute with morphisms in \(\mathscr O^{int}_q({\mathfrak {g}})\).

Lemma 2.12

Let \(u\in \ker E_i\cap \ker ( K_{\alpha _i}-q_i^l)\) and \(u^{\prime }\in \ker F_i\cap \ker (K_{\alpha _i}-q_i^{-l})\), 0 ≤ k  l. Then
$$\displaystyle \begin{aligned} \tilde e_{i,s}^{\mathbf c}(F_i^{(k)}(u))=\underline{\mathbf c}_{l,k,s}(q_i) F_i^{(k-s)}(u),\quad \tilde e_{i,s}^{\mathbf c}(E_i^{(k)}(u^{\prime})) =\underline{\mathbf c}_{l,l-k,s}(q_i) E_i^{(k+s)}(u^{\prime}), \end{aligned} $$
(2.6)

for all \((l,k,s)\in \mathbb D\), where \( \underline {\mathbf c}_{l,k^{\prime },s^{\prime }}={\mathbf c}_{l,k^{\prime },s^{\prime }}(k^{\prime }-s^{\prime })_z!/(k^{\prime })_z!\).

Proof

The first identity in (2.6) is immediate from (2.5). To prove the second, note that \(E_i^{(l+1)}(u^{\prime })=0\) and so \(u=E_i^{(l)}(u^{\prime })\in \ker E_i\cap \ker (K_{\alpha _i}-q_i^l)\). It follows from [30, §3.4.2] that \(E_i^{(k)}(u^{\prime })=F_i^{(l-k)}(u)\). Using the first identity in (2.6) we obtain
$$\displaystyle \begin{aligned} \tilde e_{i,s}^{\mathbf c}(E_i^{(k)}(u^{\prime}))=\tilde e_{i,s}^{\mathbf c}(F_i^{(l-k)}(u))=\underline{\mathbf c}_{l,l-k,s}(q_i) F_i^{(l-k-s)}(u)= \underline{\mathbf c}_{l,l-k,s}(q_i) E_i^{(k+s)}(u^{\prime}). \end{aligned}$$

The following is immediate.

Lemma 2.13

Given \(\mathbf c:\mathbb D\to \mathbb Q(z)^\times \) we have
  1. (a)

    \(\tilde e_{i,0}^{\mathbf c}= \operatorname {\mathrm {id}}_V\) for all \(V\in \mathscr O^{int}_q({\mathfrak {g}})\) if and only if cl,k,0 = 1 for all 0 ≤ k  l;

     
  2. (b)

    \(\tilde e_{i,t}^{\mathbf c}\circ \tilde e_{i,s}^{\mathbf c}=\tilde e_{i,s+t}^{\mathbf c}\) for all \(s,t\in {\mathbb Z}\) with st ≥ 0 and for all \(V\in \mathscr O^{int}_q({\mathfrak {g}})\) if and only if cl,k,s+t = cl,k,scl,ks,t for all 0 ≤ k  l, \(s,t\in {\mathbb Z}\), st ≥ 0.

     

The following is easy to deduce from [25, §3.1]

Lemma 2.14

Define \(\mathbf c^{low},\mathbf c^{up}:\mathbb D\to \mathbb Q(z)^\times \) by
$$\displaystyle \begin{aligned} \mathbf c^{low}_{l,k,s}=\frac{(k)_z!}{(k-s)_z!},\qquad \mathbf c^{up}_{l,k,s}=\frac{(l-k+s)_z!}{(l-k)_z!},\qquad (l,k,s)\in\mathbb D.\end{aligned} $$
(2.7)
We have
$$\displaystyle \begin{aligned} (\tilde e_i^{low})^s=\tilde e^{\mathbf c^{low}}_{i,s},\qquad (\tilde e_i^{up})^s=\tilde e^{\mathbf c^{up}}_{i,s},\qquad i\in I,\,s\in{\mathbb Z},\end{aligned} $$

where \(\tilde e_i^{low}\) (respectively, \(\tilde e_i^{up}\)) are lower (respectively, upper) Kashiwara’s operators as defined in [25, §3.1] .

Fix \(\mathbf c:\mathbb D\to \mathbb Q(z)^\times \) and let \(V\in \mathscr O^{int}_q({\mathfrak {g}})\). Let \(\mathbb A\) be the local subring of Open image in new window consisting of rational functions regular at 0. Generalizing well-known definitions of Kashiwara, we say that an \(\mathbb A\)-submodule L of V  is a c-crystal lattice if Open image in new window , L =⊕βP(L ∩ V (β)), and \(\tilde e_{i,s}^{\mathbf c}(L)\subset L\) for all i ∈ I, \(s\in \mathbb Z\).

We will be mostly interested in a special class of crystal lattices which we refer to as monomial. We need the following notation. Given v ∈ V  set
$$\displaystyle \begin{aligned}\mathsf M_J^{\mathbf c}(v)=\{v\}\cup \bigcup_{k\in{\mathbb Z}_{>0}} \{ \tilde e^{\mathbf c}_{i_1,m_1}\cdots\tilde e^{\mathbf c}_{i_k,m_k}(v)\,:\, (i_1,\dots,i_k)\in J^k,\, (m_1,\dots,m_k)\in{\mathbb Z}^k\}. \end{aligned}$$
We abbreviate \(\mathsf M^{\mathbf c}(v)=\mathsf M_I^{\mathbf c}(v)\). We call an \(\mathbb A\)-submodule L of V  a (c, J)-monomial lattice if
$$\displaystyle \begin{aligned} L=\sum_{v_+} \mathsf M_J^{\mathbf c}(v_+)\end{aligned} $$
where the sum is over all \(v_+\in L\cap V_+^J(\lambda _J)\), λJ ∈ P+. Clearly, L inherits a weight decomposition from V  and \(\tilde e_{j,a}^{\mathbf c}(L)\subset L\) for all j ∈ J, \(a\in {\mathbb Z}\). In particular, if L is a (c, I)-monomial lattice and Open image in new window , then L is a c-crystal lattice.
Denote \(\tilde L\) the \(\mathbb Q\)-vector space LqL. Given \(\tilde v\in \tilde L\), denote
$$\displaystyle \begin{aligned} \tilde{\mathsf M}^{\mathbf c}_J(\tilde v)&=\{\tilde v\}\cup \bigcup_{k\in{\mathbb Z}_{>0}} \{ \tilde e^{\mathbf c}_{i_1,m_1}\cdots\tilde e^{\mathbf c}_{i_k,m_k}(\tilde v)\,:\, (i_1,\dots,i_k)\in J^k,\\ & \quad (m_1,\dots,m_k)\in{\mathbb Z}^k\}\subset \tilde L.\end{aligned} $$
As before, we abbreviate \(\tilde {\mathsf M}^{\mathbf c}(v)=\tilde {\mathsf M}_I^{\mathbf c}(v)\).

By [24, Theorem 3] and [25, Theorem 3.3.1], if c = clow or c = cup, then every object in \(\mathscr O^{int}_q({\mathfrak {g}})\) admits a c-crystal lattice. Moreover, in that case for any λ ∈ P+, and any vλ ∈ Vλ(λ) the smallest \(\mathbb A\)-submodule of Vλ containing vλ and invariant with respect to the \(\tilde e_{i,s}^{\mathbf c}\), i ∈ I, \(s\in {\mathbb Z}_{<0}\) is a c-crystal lattice.

Let L be a c-crystal lattice of \(V\in \mathscr O^{int}_q({\mathfrak {g}})\). Clearly, operators \(\tilde e_{i,s}^{\mathbf c}\) commute with the action of q on L and thus factor through to \(\mathbb Q\)-linear operators on \(\tilde L=L/q L\) denoted by the same symbols. Similarly to [24, 25], we say that (L, B), where B is a weight basis of \(\tilde L\), is a c-crystal basis of V  at q = 0 if \(\tilde e_{i,s}^{\mathbf c}(B)\subset B\cup \{0\}\), i ∈ I, \(s\in {\mathbb Z}\). By [24, 25], every object in \(\mathscr O^{int}_q({\mathfrak {g}})\) admits a c-crystal basis provided that c ∈{cup, clow}.

The following is well-known (cf. [24, 25]).

Lemma 2.15

Let \(V\in \mathscr O^{int}_q({\mathfrak {g}})\), c ∈ {clow, cup} and let (L, B) be a c-crystal basis at q = 0. Then for any J  I
  1. (a)

    L is a (c, J)-monomial lattice;

     
  2. (b)

    \(\tilde {\mathsf M}^{\mathbf c}_J(b)\subset B\cup \{0\}\) for any b  B;

     
  3. (c)

    \(B=\bigcup _{b_+\in B_+^J} \tilde {\mathsf M}^{\mathbf c}_J(b_+)\setminus \{0\}\) where \(B_+^J=\bigcap _{j\in J}\ker \tilde e_{j,1}^{\mathbf c}\subset B\) ;

     

Remark 2.16

It is not hard to see that if for given \(\mathbf c:\mathbb D\to \mathbb Q(z)^\times \) and J ⊂ I Lemma 2.15(a)–(c) hold then (L, B) is a c-crystal basis at q = 0 of V  regarded as a \(U_q({\mathfrak {g}}^J)\)-module.

3 Properties of σi and Proof of Theorem 1.1

3.1 Special Monomials in \(U_q^\pm ({\mathfrak {g}})\)

Given any reduced sequence i = (i1, …, im) ∈ Im and λ ∈ P+ we define \(F_{\mathbf i,\lambda }\in U_q^-({\mathfrak {g}})\) and \(E_{\mathbf i,\lambda }\in U_q^+({\mathfrak {g}})\), λ ∈ P+ by
$$\displaystyle \begin{aligned} F_{\mathbf i,\lambda}=F_{i_1}^{(a_{1})} \cdots F_{i_m}^{(a_m)},\qquad E_{\mathbf i,\lambda}=E_{i_1}^{(a_1)} \cdots E_{i_m}^{(a_m)}, \end{aligned} $$
(3.1)
where \(a_k=a_{k}(\mathbf i,\lambda )=s_{i_{k+1}}\cdots s_{i_m}\lambda (\alpha _{i_k}^\vee )=\lambda (s_{i_m}\cdots s_{i_{k+1}}\alpha _{i_k}^\vee )\in {\mathbb Z}_{\ge 0}\).

Lemma 3.1

Let w  W, λ  P+, i  I. Then
  1. (a)

    \(F_{\mathbf i,\lambda }=F_{{\mathbf {i}}^{\prime },\lambda }\) and \(E_{\mathbf i,\lambda }=E_{{\mathbf {i}}^{\prime },\lambda }\) for any i, i R(w) and λ  P+. Thus, we can define Fw,λ := Fi,λ and Ew,λ := Ei,λ for some i ∈ R(w);

     
  2. (b)

    If ℓ(siw) = ℓ(w) + 1, then \(F_{s_iw,\lambda }=F_i^{(w\lambda (\alpha _i^\vee ))}F_{w,\lambda }\) and \(E_{s_i w,\lambda }=E_i^{(w\lambda (\alpha _i^\vee ))}E_{w,\lambda }\);

     
  3. (c)

    If ℓ(siw) = ℓ(w) − 1, then \(F_{w,\lambda }=F_i^{(-w\lambda (\alpha _i^\vee ))} F_{s_i w,\lambda }\) and \(E_{w,\lambda }=E_i^{(-w\lambda (\alpha _i^\vee ))} E_{s_i w,\lambda }\);

     
  4. (d)

    If siλ = λ, then \(F_{ws_i,\lambda }=F_{w,\lambda }\);

     
  5. (e)

    deg Fw,λ = −deg Ew,λ = wλ  λ;

     
  6. (f)

    Suppose that W is finite. Then \(\theta (F_{w,\lambda })=E_{w_\circ ww_\circ ,-w_\circ \lambda }\).

     

Proof

It is well-known that i can be obtained from i by a finite sequence of rank 2 braid moves of the form \(\underbrace {s_i s_j\cdots }_{m_{ij}}=\underbrace {s_j s_i\cdots }_{m_{ij}}\) with mij finite. Thus, it suffices to prove part (a) in case when w is the longest element in the subgroup of W generated by si, sj, i ≠ j ∈ I. But in that case it was established in [30, Proposition 39.3.7].

Parts (b), (c), and (d) are obtained from part (a) by choosing appropriate reduced decompositions. To prove parts (e) and (f) we use induction on (w), the case (w) = 0 being obvious. For the inductive step, suppose that (siw) = (w) + 1. Since \(\mu (\alpha _i^\vee )\alpha _i=\mu -s_i\mu \), μ ∈ P, by part (b) and the induction hypothesis we have \(\deg F_{s_iw,\lambda }=-w\lambda (\alpha _i^\vee )\alpha _i+w\lambda -\lambda =s_iw\lambda -\lambda \). This proves the inductive step in part (e). Since \(s_{i^\star }=w_\circ s_i w_\circ \) we have by Lemma 3.1(b)
$$\displaystyle \begin{aligned} \theta(F_{s_i w,\lambda})&=\theta(F_i^{(w\lambda(\alpha_i^\vee))}F_{w,\lambda}) {=}E_{i^\star}^{(w\lambda(\alpha_i^\vee))}E_{w_\circ w w_\circ,-w_\circ\lambda}=E_{i^{\star}}^{(-w\lambda(w_\circ\alpha^\vee_{i^\star}))}E_{w_\circ w w_\circ,-w_\circ\lambda} \\&=E_{i^{\star}}^{((w_\circ ww_\circ(-w_\circ\lambda))(\alpha^\vee_{i^\star}))}E_{w_\circ w w_\circ,-w_\circ\lambda} =E_{s_{i^\star}w_\circ ww_\circ,-w_\circ\lambda}=E_{w_\circ s_i w w_\circ,-w_\circ\lambda}. \end{aligned} $$

The inductive step in part (f) is proven. □

3.2 Extremal Vectors

For \(V\in \mathscr O^{int}_q({\mathfrak {g}})\) and v ∈ V , denote J(v) = {i ∈ I  :  Fi(v) = 0} and define J(S) =⋂vSJ(v), S ⊂ V . One can show that J(V ) also equals to {i ∈ I  :  Ei(V ) = {0}}. We will need the following basic properties of these sets.

Proposition 3.2

Let \(V\in \mathscr O^{int}_q({\mathfrak {g}})\) . Then
  1. (a)

    J(v) = Jλ for any v  V+(λ) ∖{0}, λ  P+;

     
  2. (b)

    There exists v  V+ such that \(J(V)=J(U_q({\mathfrak {g}})(v))\).

     

Proof

It is well-known (see, e.g., [30, Chap. 6]) that the annihilating ideal of v in \(U_q^-({\mathfrak {g}})\) is generated by the \(F_i^{\lambda (\alpha _i^\vee )+1}\), i ∈ I. Thus, Fi(v) = 0 if and only if \(\lambda (\alpha _i^\vee )=0\). This proves part (a). To prove part (b), note the following obvious fact.

Lemma 3.3

Let R be a ring and let M =⊕αAMα as R-modules. Let S be a subset of R. Then \( \operatorname *{\mathrm {Ann}}_S M=\bigcap _{\alpha \in A^{\prime }} \operatorname *{\mathrm {Ann}}_S M_\alpha = \operatorname *{\mathrm {Ann}}_S M^{\prime }\) where A is any subset of A such that for each α  A there exists α A such that \(M_\alpha \cong M_{\alpha ^{\prime }}\) and \(M^{\prime }=\bigoplus _{\alpha \in A^{\prime }} M_\alpha \). In particular, if S is finite, then \( \operatorname *{\mathrm {Ann}}_S M=\bigcap _{\alpha \in A_0} \operatorname *{\mathrm {Ann}}_S M_\alpha = \operatorname *{\mathrm {Ann}}_S M_0\) where A0 is a finite subset of A and \(M_0=\bigoplus _{\alpha \in A_0} M_\alpha \).

Apply this Lemma to \(R=U_q({\mathfrak {g}})\) and S = {Fi  :  i ∈ I}, which identifies with I, and M = V . Clearly, \(J(V)=\{ i\in I\,:\, F_i\in \operatorname *{\mathrm {Ann}}_S V\}\). Since S is finite and V  is a direct sum of simple modules, it follows from Lemma 3.3 that \( \operatorname *{\mathrm {Ann}}_S V= \operatorname *{\mathrm {Ann}}_S V^{\prime }\) where \(V^{\prime }=\bigoplus _{\lambda \in \Omega } U_q({\mathfrak {g}})(v_\lambda )\) for some finite \(\Omega \subset \{\lambda \in P^+\,:\,\mathcal I_\lambda (V)\neq 0\}\) and vλ ∈ V+(λ) ∖{0}, λ ∈ Ω. Since \(V^{\prime }=U_q({\mathfrak {g}})(v)\) with v =∑λ ∈ Ωvλ, part (b) follows. □

Given v ∈ V+(λ) ∖{0} and w ∈ W, define the standard extremal vectors [v]w of v by [v]w := Fw,λ(v). Furthermore, given v ∈ V+ ∖{0}, define
$$\displaystyle \begin{aligned}{}[v]_W:=\{ [v(\lambda)]_{w}\,:\, w\in W,\,\lambda\in \operatorname*{\mathrm{supp}} v\}. \end{aligned} $$
(3.2)

Proposition 3.4

Let \(V\in \mathscr O^{int}_q({\mathfrak {g}})\). Then for any v  V+ ∖{0}
  1. (a)

    if v is homogeneous, then \([v]_{w}=[v]_{w^{\prime }}\) if and only if w wWJ(v). In particular, the assignments [v]wwWJ(v) define a bijection Jv : [v]W → WWJ(v);

     
  2. (b)

    for any v  V+ ∖{0} the set [v]W is linearly independent.

     

Proof

To prove (a), let v ∈ V+(λ) ∖{0} for some λ ∈ P+ and recall that \(\operatorname {Stab}_W\lambda =W_{J_\lambda }\). It follows from Lemma 3.1(d) by an obvious induction on (w′′) that \(F_{ww^{\prime \prime },\lambda }=F_{w,\lambda }\) for all \(w^{\prime \prime }\in W_{J_\lambda }\). Since  = wλ implies that w = ww′′ for some \(w^{\prime \prime }\in W_{J_\lambda }\), it follows that \(F_{w^{\prime },\lambda }(v)=F_{w,\lambda }(v)\). Conversely, by Lemma 3.1(e) we have Fw,λ(v) ∈ V (). Thus, if  ≠ wλ, then Fw,λ(v) and \(F_{w^{\prime },\lambda }(v)\) are in different weight subspaces of V  and are linearly independent (and hence not equal).

In particular, we proved that [v(λ)]W is linearly independent for all v ∈ V+ ∖{0} and \(\lambda \in \operatorname *{\mathrm {supp}} v\). This, together with ( 2.4), proves part (b). □

Proposition 3.5

Let \(V\in \mathscr O^{int}_q({\mathfrak {g}})\). For each i  I, v  V+ ∖{0} the following are equivalent.
  1. (a)

    (λ, i) = 0 for all \(\lambda \in \operatorname *{\mathrm {supp}} v\), w  W;

     
  2. (b)

    \( \operatorname *{\mathrm {cl}}(\{i\})\in J(U_q({\mathfrak {g}})(v))\).

     

Proof

Let J be the neighborhood of i. In particular, J ∪ J = I.

(a) ⇒ (b) We need the following Lemma.

Lemma 3.6

For every i  j  I, i  j there exists w = wi,j ∈ W such that \(w_{i,j}\alpha _i\in {\mathbb Z}_{>0} \alpha _j+\sum _{k\in I\setminus \{j\}} {\mathbb Z}_{\ge 0}\alpha _k\).

Proof Let i = (i = i0, i1, …, id = j) ∈ Id+1 be an admissible sequence with \(d= \operatorname *{\mathrm {dist}}(i,j)\). In particular, this sequence is repetition free. Denote \(\beta _k=s_{i_k}\cdots s_{i_1}(\alpha _i)\). We claim that \(\beta _k\in \sum _{0\le r\le k} {\mathbb Z}_{>0} \alpha _{i_r}\). We argue by induction on k, the case k = 0 being obvious. For the inductive step, note that
$$\displaystyle \begin{aligned} \beta_k=s_{i_k}(\beta_{k-1}) \in \sum_{0\le r\le k-1}{\mathbb Z}_{>0} s_{i_k}\alpha_{i_r}=\sum_{0\le r\le k-1} {\mathbb Z}_{>0} (\alpha_{i_r}-\alpha_{i_r}(\alpha_{i_k}^\vee)\alpha_{i_k}). \end{aligned}$$
Since i is admissible, \(\alpha _{i_{k-1}}(\alpha _{i_k}^\vee )<0\) while \(\alpha _{i_r}(\alpha _{i_k}^\vee )\le 0\) for all 0 ≤ r ≤ k − 2. Therefore, \(\beta _k\in \sum _{0\le r\le k} {\mathbb Z}_{>0} \alpha _{i_r}\). In particular, \(w=s_{i_d}\cdots s_{i_1}\) is the desired wi,j ∈ W. □

Write \(\lambda \in \operatorname *{\mathrm {supp}} v\) as λ = λ +∑kIlkωk where λ∈ PW and \(l_k\in {\mathbb Z}_{\ge 0}\), k ∈ I. Let j ∈ J. In the notation of Lemma 3.6, we have wi,jαi =∑kInkαk with \(n_j\in {\mathbb Z}_{>0}\) and \(n_k\in {\mathbb Z}_{\ge 0}\), k ∈ I ∖{j}, for some wi,j ∈ W. Then 0 = 2(λ, wi,jαi) = (αj, αj)ljnj +∑kI∖{j}(αk, αk)lknk. Since nj > 0 and nk ≥ 0 for all k ≠ j this forces lj = 0.

In particular, J ⊂ J(v). Since J is closed in I, \([U_q^-({\mathfrak {g}}^{J^\perp }),F_j]=0\). Let \(V^{\prime }=U_q({\mathfrak {g}})(v)\). Then \(V^{\prime }=U_q^-({\mathfrak {g}})(v)=U_q^-({\mathfrak {g}}^{J^\perp })(v)\) and so J ⊂ J(V).

(b) ⇒ (a) Since \(J\subset J(U_q({\mathfrak {g}})(v))\) it follows that (λ, αj) = 0 for all j ∈ J. Since \(W\alpha _j\in \sum _{j^{\prime }\in J}{\mathbb Z} \alpha _{j^{\prime }}\) for any j ∈ J, the assertion follows. □

Lemma 3.7

Let \(V\in \mathscr O^{int}_q({\mathfrak {g}})\) such that \(V=U_q({\mathfrak {g}})(v)\) for some v  V+. The following are equivalent for i  I.
  1. (a)

    i  J(V ).

     
  2. (b)

    \([v(\lambda )]_{s_i w}=[v(\lambda )]_w\) for all \(\lambda \in \operatorname *{\mathrm {supp}} v\) and for all w  W;

     

Proof

The condition in part (b) implies that si =  for all \(\lambda \in \operatorname *{\mathrm {supp}} v\) and w ∈ W. Since \(s_i w\lambda =w\lambda -(w\lambda )(\alpha _i^\vee )\alpha _i\), it follows that (λ, i) = 0 for all \(\lambda \in \operatorname *{\mathrm {supp}} v\) and w ∈ W and so i ∈ J(V ) by Proposition 3.5.

Conversely, if i ∈ J(V ), then Fi([v]w) = 0 for all w ∈ W. In particular, if (siw) = (w) + 1 then, since \([v]_{s_i w}=F_i^{(w\lambda (\alpha _i^\vee ))}[v]_w\neq 0\) it follows that (, αi) = 0 and thus \([v]_{s_i w}=[v]_w\). Similarly, if (siw) = (w) − 1 applying the previous argument to w = siw, we obtain the same equality. □

3.3 Proof of Theorem 1.1

We will now express the action of the σi, i ∈ I on extremal vectors in terms of the natural action of W on WWJ.

Proposition 3.8

Let \(V\in \mathscr O^{int}_q({\mathfrak {g}})\) and v  V+ ∖{0}. Then
  1. (a)

    The set [v]W is W(V )-invariant. More precisely, \(\sigma ^i([v(\lambda )]_w)=[v(\lambda )]_{s_i w}\) for all i  I, w  W and \(\lambda \in \operatorname *{\mathrm {supp}} v\);

     
  2. (b)

    The canonical image of W(V ) in \(\operatorname {Bij}([v]_W)\) is isomorphic to \(W_{J_0}\) where \(J_0=I\setminus \operatorname *{\mathrm {cl}}(J(U_q({\mathfrak {g}})(v)))\).

     

Proof

To prove part (a), let λ ∈ P+, w ∈ W, i ∈ I and suppose first that (siw) = (w) + 1. Then \(w\lambda (\alpha _i^\vee )>0\) and \([v(\lambda )]_w\in \ker E_i\), whence \(\sigma ^i([v(\lambda )]_w)=F_i^{(w\lambda (\alpha _i^\vee ))}([v(\lambda )]_w)=F_{s_iw,\lambda }(v(\lambda ))=[v(\lambda )]_{s_iw}\) where we used Lemma 3.1(b). If (siw) = (w) − 1, then w = siw. By the above, \([v(\lambda )]_w=[v(\lambda )]_{s_i w^{\prime }}=\sigma ^i([v(\lambda )]_{w^{\prime }})=\sigma ^i([v(\lambda )]_{s_iw})\). Since σi is an involution, it follows that \(\sigma ^i([v(\lambda )]_{w})=[v(\lambda )]_{s_iw}\).

To prove (b), the bijections from Proposition 3.4(a) allow one to identify [v]W with \(\bigsqcup \limits _{\lambda \in \operatorname *{\mathrm {supp}} v} W/W_{J_\lambda }\). In particular, this induces an action of W on [v]W via \(w\cdot [v(\lambda )]_{w^{\prime }}=[v(\lambda )]_{ww^{\prime }}\), \(\lambda \in \operatorname *{\mathrm {supp}} v\), w, w∈ W. By part (a) the canonical images of W(V ) and W in \(\operatorname {Bij}([v]_W)\) coincide. We need the following general fact.

Lemma 3.9

Let G be a group acting on X =⊔αAXα. Then the canonical image of G in \(\operatorname {Bij}(X)\) is isomorphic to GK where K =⋂αAKα and Kα = {g  G  :  gx = x, ∀x  Xα} is the kernel of the action of G on Xα.

Applying this Lemma to X = [v]W, Xλ = [v(λ)]W and G = W and using the fact that \(K_\lambda =W_{J_\lambda }\) by Theorem 2.4 we conclude that \(K=\bigcap _{\lambda \in \operatorname *{\mathrm {supp}} v}W_{J_\lambda }\). By Lemma 2.1, \(K=W_{J^{\prime }}\) where J is the set of i ∈ I such that si fixes [v]W elementwise. By Lemma 3.7, J = J(V) where \(V^{\prime }=U_q({\mathfrak {g}})(v)\). Since J is closed, being an intersection of closed sets, \(W/W_{J^{\prime }}\cong W_{I\setminus J^{\prime }}\). □

Proof of Theorem 1.1

It follows from Proposition 3.4(a) that the assignments [v]wwWJ define a bijection J : [v]W → WWJ. This induces a group homomorphism \(\xi _V:\mathsf W(V)\to \operatorname {Bij}(W/W_J)\) via \(\xi _V(\sigma ^i)(w W_J)=\mathsf J(\sigma ^i([v]_w))=\mathsf J([v]_{s_i w})=s_i w W_J\). It follows that ξV(W(V )) coincides with the image of W in \(\operatorname {Bij}(W/W_J)\) given by the natural action. By Lemma 2.3, the latter is canonically isomorphic to \(W_{I\setminus J_0}\) where \(J_0= \operatorname *{\mathrm {cl}}(I\setminus J)\). □

4 Modified Lusztig Symmetries and Involutions σJ

Let \(\mathscr C\) be a Open image in new window -linear category whose objects are Open image in new window -vector spaces and let G be a group. An action of G on \(\mathscr C\) is an assignment \(g\mapsto g_\bullet =\{ g_V\,:\, V\in \mathscr C\}\), g ∈ G, where Open image in new window such that \((gg^{\prime })_V=g_V\circ g^{\prime }_V\) for all g, g∈ G and \(V\in \mathscr C\) and \(g_{V^{\prime }}\circ f=f\circ g_V\) for any g ∈ G and any morphism f : V → V in \(\mathscr C\).

Recall that the braid group \( \operatorname {\mathrm {\mathsf {Br}}}_W\) associated with a Coxeter group W is generated by the Ti, i ∈ I subject to the relations \(\underbrace {T_i T_j\cdots }_{m_{ij}}= \underbrace {T_j T_i\cdots }_{m_{ij}}\) for all i ≠ j ∈ I in the notation of Sect. 2.1. In this section we discuss modified Lusztig symmetries which provide an action of \( \operatorname {\mathrm {\mathsf {Br}}}_W\) associated with the Weyl group W of \({\mathfrak {g}}\) on the category \(\mathscr O^{int}_q({\mathfrak {g}})\) and use them to construct an action of \( \operatorname {\mathrm {\mathsf {Cact}}}_W\) on the same category.

4.1 Modified Lusztig Symmetries

Given i ∈ I and \(V\in \mathscr O^{int}_q({\mathfrak {g}})\) define Open image in new window by
$$\displaystyle \begin{aligned} T_i^+= T^{\prime}_{i,1}K_{\frac 12\alpha_i},\qquad T_i^-= T^{\prime\prime}_{i,1}K_{-\frac 12\alpha_i} \end{aligned}$$
where Open image in new window are Lusztig symmetries (see [30, §5.2]). We refer to these operators as modified Lusztig symmetries. By definition, \(T_i^\pm (V(\beta ))=V(s_i\beta )\) and so \(T_i^\pm K_\lambda =K_{s_i\lambda } T_i^\pm \), \(\lambda \in \frac 12 P\).

Lemma 4.1

The assignments \(T_i\mapsto T_i^+\) (respectively, \(T_i\mapsto T_i^-\)) define an action of \( \operatorname {\mathrm {\mathsf {Br}}}_W\) on \(\mathscr O^{int}_q({\mathfrak {g}})\).

Proof

It can be deduced from [30, Proposition 39.4.3] along the lines of [2, Lemma 5.2] that the \(T_i^+\) (and the \(T_i^-\)), i ∈ I, satisfy the defining relations of \( \operatorname {\mathrm {\mathsf {Br}}}_W\) as endomorphisms of V  for each \(V\in \mathscr O^{int}_q({\mathfrak {g}})\). To prove that this action of \( \operatorname {\mathrm {\mathsf {Br}}}_W\) commutes with morphisms, write, using [32, §3.1], for \(V\in \mathscr O^{int}_q({\mathfrak {g}})\) and v ∈ V
$$\displaystyle \begin{aligned} \begin{aligned} &T_i^+(v)=\sum_{(a,b,c)\in{\mathbb Z}_{\ge 0}^3} (-1)^b q_i^{b-a c} K_{\frac 12(a-c-1)\alpha_i}F_i^{(a)}E_i^{(b)}F_i^{(c)}K_{\frac 12(a-c)\alpha_i}(v),\\ &T_i^-(v)=\sum_{(a,b,c)\in{\mathbb Z}_{\ge 0}^3} (-1)^b q_i^{b-a c} K_{\frac 12(c-a+1)\alpha_i}E_i^{(a)}F_i^{(b)}E_i^{(c)}K_{\frac 12(c-a)\alpha_i}(v) \end{aligned} \end{aligned} $$
(4.1)
where the sum is finite since V  is integrable. It is now obvious that the \(T_i^\pm \), i ∈ I commute with homomorphisms of \(U_q({\mathfrak {g}})\)-modules. □

It is well-known (see, e.g., [30, §39.4.7]) that the element \(T_{i_1}\cdots T_{i_r}\) with i = (i1, …, ir) ∈ Ir reduced depends only on \(w=s_{i_1}\cdots s_{i_r}\) and not on i. This allows to define the canonical section of the natural group homomorphism \( \operatorname {\mathrm {\mathsf {Br}}}_W\to W\), Tisi, i ∈ I, by \(w\mapsto T_w:=T_{i_1}\cdots T_{i_r}\) where (i1, …, ir) ∈ R(w). Denote \(T^\pm _w\) the linear endomorphisms of any \(V\in \mathscr O^{int}_q({\mathfrak {g}})\) arising from Lemma 4.1 which correspond to the canonical element Tw of \( \operatorname {\mathrm {\mathsf {Br}}}_W\). The elements Tw, \(T^\pm _w\) are characterized by the following well-known property.

Lemma 4.2 ([30, §39.4.7])

Let w, w W be such that ℓ(w) + ℓ(w) = ℓ(ww). Then \(T_{ww^{\prime }}=T_w T_{w^{\prime }}\) and \(T^\pm _{ww^{\prime }}=T^\pm _w\circ T^\pm _{w^{\prime }}\) as linear endomorphisms of \(V\in \mathscr O^{int}_q({\mathfrak {g}})\).

Proposition 4.3

Let \(V\in \mathscr O^{int}_q({\mathfrak {g}})\). Then for any w, w W, λ  P+ and v  V+(λ) we have:
  1. (a)
    if ℓ(ww) = ℓ(w) + ℓ(w), then
    $$\displaystyle \begin{aligned} T_w^{+}(F_{w^{\prime},\lambda}(v))&=q^{\frac 12(w^{\prime}\lambda,\rho-w^{-1}\rho)}F_{ww^{\prime},\lambda}(v), \\ T_w^{-}(F_{w^{\prime},\lambda}(v))&=(-1)^{\rho^\vee(w^{\prime}\lambda-ww^{\prime}\lambda)}q^{\frac 12(w^{\prime}\lambda,\rho-w^{-1}\rho)}F_{ww^{\prime},\lambda}(v). \end{aligned} $$
     
  2. (b)
    if ℓ(ww) = ℓ(w) − ℓ(w), then
    $$\displaystyle \begin{aligned} T_w^{+}(F_{w^{\prime},\lambda}(v))&=(-1)^{\rho^\vee(w^{\prime}\lambda-ww^{\prime}\lambda)}q^{-\frac 12(w^{\prime}\lambda,\rho-w^{-1}\rho)}F_{ww^{\prime},\lambda}(v), \\ T_w^{-}(F_{w^{\prime},\lambda}(v))&=q^{-\frac 12(w^{\prime}\lambda,\rho-w^{-1}\rho)}F_{ww^{\prime},\lambda}(v). \end{aligned} $$
     
  3. (c)
    if ℓ(ww) = ℓ(w) − ℓ(w), then
    $$\displaystyle \begin{aligned} T_w^+(F_{w^{\prime},\lambda}(v))&=(-1)^{\rho^\vee(w^{\prime}\lambda-\lambda)}q^{\frac 12(\lambda,2\rho-w^{\prime -1}(\rho+w^{-1}\rho))} F_{ww^{\prime},\lambda}(v)\\ T_w^-(F_{w^{\prime},\lambda}(v))&=(-1)^{\rho^\vee(\lambda-ww^{\prime}\lambda)}q^{\frac 12(\lambda,2\rho-w^{\prime -1}(\rho+w^{-1}\rho))} F_{ww^{\prime},\lambda}(v). \end{aligned} $$
     

Proof

To prove (a) we argue by induction on (w), the case (w) = 0 being vacuously true. The following Lemma is the main ingredient in the proof of the inductive steps in Proposition 4.3.

Lemma 4.4

In the notation of Proposition 4.3 we have, for all i  I
Proof Clearly, \(F_{w^{\prime },\lambda }(v)\) is either a highest or a lowest weight vector in the ith simple quantum \({\mathfrak {sl}}_2\)-submodule Vm it generates where \(m=|(w^{\prime }\lambda ,\alpha _i^\vee )|\). Then (4.2) follows from [30, Propositions 5.2.2, 5.2.3]. Namely, let Vm be the standard simple \(U_{\mathbf {v}}({\mathfrak {sl}}_2)\)-module with the standard basis {zk}0≤km such that K(zk) = vm−2kzk and zk = F(k)(z0) = E(mk)(zm), 0 ≤ k ≤ m. Recall that \(T^+=T^{\prime }_1 K^{\frac 12}\) and \(T^-=T^{\prime \prime }_1 K^{-\frac 12}\). Then by [30, Propositions 5.2.2, 5.2.3] we have
$$\displaystyle \begin{aligned} T^{+}(z_k)=({-}1)^k \mathbf v^{k(m-k){+}\frac 12m} z_{m-k},\, T^-(z_k){=}({-}1)^{m{-}k} \mathbf v^{k(m-k){+}\frac 12 m}z_{m{-}k}, 0\le k\le m.\end{aligned} $$
(4.3)

Thus, (4.2) is obtained by applying (4.3) with k = 0 if \(w^{\prime }\lambda (\alpha _i^\vee )\ge 0\) and k = m if \(w^{\prime }\lambda (\alpha _i^\vee )\le 0\). □

To prove the inductive steps in part (a) of Proposition 4.3, suppose that w, w∈ W and i ∈ I satisfy (siww) = (w) + (w) + 1. In particular, (siw) = (w) + 1 and (ww) = (w) + (w). Then we have, by the induction hypothesis and (4.2)
$$\displaystyle \begin{aligned}\displaystyle T_{s_iw}^+(F_{w^{\prime},\lambda}(v))=T_i^+ T_{w}^+(F_{w^{\prime},\lambda}(v))=q^{\frac 12(w^{\prime}\lambda,\rho-w^{-1}\rho)} T_i^+(F_{ww^{\prime},\lambda}(v))\\\displaystyle =q^{\frac 12 (w^{\prime}\lambda,\rho-w^{-1}\rho)+\frac 12 (ww^{\prime}\lambda,\alpha_i)} F_{s_iww^{\prime},\lambda}(v) =q^{\frac 12 (w^{\prime}\lambda,\rho-w^{-1}\rho+w^{-1}\alpha_i)} F_{s_iww^{\prime},\lambda}(v)\\ =q^{\frac 12 (w^{\prime}\lambda,\rho-(s_iw)^{-1}\rho)} F_{s_iww^{\prime},\lambda}(v), \end{aligned} $$
where we used the W-invariance of (⋅, ⋅) and the obvious observation that αi = ρ − siρ. The second identity in part (a) is proved similarly using the observation that \(\mu (\alpha _i^\vee )=\rho ^\vee (\mu -s_i \mu )\).

The proof of part (b) is identical, the only difference being that we assume (siww) = (w) − (w) − 1 which implies that (siw) = (w) + 1 and (ww) = (w) − (w).

To prove part (c), denote w1 = ww. Then (w) = (w1) + (w−1), w = w1w−1 and so \(T^+_w=T^+_{w_1}\circ T^+_{w^{\prime }{ }^{-1}}\) by Lemma 4.2. Applying part (b) with w = w−1 and then part (a) with w = 1 and w = w1 we obtain
$$\displaystyle \begin{aligned} & T_w^+(F_{w^{\prime},\lambda}(v))=T_{w_1}^+(T^+_{w^{\prime}{}^{-1}}(F_{w^{\prime},\lambda}(v)))\\ &\quad =({-}1)^{\rho^\vee(w^{\prime}\lambda{-}\lambda)}q^{{-}\frac 12(w^{\prime}\lambda,\rho-w^{\prime}\rho)}T_{w_1}^+(v)\\ &\quad =({-}1)^{\rho^\vee(w^{\prime}\lambda{-}\lambda)}q^{{-}\frac 12(\lambda,w^{\prime}{}^{{-}1}\rho{+}w_1^{-1}\rho)} F_{w_1,\lambda}(v) \\ &\quad =(-1)^{\rho^\vee(w^{\prime}\lambda-\lambda)}q^{(\lambda,\rho)-\frac 12(\lambda,w^{\prime}{}^{-1}\rho)+\frac 12(\lambda,\rho-w^{\prime -1}w^{-1}\rho)} F_{w_1,\lambda}(v) \\ &\quad =(-1)^{\rho^\vee(w^{\prime}\lambda-\lambda)}q^{\frac 12(\lambda,2\rho-w^{\prime -1}(\rho+w^{-1}\rho))} F_{ww^{\prime},\lambda}(v). \end{aligned} $$
The identity for \(T_w^-\) is proved similarly. □

Recall from [30, Chapter 37] that \( \operatorname {\mathrm {\mathsf {Br}}}_W\) also acts on \(U_q({\mathfrak {g}})\) via Lusztig symmetries and let \(T_i^\pm \) be the automorphisms of \(U_q({\mathfrak {g}})\) defined as \(T_i^+=T^{\prime }_{i,1} \operatorname {\mathrm {ad}} K_{\frac 12\alpha _i}\) and \(T_i^-=T^{\prime \prime }_{i,1} \operatorname {\mathrm {ad}} K_{-\frac 12 \alpha _i}\), i ∈ I where \( \operatorname {\mathrm {ad}} K_{\lambda }(u)= K_\lambda u K_{-\lambda }\), \(\lambda \in \frac 12 P\), \(u\in U_q({\mathfrak {g}})\).

Remark 4.5

The operators \(T_i^\pm \), viewed as automorphisms of \(U_q({\mathfrak {g}})\), were already used in [2] for studying double canonical bases of \(U_q({\mathfrak {g}})\).

Lemma 4.6

On \(U_q({\mathfrak {g}})\) we have
  1. (a)

    \(T^\pm _i\circ \operatorname {\mathrm {ad}} K_\lambda = \operatorname {\mathrm {ad}} K_{s_i\lambda }\circ T^\pm _i\) for all \(\lambda \in \frac 12 P\), i  I;

     
  2. (b)

    \(T^+_{w}=T^{\prime }_{w,1}\circ \operatorname {\mathrm {ad}} K_{\frac 12(\rho -w^{-1}\rho )}\) and \(T^-_{w}=T^{\prime \prime }_{w,1}\circ \operatorname {\mathrm {ad}} K_{\frac 12(w^{-1}\rho -\rho )}\) for all w  W.

     

Proof

Part (a) is immediate, while part (b) follows from (a) by induction similar to that in Proposition 4.3. □

Lemma 4.7

Suppose that W is finite. Then for all i  I we have
$$\displaystyle \begin{aligned} \begin{aligned} T^\pm_{w_\circ}(E_i)=-q^{-\frac 12(\alpha_i,\alpha_i)}F_{i^\star }K_{\alpha_{i^\star }},\quad T^\pm_{w_\circ}(F_i)=-q^{-\frac 12(\alpha_i,\alpha_i)}E_{i^\star }K_{-\alpha_{i^\star }}. \end{aligned} \end{aligned}$$

Proof

By [1, Lemma 2.8] and Lemma 4.6(b) we have \(T^\pm _w(E_i)=q^{\pm \frac 12 (\rho -w^{-1}\rho ,\alpha _i)} E_j\) provided that i = αj. Since (w−1ρ, αi) = (ρ, i) = (ρ, αj) and (αi, αi) = (αj, αj) it follows that \(T^\pm _w(E_i)=E_j\). In particular, since \(w_\circ s_i\alpha _i=\alpha _{i^\star }\) we have \(T^\pm _{w_\circ s_i}(E_i)=E_{i^\star }\).

On the other hand, \(T^\pm _{w_\circ s_i}(E_i)=T^\pm _{s_{i^\star }w_\circ }(E_i)\) and \(\ell (w_\circ )=\ell (s_{i^\star }w_\circ )+1\) whence \(T^\pm _{w_\circ }(E_i)=T^\pm _{i^\star }(E_{i^\star })=-q^{-\frac 12(\alpha _i,\alpha _i)}F_{i^\star }K_{\alpha _{i^\star }}\). The argument for \(T^\pm _{w_\circ }(F_i)\) is similar. □

The following Lemmata are immediate from [30, Proposition 37.1.2].

Lemma 4.8

Let \(V\in \mathscr O_q^{int}({\mathfrak {g}})\). Then \(T_i^\pm (x(v))=T_i^\pm (x)(T_i^\pm (v))\) for all i  I, v  V , \(x\in U_q({\mathfrak {g}})\).

Lemma 4.9

We have \(T_w^\pm (\mathcal I_\lambda (V))=\mathcal I_\lambda (V)\) and \(T_w^\pm (V(\beta ))=V(w\beta )\) for any w  W, \(V\in \mathscr O^{int}_q(V)\), λ  P+ and β  P.

4.2 σ via Modified Lusztig Symmetries

Let \({\mathfrak {g}}\) be finite dimensional reductive and define
$$\displaystyle \begin{aligned} \sigma^\pm(v)=(-1)^{\rho^\vee(\lambda\mp \beta)} q^{\frac 12((\beta,\beta)-(\lambda,\lambda))-(\lambda,\rho)} T_{w_\circ}^\pm(v),\quad v\in V(\beta)\cap\mathcal I_\lambda(V). \end{aligned} $$
(4.4)
The main ingredient in our proof of Theorem 1.5 is the following result which generalizes [31, Proposition 5.5] to all reductive algebras including those whose semisimple part is not necessarily simply laced.

Theorem 4.10

Let \(V\in \mathscr O^{int}_q({\mathfrak {g}})\) . Then
  1. (a)

    σ+ = σ and is an involution which thus will be denoted by σ;

     
  2. (b)

    σ(x(v)) = θ(x)(σ(v)) for any \(x\in U_q({\mathfrak {g}})\), v  V ;

     
  3. (c)

    σ commutes with morphisms in \(\mathscr O^{int}_q({\mathfrak {g}})\).

     

Remark 4.11

For \({\mathfrak {g}}={\mathfrak {sl}}_n\) the involution σ coincides with the famous Schützenberger involution on Young tableaux which was established for the first time in [8]. Thus, we can regard σ as the generalized Schützenberger involution.

Proof

We need the following properties of σ±.

Lemma 4.12

For any \(V\in \mathscr O^{int}_q({\mathfrak {g}})\) we have
  1. (i)

    \(\sigma ^\pm (F_{w,\lambda }(v_\lambda ))=F_{w_\circ w,\lambda }(v_\lambda )\) for any \(v_\lambda \in \mathcal I_\lambda (V)(\lambda )\), λ  P+;

     
  2. (ii)

    σ±(x(v)) = θ(x)(σ±(v)) for any \(x\in U_q({\mathfrak {g}})\), v  V ;

     
  3. (iii)

    σ+ = σ and \((\sigma ^\pm )^2= \operatorname {\mathrm {id}}_V\).

     
Proof Let \(v_\lambda \in \mathcal I_\lambda (V)(\lambda )\), λ ∈ P+. Since ρ + wρ = 0, ρ(wμ) = −ρ(μ) for any μ ∈ P and (wu) = (w) − (u) for any u ∈ W, Proposition 4.3(c) with w = w and w = u yields
$$\displaystyle \begin{aligned} T_{w_\circ}^\pm(F_{u,\lambda}(v_\lambda))=(-1)^{\rho^\vee(\pm u\lambda-\lambda)}q^{(\lambda,\rho)} F_{w_\circ u,\lambda}(v_\lambda), \end{aligned}$$
Since Fu,λ(vλ) ∈ V (), (4.4) yields
$$\displaystyle \begin{aligned} \sigma^\pm(F_{u,\lambda}(v_\lambda))=(-1)^{\rho^\vee(\lambda\mp u\lambda)} q^{-(\lambda,\rho)}T_{w_\circ}^\pm(F_{u,\lambda}(v_\lambda)) =F_{w_\circ u,\lambda}(v_\lambda). \end{aligned}$$
This proves part (i). To prove part (ii), let v ∈ V (β). Then Ei(v) ∈ V (β + αi), and we obtain, by (4.4) and Lemmata 4.6, 4.7
$$\displaystyle \begin{aligned} \sigma^\pm(E_i(v))&=(-1)^{\rho^\vee(\lambda\mp(\beta+\alpha_i))}q^{\frac 12((\beta+\alpha_i,\beta+\alpha_i)-(\lambda,\lambda))-(\lambda,\rho)} T_{w_\circ}^\pm(E_i(v)) \\ &=-(-1)^{\rho^\vee(\lambda\mp \beta)}q^{(\beta,\alpha_i)+\frac 12((\alpha_i,\alpha_i)+(\beta,\beta)-(\lambda,\lambda))-(\lambda,\rho)} T_{w_\circ}^\pm(E_i)(T_{w_\circ}^\pm(v)) \\ &=(-1)^{\rho^\vee(\lambda\mp \beta)}q^{(\beta,\alpha_i)+\frac 12((\beta,\beta)-(\lambda,\lambda))-(\lambda,\rho)} F_{i^\star}K_{i^\star}(T_{w_\circ}^\pm(v)) \\ &=\theta(E_i)(q^{(\beta,\alpha_i+w_\circ\alpha_{i^\star})}\sigma^\pm(v)) =\theta(E_i)\sigma^\pm(v). \end{aligned} $$
The identity σ±(Fi(v)) = θ(Fi)(σ±(v)) is proved similarly. Finally, for any \(\mu \in \frac 12 P\) we have \(\sigma ^\pm (K_\mu (v))=q^{(\beta ,\mu )}\sigma ^\pm (v)=q^{(w_\circ \beta ,w_\circ \mu )}\sigma ^\pm (v)=\theta (K_\mu )(\sigma ^\pm (v))\).
Let 𝜖, 𝜖∈{+, −}. It follows from part (ii) that
$$\displaystyle \begin{aligned} \sigma^\epsilon\circ\sigma^{\epsilon^{\prime}}(x(v))=\sigma^\epsilon(\theta(x)(v))=\theta^2(x)(v)=x(v) \end{aligned}$$
for any \(x\in U_q({\mathfrak {g}})\) and v ∈ V  since θ is an involution. Thus, \(\sigma ^\epsilon \circ \sigma ^{\epsilon ^{\prime }}\) is an endomorphism of V  as a \(U_q({\mathfrak {g}})\)-module. By part (i) we have
$$\displaystyle \begin{aligned} \sigma^\epsilon\circ\sigma^{\epsilon^{\prime}}(F_{w,\lambda}(v_\lambda))=\sigma^\epsilon(F_{w_\circ w,\lambda}(v_\lambda)) =F_{w,\lambda}(v_\lambda) \end{aligned}$$
for any w ∈ W, λ ∈ P+ and \(v_\lambda \in \mathcal I_\lambda (V)(\lambda )\). In particular, \(\sigma ^\epsilon \circ \sigma ^{\epsilon ^{\prime }}(v_\lambda )=v_\lambda \) and so \(\sigma ^\epsilon \circ \sigma ^{\epsilon ^{\prime }}\) is the identity map on the (simple) \(U_q({\mathfrak {g}})\)-submodule of V  generated by vλ. Since V  is generated by \(\bigoplus _{\lambda \in P^+}\mathcal I_\lambda (V)(\lambda )\) as a \(U_q({\mathfrak {g}})\)-module, \(\sigma ^{\epsilon }\circ \sigma ^{\epsilon ^{\prime }}= \operatorname {\mathrm {id}}_V\). This proves part (iii). □

Parts (a) and (b) of Theorem 4.10 were established in Lemma 4.12. To prove part (b), note the following obvious fact.

Lemma 4.13

Let \(\xi _\bullet =\{ \xi _V\in \operatorname {\mathrm {End}}_{U_q({\mathfrak {g}})}V\,:\, V\in \mathscr O_q^{int}({\mathfrak {g}})\}\) and suppose that ξ commute with morphisms in \(\mathscr O^{int}_q({\mathfrak {g}})\), that is \(\xi _{V^{\prime }}\circ f=f\circ \xi _V\) for any morphism f : V  V in \(\mathscr O^{int}_q({\mathfrak {g}})\). Let Open image in new window and define \(\xi ^\chi _\bullet \) by \(\xi ^\chi _V(v)=\sum _{\beta \in \operatorname *{\mathrm {supp}} v}\chi (\lambda ,\beta ) \xi _V(v(\beta ))\), \(v\in \mathcal I_\lambda (V)\), \(V\in \mathscr O^{int}_q({\mathfrak {g}})\). Then \(\xi ^\chi _\bullet \) also commutes with morphisms in \(\mathscr O_q^{int}({\mathfrak {g}})\).

It is immediate from the definition (4.4) of σ that \(\sigma _\bullet =(T^\pm _\bullet )^{\chi _\pm }\) where
$$\displaystyle \begin{aligned}\chi_\pm(\lambda,\beta)=(-1)^{\rho^\vee(\lambda\mp \beta)} q^{\frac 12((\beta,\beta)-(\lambda,\lambda))-(\lambda,\rho)}, \quad (\lambda,\beta)\in P^+\times P. \end{aligned}$$
Then part (c) follows from Lemma 4.13. □

4.3 Parabolic Involutions and the Proof of Theorem 1.5

In view of Theorem 4.10, given \(V\in \mathscr O^{int}_q({\mathfrak {g}})\) and \(J\in \mathscr J\), let σJ : V → V  be defined by (4.4) with \(U_q({\mathfrak {g}})\) replaced by \(U_q({\mathfrak {g}}^J)\). Thus,
$$\displaystyle \begin{aligned} \sigma^J(v)=(-1)^{\rho^\vee_J(\lambda_J\mp\beta)} q^{-\frac 12((\lambda_J,\lambda_J)-(\beta,\beta))-(\lambda_J,\rho_J)}T_{w_\circ^J}^\pm(v), \end{aligned} $$
(4.5)
for any \(\lambda _J\in P^+_J\), β ∈ P and \(v\in \mathcal I_{\lambda _J}^J(V)(\beta )\). Note that \(\mathcal I_{\lambda _J}^J(V)(\beta )=0\) unless \(\lambda _J-\beta \in \sum _{j\in J}{\mathbb Z}_{\ge 0}\alpha _j\). We need the following properties of σJ.

Proposition 4.14

Let \(V\in \mathscr O^{int}_q({\mathfrak {g}})\) . Then
  1. (a)

    \(\sigma ^J(F_{w,\lambda _J}(v))=F_{w_\circ ^J w,\lambda }(v)\) for any \(v\in \mathcal I_{\lambda _J}^J(V)(\lambda _J)\), λJ ∈ P+. In particular, \(\sigma ^J(v)=v^J:=F_{w_\circ ^J,\lambda }(v)\).

     
  2. (b)

    σ J is an involution;

     
  3. (c)

    σJ commutes with morphisms in \(\mathscr O^{int}_q({\mathfrak {g}}^J)\) and satisfies σJ(x(v)) = θJ(x)(σJ(v)), \(x\in U_q({\mathfrak {g}}^J)\), v  V ;

     
  4. (d)

    \(\sigma ^J(V(\beta ))=V(w_\circ ^J\beta )\), β  P;

     
  5. (e)
    for any \(\lambda _J\in P^+_J\), β  P and \(v\in \mathcal I_{\lambda _J}^J(V)(\beta )\) we have
    $$\displaystyle \begin{aligned} \sigma^J(v) =(-1)^{\rho_J^\vee(-\lambda_J\mp\beta)} q^{\frac 12((\lambda_J,\lambda_J)-(\beta,\beta))+(\lambda_J,\rho_J)}(T_{w_\circ^J}^\pm)^{-1}(v). \end{aligned}$$
     

Proof

Replacing \({\mathfrak {g}}\) by \({\mathfrak {g}}^J\) we obtain part (a) from Lemma 4.12(i) and parts (b), (c) from Theorem 4.10. Part (d) is immediate from (4.5) and Lemma 4.9. To prove part (e), let v = σJ(v). Then v = σJ(v) and \(v^{\prime }\in \mathcal I_{\lambda _J}^J(V)(\beta ^{\prime })\) where \(\beta ^{\prime }=w_\circ ^J\beta \). Applying \((T^\pm _{w_\circ })^{-1}\) to (4.5) with v replaced by v we obtain
$$\displaystyle \begin{aligned} (T^\pm_{w_\circ})^{-1}(\sigma^J(v^{\prime}))=(-1)^{\rho^\vee_J(\lambda_J\mp\beta^{\prime})} q^{-\frac 12((\lambda_J,\lambda_J)-(\beta^{\prime},\beta^{\prime}))-(\lambda_J,\rho_J)}v^{\prime}. \end{aligned}$$
Since \(\rho ^\vee _J(\beta ^{\prime })=\rho ^\vee _J(w_\circ ^J\beta )=-\rho ^\vee _J(\beta )\) and (⋅, ⋅) is W-invariant, it follows that
$$\displaystyle \begin{aligned} (T^\pm_{w_\circ})^{-1}(v)=(-1)^{\rho^\vee_J(\lambda_J\pm\beta)} q^{-\frac 12((\lambda_J,\lambda_J)-(\beta,\beta))-(\lambda_J,\rho_J)}\sigma^J(v). \end{aligned}$$
The assertion is now immediate. □

We need the following results.

Proposition 4.15

For any \(J\subset J^{\prime }\in \mathscr J\), \(\sigma ^{J^{\prime }}\circ \sigma ^J=\sigma ^{J^\star }\circ \sigma ^{J^{\prime }}\) where  : J J is the unique involution satisfying \(\alpha _{j^\star }=-w_\circ ^{J^{\prime }}\alpha _j\), j  J.

Proof

We may assume, without loss of generality, that J = I (and so \({\mathfrak {g}}\) is reductive finite dimensional). Let \(w_J=w_\circ w_\circ ^J\). Note that \(w_\circ =w_J w_\circ ^J=w_\circ ^{J^\star } w_J\) and \(\ell (w_\circ )=\ell (w_J)+\ell (w_\circ ^J)=\ell (w_J)+\ell (w_\circ ^{J^\star })\). Then by Lemma 4.2
$$\displaystyle \begin{aligned} T^\pm_{w_J}=T^\pm_{w_\circ}\circ (T^\pm_{w_\circ^J})^{-1} =(T^\pm_{w_\circ^{J^\star}})^{-1}\circ T^\pm_{w_\circ}. \end{aligned} $$
(4.6)
Let \(v\in V(\beta )\cap \mathcal I_\lambda (V)\cap \mathcal I^J_{\lambda _J}(V)\), λ ∈ P+, \(\lambda _J\in P^+_J\), β ∈ P. Using Lemma 4.14(e), (4.4), Lemma 4.9 and (4.6) we obtain
$$\displaystyle \begin{aligned}\displaystyle \sigma\circ \sigma^J(v)=(-1)^{\rho^\vee_J(-\lambda_J\mp\beta)} q^{\frac 12((\lambda_J,\lambda_J)-(\beta,\beta))+(\lambda_J,\rho_J)}\sigma( (T^\pm_{w_\circ^J})^{-1}(v)) \\\displaystyle =(-1)^{\rho^\vee(\lambda\mp w_\circ^J\beta)+\rho^\vee_J(-\lambda_J\mp\beta)} q^{\frac 12((\lambda_J,\lambda_J)-(\lambda,\lambda))+(\lambda_J,\rho_J)-(\lambda,\rho)}T_{w_\circ}^\pm (T^\pm_{w_\circ^J})^{-1}(v)\\ =(-1)^{\rho^\vee(\lambda\mp w_\circ^J\beta)+\rho^\vee_J(-\lambda_J\mp\beta)} q^{\frac 12((\lambda_J,\lambda_J)-(\lambda,\lambda))+(\lambda_J,\rho_J)-(\lambda,\rho)} T^\pm_{w_J}(v). \end{aligned} $$
Similarly,
$$\displaystyle \begin{aligned}\displaystyle \sigma^{J^\star}\circ \sigma(v)=(-1)^{\rho^\vee(\lambda\mp\beta)} q^{-\frac 12((\lambda,\lambda)-(\beta,\beta))-(\lambda,\rho)}\sigma^{J^\star}(T_{w_\circ}^\pm(v))\\\displaystyle =(-1)^{\rho^\vee(\lambda\mp\beta)+\rho^\vee_{J^\star}(w_\circ\lambda_J\mp w_\circ\beta)} q^{\frac 12((\lambda_J,\lambda_J)-(\lambda,\lambda)) -(w_\circ\lambda_J,\rho_{J^\star})-(\lambda,\rho)} (T_{w_\circ^{J^\star}}^\pm)^{-1}(T_{w_\circ}^\pm(v)) \\ =(-1)^{\rho^\vee(\lambda\mp\beta)+\rho^\vee_{J}(-\lambda_J\pm \beta)} q^{\frac 12((\lambda_J,\lambda_J)-(\lambda,\lambda)) +(\lambda_J,\rho_{J})-(\lambda,\rho)} T_{w_J}^\pm(v) \end{aligned} $$
since \(\rho ^\vee _{J^\star }(w_\circ \mu )=-\rho ^\vee _J(\mu )\), μ ∈ P and \(w_\circ \rho _{J^\star }=-\rho _J\). Since \(-\rho ^\vee _J(\beta )=\rho ^\vee _J(w_\circ ^J\beta )\) and \(\rho ^\vee (\beta -w\beta )=\rho ^\vee _J(\beta -w\beta )\) for any w ∈ WJ, it follows that \(\sigma ^{J^\star }\circ \sigma =\sigma \circ \sigma ^J\). In particular, since \(\rho ^\vee _J(\lambda _J-\beta )=\rho ^\vee (\lambda _J-\beta )\) we have
$$\displaystyle \begin{aligned} \sigma\circ\sigma^J=\sigma^{J^\star}\circ\sigma=(-1)^{\rho^\vee(\lambda\pm\lambda_J)+\rho_J^\vee(\pm\lambda_J-\lambda_J)}q^{\frac 12((\lambda_J,\lambda_J)-(\lambda,\lambda)) +(\lambda_J,\rho_{J})-(\lambda,\rho)} T_{w_J}^\pm(v), \end{aligned} $$
(4.7)
that is, the right-hand side does not explicitly depend on β. □

Proposition 4.16

Let \(J,J^{\prime }\in \mathscr J\) be orthogonal. Then \(\sigma ^{J\sqcup J^{\prime }}=\sigma ^J\circ \sigma ^{J^{\prime }}\).

Proof

As before we may assume, without loss of generality, that I = J ⊔ J. Then \(w_\circ =w_\circ ^{J}w_\circ ^{J^{\prime }}= w_\circ ^{J^{\prime }}w_\circ ^J\) and hence \(T^\pm _{w_\circ }=T^\pm _{w_\circ ^J}\circ T^\pm _{w_\circ ^{J^{\prime }}}=T^\pm _{w_\circ ^{J^{\prime }}}\circ T^\pm _{w_\circ ^J}\) by Lemma 4.2. Let λ ∈ P+, \(\lambda _J\in P^+_J\), \(\lambda _{J^{\prime }}\in P^+_{J^{\prime }}\), β ∈ P and \(v\in \mathcal I_\lambda (V)\cap \mathcal I^J_{\lambda ^J}(V)\cap \mathcal I^J_{\lambda ^{J^{\prime }}}(V)\). Then \(\gamma _J=\lambda _J-\beta \in \sum _{j\in J}{\mathbb Z}_{\ge 0}\alpha _j\), \(\gamma _{J^{\prime }}=\lambda _{J^{\prime }}-\beta \in \sum _{j\in J^{\prime }}{\mathbb Z}_{\ge 0}\alpha _{j^{\prime }}\), and \(\gamma =\lambda -\beta =\gamma _{J}+\gamma _{J^{\prime }}\). Then we can rewrite (4.5) and (4.4) as
$$\displaystyle \begin{aligned} &\sigma^{J}(v)=(-1)^{\rho^\vee_{J}(\gamma_{J})} q^{-\frac 12(\gamma_{J},\gamma_{J})+(\lambda_{J},\gamma_{J})-(\lambda_{J},\rho_{J})} T^+_{w_\circ^{J}}(v) \\ &\sigma^{J^{\prime}}(v)=(-1)^{\rho^\vee_{J^{\prime}}(\gamma_{J^{\prime}})} q^{-\frac 12(\gamma_{J^{\prime}},\gamma_{J^{\prime}})+(\lambda_{J^{\prime}},\gamma_{J^{\prime}})-(\lambda_{J^{\prime}},\rho_{J^{\prime}})} T^+_{w_\circ^{J^{\prime}}}(v) \\ &\sigma(v)=(-1)^{\rho^\vee(\gamma)} q^{-\frac 12(\gamma,\gamma)+(\lambda,\gamma)-(\lambda,\rho)} T^+_{w_\circ}(v). \end{aligned} $$
Since \(w_\circ ^J(\gamma _{J^{\prime }})=\gamma _{J^{\prime }}\) we have
$$\displaystyle \begin{aligned} &\sigma^{J}(\sigma^{J^{\prime}}(v)) \\ &=({-}1)^{\rho^\vee_{J}(\gamma_J){+}\rho^\vee_{J^{\prime}}(\gamma_{J^{\prime}})} q^{{-}\frac 12((\gamma_J,\gamma_{J}){+}(\gamma_{J^{\prime}},\gamma_{J^{\prime}})){-}(\lambda_{J},\rho_{J}) {-}(\lambda_{J^{\prime}},\rho_{J^{\prime}}){+}(\lambda_J,\gamma_J){+}(\lambda_{J^{\prime}},\gamma_{J^{\prime}})} T^+_{w_\circ}(v) \\ &=(-1)^{\rho^\vee(\gamma)} q^{-\frac 12 (\gamma,\gamma)-(\lambda,\rho)+(\lambda,\gamma)}T^+_{w_\circ}(v)=\sigma(v), \end{aligned} $$

since \(\rho ^\vee (\gamma )=\rho ^\vee _J(\gamma _J)+\rho ^\vee _{J^{\prime }}(\gamma _{J^{\prime }})\), \((\gamma ,\gamma )=(\gamma _J,\gamma _J)+(\gamma _{J^{\prime }},\gamma _{J^{\prime }})\), \((\lambda _J,\zeta )+(\lambda _{J^{\prime }},\zeta ^{\prime })=(\lambda _J+\lambda _{J^{\prime }},\zeta +\zeta ^{\prime })=(\lambda ,\zeta +\zeta ^{\prime })\) for any \(\zeta \in \sum _{j\in J}\mathbb Q\alpha _j\), \(\zeta ^{\prime }\in \sum _{j\in J^{\prime }}\mathbb Q\alpha _{j^{\prime }}\). □

Proof of Theorem 1.5

Parts (a) (respectively, (b), (c)) of Theorem 1.5 were established in Lemma 4.14(b) (respectively, Proposition 4.16, Proposition 4.15). □

4.4 Kernels of Actions of Cactus Groups

For any \(V\in \mathscr O^{int}_q({\mathfrak {g}})\), denote ΦV be the subgroup of Open image in new window generated by the \(\sigma ^J_V\), \(J\in \mathscr J\). We need the following basic properties of ΦV.

Lemma 4.17

For any injective morphism f : V V  in \(\mathscr O^{int}_q({\mathfrak {g}})\) the assignments \(\sigma ^J_V\mapsto \sigma ^J_{V^{\prime }}\), \(J\in \mathscr J\) define a surjective homomorphism \(f^*:\Phi _{V}\to \Phi _{V^{\prime }}\). In particular, if f is an isomorphism then so is f.

Proof

Let V′′ = f(V). By Theorem 4.10(c), we have \(\sigma ^J_V\circ f=f\circ \sigma ^J_{V^{\prime }}\) for all \(J\in \mathscr J\) and so the group ΦV acts on V′′, that is, there is a canonical homomorphism of groups Open image in new window . Clearly, the assignments gf−1 ∘ g ∘ f, Open image in new window define an isomorphism Open image in new window . Let Open image in new window . We claim that \(f^*(\Phi _V)=\Phi _{V^{\prime }}\). Indeed, \(f^*(\sigma ^J_V)=\sigma ^J_{V^{\prime }}\) for all \(J\in \mathscr J\). Since \(\Phi _{V^{\prime }}\) is generated by the \(\sigma ^J_{V^{\prime }}\), ΦV is generated by the \(\sigma ^J_V\), \(J\in \mathscr J\), and f is a homomorphism of groups, the assertion follows. □

Proposition 4.18

Let \(V\in \mathscr O^{int}_q({\mathfrak {g}})\). Then \(\Phi _V\cong \Phi _{ \underline V}\) where \( \underline V=\bigoplus \limits _{\lambda \in P^+\,:\, \operatorname {\mathrm {Hom}}_{U_q({\mathfrak {g}})}(V_\lambda ,V)\neq 0} V_\lambda \). In particular, for any \(V\in \mathscr O^{int}_q({\mathfrak {g}})\) the group ΦV is a quotient of \(\Phi _{\mathcal C_q({\mathfrak {g}})}\) where \(\mathcal C_q({\mathfrak {g}})=\bigoplus _{\lambda \in P^+} V_\lambda \).

Proof

Fix \(f_\lambda \in \operatorname {\mathrm {Hom}}_{U_q({\mathfrak {g}})}(V_\lambda ,V)\setminus \{0\}\) for all λ ∈ P+ with \( \operatorname {\mathrm {Hom}}_{U_q({\mathfrak {g}})}(V_\lambda ,V)\neq 0\) and let \(f: \underline V\to V\) be the direct sum of these fλ. Then f is injective. Applying Lemma 4.17 with \(V^{\prime }= \underline V\) we obtain a surjective group homomorphism \(f^*:\Phi _V\to \Phi _{ \underline V}\). It remains to prove that its kernel is trivial. We apply Lemma 3.3 with Open image in new window and S = {g − 1  :  g ∈ ΦV}⊂ R. Since ΦV is a subgroup of Open image in new window , \( \operatorname *{\mathrm {Ann}}_S V=\{0\}\). By our choice of \( \underline V\), M = V  and \(M^{\prime }=f( \underline V)\) satisfy the assumptions of Lemma 3.3 and so \( \operatorname *{\mathrm {Ann}}_S f( \underline V)=\{0\}\). Since \(\ker f^*=\{ g\in \Phi _V\,:\, g\circ f= \operatorname {\mathrm {id}}_{ \underline V}\}\), it follows that \(\ker f^*\) is trivial. The second assertion is immediate from the first one and Lemma 4.17. □

5 An Action of \( \operatorname {\mathrm {\mathsf {Cact}}}_W\) on c-Crystal Bases and Proof of Theorem 1.8

Retain the notation of Sect. 2.5 and observe that the assignment (l, k, s)↦(l, l − k, −s), \((l,k,s)\in \mathbb D\), defines an involution on \(\mathbb D\). The following is the main result of this section.

Theorem 5.1

Let \({\mathfrak {g}}\) be reductive. Suppose that \(\mathbf c:\mathbb D\to \mathbb Q(z)^\times \) satisfies
$$\displaystyle \begin{aligned} \underline{\mathbf c}_{l,k,s}= \underline{\mathbf c}_{l,l-k,-s},\qquad \underline{\mathbf c}_{l,0,-l}=1,\qquad (l,k,s)\in\mathbb D \end{aligned} $$
(5.1)
in the notation of Lemma 2.12 . Then for any \(V\in \mathscr O^{int}_q({\mathfrak {g}})\)
  1. (a)

    σI(L) = L for any (c, I)-monomial lattice L in V ;

     
  2. (b)

    If (L, B) is a c-crystal basis such that \(B_+=\{b\in B\,:\, \tilde e_{i,1}^{\mathbf c}(b)=0,\, i\in I\}\) is a basis of L+qL+ where L+ = L  V+, then the induced \(\mathbb Q\)-linear map \(\tilde \sigma ^I: L/qL\to L/qL\) preserves B.

     

Proof

We abbreviate σ = σI, \(V_+^I=V_+\) and \(\mathsf M^{\mathbf c}(v_+)=\mathsf M_I^{\mathbf c}(v_+)\) for any homogeneous v+ ∈ V+. The key ingredient of our argument is the following

Proposition 5.2

Let \({\mathfrak {g}}\) be reductive, let \(V\in \mathscr O^{int}_q({\mathfrak {g}})\) and let \(\mathbf c:\mathbb D\to \mathbb Q(z)^\times \) . Then
  1. (a)

    If c satisfies the first condition in (5.1) then \(\sigma \circ \tilde e_{i,s}^{\mathbf c}=\tilde e_{i^\star ,-s}^{\mathbf c}\circ \sigma \) in Open image in new window for any i  I, \(s\in {\mathbb Z}\).

     
  2. (b)

    If c satisfies (5.1), then

    σ(Mc(v+)) = Mc(v+) for any homogeneous v+ ∈ V+.

     
Proof In view of Lemma 2.11 and (2.6), to prove (a) it suffices to verify the identity for all v ∈ V  of the form \(v=F_i^{(k)}(u)\), \(u\in \ker E_i\cap \ker (K_{\alpha _i}-q_i^l)\), 0 ≤ k ≤ l. We have
$$\displaystyle \begin{aligned} \sigma\circ \tilde e_{i,s}^{\mathbf c}(v)=\underline{\mathbf c}_{l,k,s}(q_i)\sigma(F_i^{(k-s)}(u))=\underline{\mathbf c}_{l,k,s}(q_i) E_{i^\star}^{(k-s)}(\sigma(u)). \end{aligned} $$
(5.2)
We need the following

Lemma 5.3

\(\sigma (u)\in \ker F_{i^\star }\cap \ker (K_{\alpha _{i^\star }}-q_{i^{\star }}^{-l})\).

Proof Indeed, \(F_{i^\star }(\sigma (u))=\sigma (E_i(u))=0\) and \(K_{\alpha _i^\star }(\sigma (u))=\sigma (K_{-\alpha _i}(u))=q_i^{-l}\sigma (u)=q_{i^\star }^{-l}\sigma (u)\). □

Using Lemmata 5.3 and 2.12, we obtain
$$\displaystyle \begin{aligned} \tilde e_{i^\star,-s}^{\mathbf c}(\sigma(v))=\tilde e_{i^\star,-s}^{\mathbf c}(E_{i^\star}^{(k)}(\sigma(u))) =\underline{\mathbf c}_{l,l-k,-s}(q_{i^\star}) E_{i^\star}^{(k-s)}(\sigma(u)). \end{aligned} $$
(5.3)
Since \(q_{i^\star }=q_i\), by the assumptions of Proposition 5.2 we have \( \underline {\mathbf c}_{l,l-k,-s}(q_{i^\star })= \underline {\mathbf c}_{l,k,s}(q_i)\). Then (5.2) and (5.3) imply that \( \tilde e_{i^\star ,-s}^{\mathbf c}(\sigma (v))= \sigma (\tilde e_{i,s}^{\mathbf c}(v))\).

To prove part (b), we need the following

Lemma 5.4

Suppose that cl,0,−l = 1∕(l)z! for all \(l\in {\mathbb Z}_{\ge 0}\) (that is, \( \underline {\mathbf c}_{l,0,-l}=1\) in the notation of Lemma 2.12). Then for any λ  P+, v+ ∈ V+(λ) and i = (i1, …, im)  ∈ Im reduced \(F_{\mathbf i,\lambda }(v_+)=\tilde e_{i_1,-a_1(\mathbf i,\lambda )}^{\mathbf c}\cdots \tilde e_{i_m,-a_m(\mathbf i,\lambda )}^{\mathbf c}(v_+)\) in the notation of (3.1). In particular,
$$\displaystyle \begin{aligned} \sigma(v_+)=\tilde e_{i_1,-a_1(\mathbf i,\lambda)}^{\mathbf c}\cdots \tilde e_{i_N,-a_N(\mathbf i,\lambda)}^{\mathbf c}(v_+) \end{aligned}$$

where i = (i1, …, iN) ∈ R(w).

Proof We use induction on m, the case m = 0 being trivial. For i and λ ∈ P+ fixed we abbreviate ak = ak(i, λ). For the inductive step, note that \(F_{\mathbf i,\lambda }=F_{i_1}^{(a_1)}F_{{\mathbf {i}}^{\prime },\lambda }\) where i = (i2, …, im) and so \(F_{\mathbf i,\lambda }(v_+)=F_{i_1}^{(a_1)}(v^{\prime })\) where \(v^{\prime }=F_{{\mathbf {i}}^{\prime },\lambda }(v_+)= \tilde e_{i_2,-a_2}^{\mathbf c}\cdots \tilde e_{i_m,-a_m}^{\mathbf c}(v_+)\) by the induction hypothesis. Since \(v^{\prime }\in \ker E_{i_1}\), it follows from assumptions of the lemma and the first identity in (2.6) with i = i1, k = 0 and l = a1 = −s that \(F_{i_1}^{(a_1)}(v^{\prime })=\tilde e_{i_1,-a_1}^{\mathbf c}(v^{\prime })=\tilde e_{i_1,-a_1}^{\mathbf c}\cdots \tilde e_{i_m,-a_m}^{\mathbf c}(v_+)\). Since \(\sigma (v_+)=F_{w_\circ ,\lambda }(v_+)\) by Lemma 4.12(i) with w = 1, the second assertion follows from the first and Lemma 3.1(a). □

Suppose now that v ∈Mc(v+) that is \(v=\tilde e_{j_1,m_1}^{\mathbf c}\cdots \tilde e_{j_r,m_r}^{\mathbf c}(v_+)\in \mathsf M^{\mathbf c}(v_+)\), for some (j1, …, jr) ∈ Ir and \((m_1,\dots ,m_r)\in {\mathbb Z}\). Using Lemma 5.4 and Proposition 5.2(a), we obtain
$$\displaystyle \begin{aligned} \sigma(v)&=\tilde e_{j_1^\star,-m_1}^{\mathbf c}\cdots \tilde e_{j_r^\star,-m_r}^{\mathbf c}(\sigma(v_+))\\ &=\tilde e_{j_1^\star,-m_1}^{\mathbf c}\cdots \tilde e_{j_r^\star,-m_r}^{\mathbf c}\tilde e_{i_1,-a_1}^{\mathbf c}\cdots \tilde e_{i_N,-a_N}^{\mathbf c}(v_+)\in\mathsf M^{\mathbf c}(v_+), \end{aligned} $$
where i = (i1, …, iN) ∈ R(w) and ak = ak(i, λ), 1 ≤ k ≤ N. Thus, σ(Mc(v+)) ⊂Mc(v+). Since σ is an involution, it follows that σ(Mc(v+)) = Mc(v+). □
Part (a) of Theorem 5.1 is immediate from Proposition 5.2(b). In particular, for each (c, I)-monomial lattice L in V  the involution σV induces an involution \(\tilde \sigma \) on the \(\mathbb Q\)-vector space \(\tilde L=L/q L\) satisfying
$$\displaystyle \begin{aligned} \tilde \sigma\circ \tilde e_{i,s}^{\mathbf c}=\tilde e_{i^\star,-s}^{\mathbf c}\circ\tilde\sigma. \end{aligned} $$
(5.4)
The following is immediate from Proposition 5.2(b).

Corollary 5.5

Let L be a (c, I)-monomial lattice in V . Then \(\tilde \sigma (\tilde {\mathsf M}^{\mathbf c}(\tilde v_+))=\tilde {\mathsf M}^{\mathbf c}(\tilde v_+)\) for any \(\tilde v_+\in L_+/qL_+\).

Using the assumptions of part (b) of Theorem 5.1 we conclude that \(\bigcup _{b_+\in B^+}\tilde {\mathsf M}^{\mathbf c}(b_+)=B\cup \{0\}\). Then it follows from Corollary 5.5 that \(\tilde \sigma \) preserves B ∪{0}. Since \(\tilde \sigma \) is an involution, it follows that \(\tilde \sigma (B)=B\). This completes the proof of Theorem 5.1(b). □

Note that (5.4) implies that for any upper crystal basis (L, B) of \(V\in \mathscr O^{int}_q({\mathfrak {g}})\) the operator \(\tilde \sigma ^I_V\) satisfies
$$\displaystyle \begin{aligned} \tilde\sigma\circ (\tilde e_i^{up})^s=(\tilde e_{i^\star}^{up})^{-s}\circ\tilde\sigma^I_V. \end{aligned} $$
(5.5)
In particular, we obtain the following

Corollary 5.6

Let λ  P+ and (Lλ, Bλ) be an upper crystal basis of Vλ. If f is any non-zero map Bλ ∪{0}→ Bλ ∪{0} satisfying (5.5), then \(f=\tilde \sigma ^I_{V_\lambda }|{ }_{B_\lambda \cup \{0\}}\).

Proof of Theorem 1.8

Note that
$$\displaystyle \begin{aligned} \underline{\mathbf c}^{low}_{l,k,s}=1,\qquad \underline{\mathbf c}^{up}_{l,k,s}=\frac{(l-k+s)_z! (k-s)_z!}{(l-k)_z! (k)_z!},\qquad (l,k,s)\in\mathbb D. \end{aligned} $$
(5.6)
It is now immediate that (5.1) holds for c ∈ {cup, clow}.

Furthermore, by Lemma 2.15 and Remark 2.16, Theorem 5.1 applies to every c-crystal basis at q = 0 for any \(J\in \mathscr J\) with \({\mathfrak {g}}\) replaced by \({\mathfrak {g}}^J\) and c ∈ {cup, clow}. Thus, σJ preserves a lower or upper crystal lattice L and \(\tilde \sigma ^J\) preserves B.

In particular, ΦV acts on L and this action factors through to an action on LqL and induces an action on B by permutations. □

Remark 5.7

Let L be an upper crystal lattice for \(V\in \mathscr O^{int}_q({\mathfrak {g}})\). It follows from the definition of \(\sigma ^i_V\) that for any v ∈ L(β), β ∈ P we have \(\sigma ^i_V(v)=\tilde e_i^{-\beta (\alpha _i^\vee )}(v)\). In particular, the action of \(\tilde \sigma ^i_V\) on an upper crystal basis (L, B) of V  coincides with Kashiwara’s crystal Weyl group action (see [26]).

We conclude this section with a discussion of the action of \( \operatorname {\mathrm {\mathsf {Cact}}}_W\) on upper global crystal bases. Let \(\bar \cdot \) be any field involution of Open image in new window such that \(\overline {q^{\frac 1{2d}}}=q^{-\frac 1{2d}}\).

Proposition 5.8

Let (L, B) be an upper crystal basis of \(V\in \mathscr O^{int}_q({\mathfrak {g}})\) and let Gup(B) be the corresponding upper global crystal basis. Denote by \(\bar \cdot \) the (unique) additive map V  V  satisfying \(\overline {f\cdot b}=\overline f\cdot \overline b\) for all Open image in new window , b  Gup(B). Then \(\overline {\sigma ^J(b)}=\sigma ^J(b)\) for any \(J\in \mathscr J\).

Proof

Denote by \(\bar \cdot \) the ring automorphism of \(U_q({\mathfrak {g}})\) satisfying \(\overline {E_i}=E_i\), \(\overline {F_i}=F_i\), i ∈ I, \(\overline {K_\lambda }=K_{-\lambda }\), \(\lambda \in \frac 12P\) and \(\overline {f u}=\overline f\cdot \overline u\) for all Open image in new window , \(u\in U_q({\mathfrak {g}})\). The following is immediate from the properties of the upper global crystal basis [25].

Lemma 5.9

The map \(\bar \cdot :V\to V\) defined in Proposition 5.8 satisfies
$$\displaystyle \begin{aligned} \overline{u(v)}=\overline u(\overline v),\qquad v\in V,\, u\in U_q({\mathfrak{g}}). \end{aligned} $$
(5.7)

The following is immediate.

Lemma 5.10

Let \(\eta :U_q({\mathfrak {g}})\to U_q({\mathfrak {g}})\) be any algebra automorphism commuting with \(\bar \cdot :U_q({\mathfrak {g}})\to U_q({\mathfrak {g}})\) and let \(V\in \mathscr O^{int}_q({\mathfrak {g}})\) with a fixed set V0 ⊂ V  generating V  as a \(U_q({\mathfrak {g}})\)-module. Let \(\bar \cdot :V\to V\) be any map satisfying (5.7) and let Open image in new window be such that:
  1. (i)

    σ(u(v)) = η(u)(σ(v)), \(u\in U_q({\mathfrak {g}})\), v  V ;

     
  2. (ii)

    \(\sigma (\overline v)=\overline {\sigma (v)}\) for any v  V0.

     

Then \(\sigma (\overline v)=\overline {\sigma (v)}\) for all v  V .

The set \(V_0=G^{up}(B)\cap V_+^J\) generates V  as a \(U_q({\mathfrak {g}}^J)\)-module. Clearly, θJ commutes with \(\bar \cdot \)-involution on \(U_q({\mathfrak {g}}^J)\). The condition (i) of Lemma 5.10 holds with η = θJ by Proposition 4.14(d). By Proposition 4.14(a), \(\sigma ^J(b)=F_{w_\circ ^J,\lambda }(b)\) for any b ∈ V0(λ), \(\lambda \in P^+_J\). Since \(\overline {F_{w_\circ ^J,\lambda }}= F_{w_\circ ^J,\lambda }\) and \(\overline b=b\), the condition (ii) of Lemma 5.10 is also satisfied. The assertion follows by Lemma 5.10. □

6 σI and the Canonical Basis

6.1 Automorphisms and Skew Derivations of Localizations

Let R be a unital Open image in new window -algebra. Given a monoid Γ written multiplicatively and acting on R by algebra automorphisms, define the semidirect product of R with the monoidal algebra Open image in new window of Γ as Open image in new window with the multiplication defined by where ⊲ denotes the action of Γ on R. Since (r ⊗ 1)(1 ⊗ γ) = r ⊗ γ, we will henceforth omit the symbol ⊗ when writing elements of Open image in new window . In other words, Open image in new window is generated by R as a subalgebra and Γ subject to the relations The following characterization of cross products is immediate.

Lemma 6.1

Let \(f:R\to \widehat R\) be a homomorphism of Open image in new window -algebras and let \(g:\Gamma \to \widehat R\) be a homomorphism of multiplicative monoids. Suppose that R is a Open image in new window -module algebra. Then assignments r  γf(r) ⋅ g(γ), r  R, γ  Γ define a homomorphism of Open image in new window -algebras if and only if
Let S be a submonoid of R ∖{0}. Denote Sop the opposite monoid of S and denote its elements by [s], s ∈ S. Suppose that R is a Open image in new window -module algebra with [s]⊲r = Σs(r), s ∈ S where Σs is an algebra automorphism of R and assume that
$$\displaystyle \begin{aligned} \Sigma_s(s)=s,\qquad s\in S. \end{aligned} $$
(6.2)
Denote Open image in new window . We say that S as above is an Ore submonoid if
$$\displaystyle \begin{aligned}r s=s\Sigma_s(r),\qquad r\in R,\, s\in S. \end{aligned} $$
(6.3)
We use the convention that Σλs = Σs for all Open image in new window . This notation is justified by the following

Lemma 6.2

Suppose that (6.2) holds. Then the following are equivalent:
  1. (i)

    the natural homomorphism 1R,S : R  R[S−1] is injective;

     
  2. (ii)

    S is an Ore submonoid of R, and the assignments r ⋅ [s]↦rs−1, r  R, s  S define an isomorphism \(R[S^{-1}]\to \underline {R[S^{-1}]}\) where \( \underline {R[S^{-1}]}\) is the Ore localization of R by S;

     

Proof

In Open image in new window we have
$$\displaystyle \begin{aligned}{}[s]\cdot r=\Sigma_s(r)\cdot [s],\qquad s\in S,\, r\in R. \end{aligned} $$
(6.4)
In particular, [s] ⋅ s = s ⋅ [s], s ∈ S. Multiplying both sides of (6.4) by s on the left and on the right we conclude that rs = s Σs(r) in R[S−1] for all r ∈ R, s ∈ S. This identity clearly holds in 1R,S(R). Since 1R,S is injective, this implies that the corresponding identity holds in R and so S satisfies the two-sided Ore condition and so R admits the Ore localization \( \underline {R[S^{-1}]}\). The assignments r[s]↦rs−1, r ∈ R, s ∈ S define a surjective homomorphism from \(R[S^{-1}]\to \underline {R[S^{-1}]}\) which is easily seen to be injective. Thus, (a) implies (b).

Conversely, the natural homomorphism \(R\to \underline {R[S^{-1}]}\), rr ⋅ 1, r ∈ R is injective. Since it equals the composition of 1R,S and the isomorphism \(R[S^{-1}]\to \underline {R[S^{-1}]}\), it follows that 1R,S is injective. □

The following Lemma is immediate.

Lemma 6.3

Let R be a Open image in new window -algebra and S  R ∖{0} be an Ore submonoid. Let R be a Open image in new window -subalgebra of R and suppose that S R S is an Ore submonoid of R. Then R[S−1] is isomorphic to the subalgebra of R[S−1] generated by R and {s−1  :  s S}.

Lemma 6.4

Suppose that (6.2) and the assumptions of Lemma 6.2(b) hold. Let φ : R  R be any Open image in new window -algebra homomorphism, S be an Ore submonoid of R and S be an Ore submonoid of R such that φ(S) ⊂ S. Suppose that \(\Sigma ^{\prime }_{\varphi (s)}\circ \varphi =\varphi \circ \Sigma _s\) for all s  S. Then there exists a unique homomorphism \(\widehat \varphi :R[S^{-1}]\to R^{\prime }[{S^{\prime }}^{-1}]\) such that \(\widehat \varphi |{ }_R=\varphi \).

Proof

We apply Lemma 6.1 with Γ = Sop, Open image in new window and \(g:S^{op}\to \widehat R\) defined by g([s]) = [φ(s)]. Then and so (6.1) holds. Therefore, the assignments r[s]↦φ(r)[φ(s)], r ∈ R, s ∈ S, define a homomorphism Open image in new window . Since \(\widehat {\widehat \varphi }(s[s])=\varphi (s)[\varphi (s)]\) it follows that the image of the defining ideal of R[S−1] under \(\widehat {\widehat \varphi }\) is contained in the defining ideal of R[S−1]. Thus, \(\widehat {\widehat \varphi }\) factors through to the desired homomorphism \(\widehat \varphi :R[S^{-1}]\to R^{\prime }[{S^{\prime }}^{-1}]\). □

Let L± : R → R be Open image in new window -algebra homomorphisms and E : R → R be a Open image in new window -linear map. We say that E is an (L+, L)-derivation from R to R if E(rr) = E(r)L+(r) + L(r)E(r) for all r, r∈ R. Denote \(\operatorname {Der}_{L_+,L_-}(R,R^{\prime })\) the Open image in new window -subspace of Open image in new window of (L+, L)-derivations from R to R. We refer to an (L, L−1)-derivation as an L-derivation and abbreviate \(\operatorname {Der}_{L_+,L_-}R=\operatorname {Der}_{L_+,L_-}(R,R)\). The following is immediate.

Lemma 6.5

Let R0 be a generating subset of R and let \(D,D^{\prime }\in \operatorname {Der}_{L_+,L_-}(R,R^{\prime })\). Then \(D|{ }_{R_0}=D^{\prime }|{ }_{R_0}\) implies that D = D.

Given r∈ R, denote by \(D^\pm _{r^{\prime }}\) the linear maps R → R
$$\displaystyle \begin{aligned} D^-_{r^{\prime}}(x)=r^{\prime} L_+(x)-L_-(x)r^{\prime},\qquad D^+_{r^{\prime}}(x)=L_-(x)r^{\prime}-r^{\prime}L_+(x),\qquad x\in R. \end{aligned} $$
(6.5)

Lemma 6.6

Let L± : R  R be Open image in new window -algebra homomorphisms. The assignments \(r^{\prime }\mapsto D^+_{r^{\prime }}\) (respectively, \(r^{\prime }\mapsto D^-_{r^{\prime }}\)), r R define Open image in new window -linear maps \(R^{\prime }\to \operatorname {Der}_{L_+,L_-}(R,R^{\prime })\).

Proof

For any x, x∈ R we have
$$\displaystyle \begin{aligned}\displaystyle D^-_{r^{\prime}}(xx^{\prime})=r^{\prime} L_+(xx^{\prime})-L_-(xx^{\prime})r^{\prime}\\\displaystyle =(r^{\prime}L_+(x)-L_-(x)r^{\prime})L_+(x^{\prime})+L_-(x)(r^{\prime}L_+(x^{\prime})-L_-(x^{\prime})r^{\prime})\\=D^-_{r^{\prime}}(x)L_+(x^{\prime})+L_-(x)D^-_{r^{\prime}}(x^{\prime}). \end{aligned} $$
Thus, \(D^-_{r^{\prime }}\in \operatorname {Der}_{L_+,L_-}(R,R^{\prime })\). The computation for \(D^+_{r^{\prime }}\) is similar and is omitted. The linearity of both maps in r is obvious. □

6.2 The Gelfand–Kirillov Model for the Category \(\mathscr O^{int}_q({\mathfrak {g}})\)

Throughout this section we mostly follow the notation from [6, Section 6]. Let Γ be the monoid P+ written multiplicatively, with its elements denoted by vλ, λ ∈ P+. Let \(\mathcal A_q({\mathfrak {g}})\) be an isomorphic copy of \(U_q^-({\mathfrak {g}})\) whose generators are denoted by xi, i ∈ I. We denote the degree of a homogeneous element \(x\in \mathcal A_q({\mathfrak {g}})\) with respect to its Q-grading by |x|∈ − Q+. Define an action of Γ on \(\mathcal A_q({\mathfrak {g}})\) by vλx = q(λ, |x|)x for \(x\in \mathcal A_q({\mathfrak {g}})\) homogeneous. Let Open image in new window . In particular, we have
$$\displaystyle \begin{aligned} v_\lambda x=q^{(\lambda,|x|)}x v_\lambda \end{aligned} $$
(6.6)
for all λ ∈ P+ and for all \(x\in \mathcal A_q({\mathfrak {g}})\) homogeneous. We extend the Q-grading on \(\mathcal A_q({\mathfrak {g}})\) to a P-grading on \(\mathcal B_q({\mathfrak {g}})\) via |vλ| = λ for λ ∈ P+. Let \(\mathscr O_q({\mathfrak {g}})\) be the category of \(U_q({\mathfrak {g}})\)-modules whose objects satisfy all assumptions on objects of \(\mathscr O^{int}_q({\mathfrak {g}})\) except that we do not assume that the Ei, Fi, i ∈ I, act locally nilpotently while assuming that all weight subspaces are finite dimensional. The following essentially coincides with [6, Lemma 6.1].

Lemma 6.7

The algebra \(\mathcal B_q({\mathfrak {g}})\) is a module algebra in the category \(\mathscr O_q({\mathfrak {g}})\) with respect to the action given by the following formulae for all \(\lambda \in \frac 12P\), i  I
  • Kλ(y) = q(λ, |y|)y for all homogeneous elements \(y\in \mathcal B_q({\mathfrak {g}})\) and \(\lambda \in \frac 12 P\);

  • \(\displaystyle F_i(y)=\frac {x_i K_{\frac 12\alpha _i}(y)-K_{-\frac 12\alpha _i}(y)x_i}{q_i-q_i^{-1}}\) for all \(y\in \mathcal B_q({\mathfrak {g}})\) and thus is a \(K_{\frac 12\alpha _i}\) -derivation of \(\mathcal B_q({\mathfrak {g}})\) ;

  • Ei is the unique \(K_{\frac 12\alpha _i}\)-derivation of \(\mathcal B_q({\mathfrak {g}})\) such that Ei(xj) = δi,j and Ei(vμ) = 0 for all i, j  I, μ  P+.

Thus for all \(x,y\in \mathcal B_q({\mathfrak {g}})\), i ∈ I we have
$$\displaystyle \begin{aligned} X_i(xy)=X_i(x)K_{\frac 12\alpha_i}(y)+K_{-\frac 12\alpha_i}(x) X_i(y) \end{aligned} $$
(6.7)
and more generally, for all n ≥ 0
$$\displaystyle \begin{aligned} X_i^{(n)}(xy)=\sum_{r+s=n}X_i^{(r)}K_{-\frac s2\alpha_i}(x)X_i^{(s)}K_{\frac r2\alpha_i}(y) \end{aligned} $$
(6.8)
where Xi is either Ei or Fi. The following is immediate from the definition of \(\mathcal B_q({\mathfrak {g}})\) and its \(U_q({\mathfrak {g}})\)-module structure.

Corollary 6.8

\(\mathcal B_q({\mathfrak {g}})=\sum _{\lambda \in P^+} \mathcal A_q({\mathfrak {g}})v_\lambda \) where \(\mathcal A_q({\mathfrak {g}})v_\lambda \) is a \(U_q({\mathfrak {g}})\)-submodule of \(\mathcal B_q({\mathfrak {g}})\) for each λ  P+ and the sum is direct.

In the sequel we will also use \(E_i^*\) which is defined as the unique \(K_{-\frac 12\alpha _i}\)-derivation of \(\mathcal B_q({\mathfrak {g}})\) satisfying \(E_i^*(x_j)=\delta _{i,j}\), \(E_i^*(v_\lambda )=0\) for all λ ∈ P+, j ∈ I. It is easy to check that \(E_i^*(x)=(E_i(x^*))^*\), \(x\in \mathcal A_q({\mathfrak {g}})\), where \({ }^*:\mathcal A_q({\mathfrak {g}})\to \mathcal A_q({\mathfrak {g}})\) is the unique anti-involution preserving the xi, i ∈ I.

In the spirit of [22], using the decomposition from Corollary 6.8 we can define a linear map \(\mathbf j:\mathcal B_q({\mathfrak {g}})\to \mathcal A_q({\mathfrak {g}})\) by
$$\displaystyle \begin{aligned}\mathbf j(x\cdot v_\lambda)=q^{-\frac 12(\lambda,|x|)} x \end{aligned} $$
(6.9)
for all λ ∈ P+ and \(x\in \mathcal A_q({\mathfrak {g}})\) homogeneous. Clearly, \(\mathbf j|{ }_{\mathcal A_q({\mathfrak {g}})v_\lambda }\) is a bijection onto \(\mathcal A_q({\mathfrak {g}})\).

Lemma 6.9

For any symmetrizable Kac–Moody \({\mathfrak {g}}\) we have:
  1. (a)

    j is a surjective homomorphism of \(U_q^+({\mathfrak {g}})\)-modules, with respect to the action defined in Lemma 6.7.

     
  2. (b)

    \( \mathbf j(x\cdot y) =q^{\frac 12(\lambda ,|\mathbf j(y)|)-\frac 12(\mu ,\mathbf j(|x|))} \mathbf j(x)\cdot \mathbf j(y) \) for all \(x\in \mathcal A_q(\mathfrak {g})v_\lambda \) , \(y\in \mathcal A_q(\mathfrak {g})v_\mu \) homogeneous.

     

Proof

Part (a) is easily checked using Corollary 6.8. To prove part (b) note that \(x=q^{\frac 12(\lambda ,|\mathbf j(x)|)}\mathbf j(x)v_\lambda \) for all \(x\in \mathcal A_q(\mathfrak {g})v_\lambda \) homogeneous and so we can write
$$\displaystyle \begin{aligned}\displaystyle x\cdot y=q^{\frac 12(\lambda+\mu,|\mathbf j(x)|+|\mathbf j(y)|)} \mathbf j(x\cdot y)v_{\lambda+\mu}=q^{\frac 12(\lambda,|\mathbf j(x)|)+\frac 12(\mu,|\mathbf j(y)|)} \mathbf j(x)v_\lambda \mathbf j(y)v_\mu \\\displaystyle =q^{\frac 12(\lambda,|\mathbf j(x)|)+\frac 12(\mu,|\mathbf j(y)|)+(\lambda,|\mathbf j(y)|)} \mathbf j(x)\cdot\mathbf j(y)v_{\lambda+\mu}. \end{aligned} $$
The assertion is now immediate. □

Given a \(U_q({\mathfrak {g}})\)-module M, denote by Mint the set of all m ∈ M such that \(U_q({\mathfrak {g}})(m)\in \mathscr O^{int}_q({\mathfrak {g}})\). The following is well-known and in fact is easy to check.

Lemma 6.10

The assignment MMint for every \(U_q({\mathfrak {g}})\)-module M and ff for any morphism of \(U_q({\mathfrak {g}})\)-modules defines an additive submonoidal functor from the tensor category of \(U_q({\mathfrak {g}})\)-modules to \(\mathscr O^{int}_q({\mathfrak {g}})\), that is Mint ⊗ Nint ⊂ (MN)int for any \(U_q({\mathfrak {g}})\)-modules M, N. In particular, if M is an algebra object in the category of \(U_q({\mathfrak {g}})\)-modules, then Mint is its \(U_q({\mathfrak {g}})\)-module subalgebra and an algebra object in the category \(\mathscr O^{int}_q({\mathfrak {g}})\).

Proposition 6.11

For any λ  P+ the \(U_q({\mathfrak {g}})\)-submodule of \(\mathcal A_q({\mathfrak {g}})v_\lambda \) generated by vλ is naturally isomorphic to Vλ and coincides with \((\mathcal A_q({\mathfrak {g}})v_\lambda )^{int}\).

Proof

Given \(M\in \mathscr O_q({\mathfrak {g}})\), define M =⊕βPM(β) where Open image in new window . Endow M with a \(U_q({\mathfrak {g}})\)-module structure via (u ⋅ f)(m) = f(uT(m)), \(u\in U_q({\mathfrak {g}})\), f ∈ M, m ∈ M, where uuT, \(u\in U_q({\mathfrak {g}})\) is the unique anti-involution of \(U_q({\mathfrak {g}})\) such that EiT = Fi and \(K_{\mu { }}^T=K_\mu \), \(\mu \in \frac 12 P\). The following is well-known

Lemma 6.12

For any symmetrizable Kac–Moody \({\mathfrak {g}}\) , we have:
  1. (a)

    The assignments M  M, \(M\in \mathscr O_q({\mathfrak {g}})\) define an involutive contravariant functor on \(\mathscr O_q({\mathfrak {g}})\).

     
  2. (b)

    For any \(M\in \mathscr O_q({\mathfrak {g}})\), (Mint) = (M)int.

     
  3. (c)

    For any \(V\in \mathscr O^{int}_q({\mathfrak {g}})\) , V is naturally isomorphic to V .

     

We need the following well-known fact which essentially coincides with [3, Lemma 2.10].

Lemma 6.13

There exists a unique non-degenerate pairing Open image in new window such that Open image in new window , i, j  I, Open image in new window , Open image in new window , \(u\in U_q^-({\mathfrak {g}})\), \(x\in \mathcal A_q({\mathfrak {g}})\) and Open image in new window for \(u\in U_q^-({\mathfrak {g}})\), \(x\in \mathcal A_q({\mathfrak {g}})\) homogeneous unless deg u = |x|.

Denote Mλ, λ ∈ P, the Verma module with highest weight λ (see, e.g., [30, §3.4.5]). For every λ ∈ P+ we fix mλ ∈ Mλ(λ) ∖{0}. Since Mλ is free as a \(U_q^-({\mathfrak {g}})\)-module, every element of Mλ can be written, uniquely, as umλ for some \(u\in U_q^-({\mathfrak {g}})\). Let \(\mathcal M_q({\mathfrak {g}})=\bigoplus _{\lambda \in P^+} M_\lambda \). Define Open image in new window by Open image in new window , \(u\in U_q^-({\mathfrak {g}})\), \(x\in \mathcal A_q({\mathfrak {g}})\), λ, μ ∈ P+. It is immediate from the definition that Open image in new window , β, β∈ P, unless β = β.

The following Lemma seems to be well-known. We provide a proof for reader’s convenience.

Lemma 6.14

The pairing Open image in new window is non-degenerate and contragredient, that is

In particular, \(\mathcal A_q({\mathfrak {g}})v_\lambda \), λ  P+ naturally identifies with \(M_\lambda ^\vee \).

Proof The pairing Open image in new window is non-degenerate as a direct sum of non-degenerate (in view of Lemma 6.13) pairings Open image in new window . To prove that it is contragredient, it suffices to prove (6.10) for u∈{Kμ, Ei, Fi}, \(\mu \in \frac 12P\), i ∈ I. for all i ∈ I. Moreover, we may assume without loss of generality that m = u(mλ) and b = xvλ with \(u\in U_q^-({\mathfrak {g}})\), \(x\in \mathcal A_q({\mathfrak {g}})\) homogeneous. We have Furthermore, by Lemmata 6.9(a) and 6.13 we obtain In particular, we proved (6.10) for u∈{Kμ, Fi}, \(\mu \in \frac 12 P\), i ∈ I for all \(m\in \mathcal M_q({\mathfrak {g}})\) and \(b\in \mathcal B_q({\mathfrak {g}})\).
It remains to prove that for all m ∈ Mλ and for all \(b\in \mathcal A_q({\mathfrak {g}})v_\lambda \) homogeneous. We argue by induction on ρ(λ − β) where m ∈ Mλ(β). If β = λ, then Ei(m) = 0 while Fi(b) = b with |b| = |b|− αi. Since |b|∈ λ − Q+, |b|≠ β and so Open image in new window . For the inductive step, it suffices to assume that m = Fj(m) where m is homogeneous. We have where we used (6.10) for u = Fj and \(u^{\prime }=[E_i,F_j]=\delta _{i,j}(q_i-q_i^{-1})^{-1}(K_{\alpha _i}-K_{-\alpha _i})=[E_j,F_i]\).

The second assertion of the Lemma is immediate from the first. □

By [30, Proposition 3.5.6], Mλ has a unique integrable quotient isomorphic to Vλ. Applying int to the surjection Mλ → Vλ we obtain, by Lemmata 6.12 and 6.14, the desired isomorphism \(V_\lambda \cong (M_\lambda ^\vee )^{int}=(\mathcal A_q({\mathfrak {g}})v_\lambda )^{int}\) such that vλvλ. □

In view of Proposition 6.11, from now on we identify Vλ, λ ∈ P+, with the \(U_q({\mathfrak {g}})\)-submodule of \(\mathcal A_q({\mathfrak {g}})v_\lambda \) generated by vλ.

Lemma 6.15

For any λ, μ  P+ we have Vλ ⋅ Vμ = Vλ+μ in \(\mathcal B_q({\mathfrak {g}})\).

Proof

It is immediate from the definition of \(\mathcal B_q({\mathfrak {g}})\) and Corollary 6.8 that \(V_\lambda \cdot V_\mu \subset \mathcal A_q({\mathfrak {g}})v_{\lambda +\mu }\). Furthermore, since Vλ ⋅ Vμ is the image of Vλ ⊗ Vμ which is integrable, by Proposition 6.11 we have \(V_\lambda \cdot V_\mu \subset (\mathcal A_q(\mathfrak {g})v_{\lambda +\mu })^{int}=V_{\lambda +\mu }\). As the latter is a simple \(U_q(\mathfrak {g})\)-module, Vλ ⋅ Vμ = Vλ+μ. □

Denote \(\mathcal C_q({\mathfrak {g}})=\mathcal B_q({\mathfrak {g}})^{int}\). The following is an immediate corollary of Proposition 6.11 and Lemma 6.15.

Corollary 6.16

\(\mathcal C_q({\mathfrak {g}})\) decomposes as \(\mathcal C_q({\mathfrak {g}})=\sum _{\lambda \in P^+} V_\lambda \) as a \(U_q({\mathfrak {g}})\) -module algebra.

Proposition 6.17

For any symmetrizable Kac–Moody \({\mathfrak {g}}\) and λ  P+ we have:
$$\displaystyle \begin{aligned} \mathbf j(V_\lambda)=\bigcap_{i\in I} \ker {E_i^*}^{\lambda(\alpha_i^\vee)+1}. \end{aligned} $$
(6.11)

Proof

Note that Lemma 6.13 yields an isomorphism of Open image in new window -vector spaces Open image in new window defined by Open image in new window , \(x\in \mathcal A_q({\mathfrak {g}})\), \(u\in U^-_q({\mathfrak {g}})\). Define \(\phi _\lambda ^\vee :M_\lambda ^\vee \to U_q^-({\mathfrak {g}})^\vee \) by \(\phi _\lambda ^\vee (f)(u)=f(u(m_\lambda ))\) for all \(f\in M_\lambda ^\vee \), \(u\in U_q^-({\mathfrak {g}})\).

Define an action of \(U_q^+({\mathfrak {g}})\) on \(U_q^-({\mathfrak {g}})^\vee \) by \((u_+\cdot f)(u_-):=f(u_+^T u_-)\), \(u_\pm \in U_q^\pm ({\mathfrak {g}})\). The following is an immediate consequence of Lemmata 6.13 and 6.14.

Lemma 6.18

For any λ  P+, \(\phi _\lambda ^\vee \) is an isomorphism of \(U_q^+({\mathfrak {g}})\)-modules. Moreover, the following diagram in the category of \(U_q^+({\mathfrak {g}})\)-modules commutes

where the left vertical arrow is obtained by the identification \(M_\lambda ^\vee \cong \mathcal A_q({\mathfrak {g}})v_\lambda \) from Proposition 6.11.

Let \(\mathcal J_\lambda \), λ ∈ P+, be the kernel of the canonical projection of Mλ on Vλ. It is well-known (see, e.g., [30, Proposition 3.5.6]) that \(\mathcal J_\lambda =\sum _{i\in I} U^-_q({\mathfrak {g}}) F_i^{\lambda (\alpha _i^\vee )+1}(m_\lambda )\). Applying to the projection Mλ → Vλ and using that \(V_\lambda \cong V_\lambda ^\vee \) we obtain an embedding \(V_\lambda \to M_\lambda ^\vee \). Note that
$$\displaystyle \begin{aligned} \phi_\lambda^\vee(V_\lambda)=\{ f\in U_q^-({\mathfrak{g}})^\vee\,:\, f\Big(\sum_{i\in I} U_q^-({\mathfrak{g}})F_i^{\lambda(\alpha_i^\vee)+1}\Big)=\{0\}\}. \end{aligned}$$
Therefore, \(\phi _\lambda ^\vee (V_\lambda )=\bigcap _{i\in I}\mathcal K_i\) where \(\mathcal K_i=\{ f\in U_q^-({\mathfrak {g}})^\vee \,:\,f(U_q^-({\mathfrak {g}})F_i^{\lambda (\alpha _i^\vee )+1})=0\}\). By Lemma 6.13, \(\xi ^{-1}(\mathcal K_i)=\ker E_i^*{ }^{\lambda (\alpha _i^\vee )+1}\). Using (6.12) we obtain \(\mathbf j(V_\lambda ) =\bigcap _{i\in I}\xi ^{-1}(\mathcal K_i)=\bigcap _{i\in I}\ker E_i^*{ }^{\lambda (\alpha _i^\vee )+1}\). □

6.3 Realization of σI via Quantum Twist

Let v = Fw,λ(vλ), λ ∈ P+ where we use the notation from Sect. 3.1 (see also Sect. 3.2). This notation agrees with that in [6, (6.3)]. We need the following

Lemma 6.19

Let \({\mathfrak {g}}\) be a symmetrizable Kac–Moody algebra. Then for any w, w W and λ, μ  P+ we have
  1. (a)

    v ⋅ v = vw(λ+μ). In particular, for any w  W, the assignments vλv, λ  P+, define a homomorphism of monoids \(g_w:\Gamma \to \mathcal B_q({\mathfrak {g}})\);

     
  2. (b)
    if ℓ(ww) = ℓ(w) + ℓ(w), then we have
    $$\displaystyle \begin{aligned}v_{w^{\prime}\mu}\cdot v_{w^{\prime} w\lambda}=q^{(w\lambda-\lambda,\mu)}v_{w^{\prime} w\lambda}\cdot v_{w^{\prime}\mu}; \end{aligned}$$
     
  3. (c)

    if ℓ(siw) = ℓ(w) − 1, i  I, then \(v_{w\lambda } x_i= q^{(w\lambda ,\alpha _i)}x_i v_{w\lambda }\) for all λ  P+;

     
  4. (d)

    if \({\mathfrak {g}}\) is finite dimensional, then \(v_{w_\circ \lambda } x= q^{-(w_\circ \lambda ,|x|)} x v_{w_\circ \lambda }\) for all \(x\in \mathcal A_q({\mathfrak {g}})\) homogeneous.

     

Proof

To prove (a) we use induction on (w), the induction base being trivial. For the inductive step, suppose that (siw) = (w) + 1. Then by Lemma 3.1(b) and the induction hypothesis Using (6.8) and observing that \(F_i^{(r)}(v_{w\lambda })F_i^{(s)}(v_{w\mu })=0\) if \(r>w\lambda (\alpha _i^\vee )\) or \(s>w\mu (\alpha _i^\vee )\) we obtain by Lemma 3.1(b)
$$\displaystyle \begin{aligned}\displaystyle v_{s_iw(\lambda+\mu)}=\sum_{r+t=w(\lambda+\mu)(\alpha_i^\vee)} q^{\frac 12(rw\mu-tw\lambda,\alpha_i)} F_i^{(r)}(v_{w\lambda})\cdot F_i^{(t)}(v_{w\mu})\\\displaystyle = F_i^{(w\lambda(\alpha_i^\vee))}(v_{w\lambda})\cdot F_i^{(w\mu(\alpha_i^\vee))}(v_{w\mu})=v_{s_i w\lambda}\cdot v_{s_i w\mu}. \end{aligned} $$
Part (b) was established in [6, Lemma 6.4]. To prove part (c), note that if (siw) = (w) − 1 then Fi(v) = 0. Then \(x_i K_{\frac 12\alpha _i}(v_{w\lambda })-K_{-\frac 12\alpha _i}(v_{w\lambda })x_i=0\), whence \(x_i v_{w\lambda }=q^{-(\alpha _i,w\lambda )} v_{w\lambda } x_i=q^{(w\lambda ,|x_i|)} v_{w\lambda }x_i\). In particular, applying part (c) with w = w we obtain, using an obvious induction on − ρ(|x|),
$$\displaystyle \begin{aligned} v_{w_\circ\lambda} x =q^{-(w_\circ\lambda,|x|)} x v_{w_\circ\lambda}, \end{aligned}$$
which yields part (d). □
Following [6, §6.1], define generalized quantum minors \(\Delta _{w\lambda }\in \mathcal A_q({\mathfrak {g}})\), w ∈ W, λ ∈ P+ by Δ := j(v). In particular,
$$\displaystyle \begin{aligned} v_{w\lambda}=q^{\frac 12(w\lambda-\lambda,\lambda)}\Delta_{w\lambda} v_\lambda. \end{aligned} $$
(6.13)

We list some properties of generalized quantum minors which will be used in the sequel.

Lemma 6.20

Let w, w W, λ, μ  P+. Then
  1. (a)

    \(\Delta _{w\lambda }\cdot \Delta _{w\mu }=q^{\frac 12 (w\mu -w^{-1}\mu ,\lambda )}\Delta _{w(\lambda +\mu )}\) ;

     
  2. (b)

    \(\Delta _{w\mu }\cdot \Delta _{w w^{\prime }\lambda } =q^{(w\mu -\mu ,ww^{\prime }\lambda +\lambda )}\Delta _{w w^{\prime }\lambda }\cdot \Delta _{w\mu }\) ;

     
  3. (c)

    If \({\mathfrak {g}}\) is finite-dimensional reductive, then \(\Delta _{w_\circ \lambda }\cdot \Delta _{w_\circ \mu }=\Delta _{w_\circ (\lambda +\mu )}\) and \( \Delta _{w_\circ \lambda }x=q^{-(w_\circ \lambda +\lambda ,|x|)} x\Delta _{w_\circ \lambda }\) for any \(x\in \mathcal A_q({\mathfrak {g}})\) homogeneous.

     

Proof

Parts (a) and (b) follow immediately from Lemma 6.19(a) and (b), respectively, by applying Lemma 6.9(b). The first assertion of part (c) is a special case of (a). Finally, using (6.6), (6.13) and Lemma 6.19(d) we can write
$$\displaystyle \begin{aligned} q^{(\lambda,|x|)}\Delta_{w_\circ\lambda}x v_\lambda=\Delta_{w_\circ\lambda}v_\lambda x= q^{-(w_\circ\lambda,|x|)}x \Delta_{w_\circ\lambda} v_\lambda. \end{aligned}$$
It remains to apply j and use the fact that \(\mathbf j|{ }_{\mathcal A_q({\mathfrak {g}})v_\lambda }\) is injective. □

Let \(\mathcal S_{w}=\{ \Delta _{w\lambda }\,:\, \lambda \in P^+\}\). It follows from Lemma 6.20(c) that \(\mathcal S_{w_\circ }\) is an abelian submonoid of \(\mathcal A_q({\mathfrak {g}})\) and in fact is an Ore submonoid with \(\Sigma _{\Delta _{w_\circ \lambda }}(x_i)=q^{(\lambda ,\alpha _{i^\star }-\alpha _i)} x_i\) for λ ∈ P+, i ∈ I.

Define \(\widehat {\mathcal B}_q({\mathfrak {g}}):=\mathcal B_q({\mathfrak {g}})[\mathcal S_{w_\circ }^{-1}]\) and let \(\widehat {\mathcal A}_q({\mathfrak {g}})\) be the subalgebra of \(\widehat {\mathcal B}_q({\mathfrak {g}})\) generated by \(\mathcal A_q({\mathfrak {g}})\), as a subalgebra, and the \(\Delta _{w_\circ \lambda }^{-1}\), λ ∈ P+. Clearly, \(\widehat {\mathcal A}_q({\mathfrak {g}})\) is isomorphic to \(\mathcal A_q({\mathfrak {g}})[\mathcal S_{w_\circ }^{-1}]\). The following is the main result of Sect. 6.

Theorem 6.21

Let \({\mathfrak {g}}\) be finite dimensional. Then
  1. (a)

    the assignments \(x_i\mapsto q_i^{\frac 12(\delta _{i,i^\star }-1)} E_{i^\star }(\Delta _{w_\circ \omega _i})\Delta _{w_\circ \omega _{i}}^{-1}\), \(v_\lambda \mapsto v_{w_\circ \lambda }\), λ  P+, define an injective algebra homomorphism \(\widehat \sigma :\mathcal B_q({\mathfrak {g}})\to \widehat {\mathcal B}_q({\mathfrak {g}})^{op}\);

     
  2. (b)

    \(\widehat \sigma (V_\lambda )=V_\lambda \) and \(\widehat \sigma |{ }_{V_\lambda }=\sigma ^I_{V_\lambda }\). In particular, the restriction of \(\widehat \sigma \) to \(\mathcal C_q({\mathfrak {g}})\) is an anti-involution on \(\mathcal C_q({\mathfrak {g}})\).

     

Proof

The first step is to construct a homomorphism of algebras \(\sigma _0:\mathcal A_q({\mathfrak {g}})\to \widehat {\mathcal A}_q({\mathfrak {g}})^{op}\).

Proposition 6.22

The assignments
$$\displaystyle \begin{aligned}x_i\mapsto q_i^{-\frac 12(1-\delta_{i,i^\star})} E_{i^\star}(\Delta_{w_\circ\omega_i})\Delta_{w_\circ\omega_i}^{-1}=q_i^{\frac 12(1-\delta_{i,i^\star})}\Delta_{w_\circ\omega_i}^{-1}E_{i^\star}(\Delta_{w_\circ\omega_i}), \quad i\in I, \end{aligned}$$

define a homomorphism \(\sigma _0:\mathcal A_q({\mathfrak {g}})\to \widehat {\mathcal A}_q({\mathfrak {g}})^{op}\) such that \(\sigma _0(\mathcal A_q({\mathfrak {g}})(-\gamma )) \subset \widehat {\mathcal A}_q({\mathfrak {g}})(-w_\circ \gamma )\), γ  Q+.

Proof Let δ be the unique involution of \(\mathcal A_q({\mathfrak {g}})\) defined by \(\delta (x_i)=x_{i^\star }\), i ∈ I. Then \(\kappa :\mathcal A_q(\mathfrak {g})\to \mathcal A_q(\mathfrak {g})\) defined by κ(x) = δ(x) = (δ(x)), \(x\in \mathcal A_q(\mathfrak {g})\) is an anti-involution. We need the following

Lemma 6.23

For any λ  P+, \(\kappa (\Delta _{w_\circ \lambda })=\epsilon _\lambda \Delta _{w_\circ \lambda }\) where 𝜖λ ∈{±1}.

Remark 6.24

Later we will show that 𝜖λ = 1. However, for the purposes of proving Proposition 6.22 this is irrelevant.

Proof Define
$$\displaystyle \begin{aligned} \mathcal A_q({\mathfrak{g}})^\lambda:=\{ x\in \mathcal A_q({\mathfrak{g}})_{w_\circ\lambda-\lambda}\,:\, (E_i^*)^{\lambda(\alpha_i^\vee)+1}(x)=0,\,\forall\,i\in I\}. \end{aligned}$$
It follows from the definition and Lemma 6.9 that In particular, \(E_i^{1-w_\circ \lambda (\alpha _i^\vee )}(\mathcal A_q({\mathfrak {g}})^\lambda )=0\) for all i ∈ I. Since \(\kappa (E_i(x))=E_{i^\star }^*(\kappa (x))\), it follows that \(\kappa (\Delta _{w_\circ \lambda })\in \mathcal A_q({\mathfrak {g}})^\lambda \) and so is a multiple of \(\Delta _{w_\circ \lambda }\). Since κ is an involution, the assertion follows. □
It follows from Lemmata 6.4 and 6.23 that κ lifts to an anti-involution \(\widehat \kappa \) on \(\widehat {\mathcal A}_q({\mathfrak {g}})\). By [27, Theorem 5.4], for any Open image in new window the assignments
$$\displaystyle \begin{aligned}x_i\mapsto c_i E_{i}^*(\Delta_{w_\circ\omega_i})\Delta_{w_\circ\omega_i}^{-1}=c_i q_i^{\delta_{i,i^\star}-1}\Delta_{w_\circ\omega_i}^{-1}E_{i}^*(\Delta_{w_\circ\omega_i}), \quad i\in I, \end{aligned}$$
define a homomorphism of algebras \(\zeta _{\mathbf c}:\mathcal A_q({\mathfrak {g}})\to \widehat {\mathcal A}_q({\mathfrak {g}})^{op}\). Let \(\mathbf c_0=(q_i^{\frac 12(1-\delta _{i,i^\star })})_{i\in I}\) and set \(\sigma _0:=\widehat \kappa \circ \zeta _{\mathbf c_0}\). Since \(\widehat \kappa \) is an anti-involution, we have
$$\displaystyle \begin{aligned} \sigma_0(x_i)=q_i^{\frac 12(1-\delta_{i,i^\star})} (\kappa(\Delta_{w_\circ\omega_i}))^{-1}\kappa(E_{i}^*(\Delta_{w_\circ\omega_{i}})) =q_i^{\frac 12(1-\delta_{i,i^\star})} \Delta_{w_\circ\omega_i}^{-1}E_{i^\star}(\Delta_{w_\circ\omega_{i}}). \end{aligned}$$
Thus, σ0 is the desired homomorphism \(\mathcal A_q({\mathfrak {g}})\to \widehat {\mathcal A}_q({\mathfrak {g}})^{op}\). Since \(|\sigma _0(x_i)|=\alpha _{i^\star }=-w_\circ \alpha _i\), it follows that |σ0(x)| = w|x| for all \(x\in \mathcal A_q({\mathfrak {g}})\) homogeneous. □
Now we have all the necessary ingredients to prove Theorem 6.21(a). We apply Lemma 6.1 with \(R=\mathcal A_q({\mathfrak {g}})\), \(\widehat R=\widehat {\mathcal B}_q({\mathfrak {g}})^{op}\), f = σ0 and \(g=g_{w_\circ }\) viewed as a homomorphism \(\Gamma \to \widehat R\) since Γ is abelian. Take \(x\in \mathcal A_q({\mathfrak {g}})\) homogeneous. Then the following holds in \(\widehat {\mathcal B}_q({\mathfrak {g}})\)

which is (6.1) in \(\widehat R\). Then by Lemma 6.1, \(\widehat \sigma :\mathcal B_q({\mathfrak {g}})\to \widehat {\mathcal B}_q({\mathfrak {g}})^{op}\), \(v_\lambda x\mapsto \sigma _0(x)v_{w_\circ \lambda }\), \(x\in \mathcal A_q({\mathfrak {g}})\), λ ∈ P+, is a well-defined homomorphism of algebras. Part (a) of Theorem 6.21 is proven.

Note that the Kλ, \(\lambda \in \frac 12 P\), satisfy the assumptions of Lemma 6.4 and so can be lifted to automorphisms \(\widehat K_{\lambda }\) of \(\widehat {\mathcal B}_q({\mathfrak {g}})\). Define
$$\displaystyle \begin{aligned} \widehat F_i(x)=\frac{ x_i \widehat K_{\frac 12\alpha_i}(x)-\widehat K_{-\frac 12\alpha_i}(x) x_i}{q_i-q_i^{-1}},\qquad \widehat E_i(x)=\frac{ \widehat K_{-\frac 12\alpha_i}(x) z_i-z_i\widehat K_{\frac 12\alpha_i}(x)}{q_i-q_i^{-1}}, \end{aligned} $$
(6.14)
where \(z_i=\widehat \sigma (x_{i^\star })=q_i^{\frac 12(\delta _{i,i^\star }-1)} E_i(\Delta _{w_\circ \omega _{i^\star }})\Delta _{w_\circ \omega _{i^\star }}^{-1}\).

Proposition 6.25

We have for all \(\lambda \in \frac 12 P\), i  I:
  1. (a)

    \(\widehat K_\lambda |{ }_{\mathcal B_q({\mathfrak {g}})}=K_\lambda \) , \(\widehat F_i|{ }_{\mathcal B_q({\mathfrak {g}})}=F_i\) and \(\widehat E_i|{ }_{\mathcal B_q({\mathfrak {g}})}=E_i\) ;

     
  2. (b)

    \(\widehat K_\lambda \circ \widehat \sigma =\widehat \sigma \circ K_{w_\circ \lambda }\), \(\widehat F_i\circ \widehat \sigma =\widehat \sigma \circ E_{i^\star }\) and \(\widehat E_i\circ \widehat \sigma = \widehat \sigma \circ F_{i^\star }\).

     

Proof The first and the second assertions in part (a) are obvious. Furthermore, since \(\widehat F_i=D^-_{(q_i-q_i^{-1})^{-1}x_i}\) and \(\widehat E_i=D^+_{(q_i-q_i^{-1})^{-1}z_i}\) with \(L_\pm =\widehat K_{\pm \frac 12\alpha _i}\) in the notation of (6.5), we immediately obtain the following

Lemma 6.26

\(\widehat F_i\) and \(\widehat E_i\), i  I are \(\widehat K_{\frac 12\alpha _i}\)-derivations of \(\widehat {\mathcal B}_q({\mathfrak {g}})\).

Thus, by Lemma 6.5 the last assertion in part (a) is equivalent to
$$\displaystyle \begin{aligned} \widehat E_i(v_\lambda)=0,\qquad \widehat E_i(x_j)=\delta_{i,j},\qquad \lambda\in P^+,\,i,j\in I. \end{aligned}$$
Since |zi| = αi and \(z_i\in \widehat {\mathcal A}_q({\mathfrak {g}})\), we have
$$\displaystyle \begin{aligned} K_{-\frac 12\alpha_i}(v_\lambda)z_i-z_i K_{\frac 12\alpha_i}(v_\lambda)=q^{-\frac 12(\alpha_i,\lambda)} v_\lambda z_i-q^{\frac 12(\lambda,\alpha_i)}z_i v_\lambda =0. \end{aligned}$$
Thus, \(\widehat E_i(v_\lambda )=0\) for all λ ∈ P+. We need the following

Lemma 6.27

The following identity holds in \(\mathcal A_q({\mathfrak {g}})\) for all i, j  I
$$\displaystyle \begin{aligned} &q_i^{\frac 12(\delta_{i,i^\star}-1)}(q^{\frac 12(\alpha_i,\alpha_j)}x_j E_i(\Delta_{w_\circ\omega_{i^\star}})-q^{-\frac 12(\alpha_i,\alpha_j)} q_j^{\delta_{i,j}-\delta_{i^\star,j}} E_i(\Delta_{w_\circ\omega_{i^\star}}) x_j)\\ &\quad =\delta_{i,j}(q_i-q_i^{-1})\Delta_{w_\circ\omega_{i^\star}}. \end{aligned} $$
Proof This results from a straightforward computation by applying Ei to the identity
$$\displaystyle \begin{aligned}x_j\Delta_{w_\circ\omega_{i^\star}}= q_j^{\delta_{i,j}-\delta_{i^\star,j}}\Delta_{w_\circ\omega_{i^\star}}x_j\end{aligned}$$
which is a special case of Lemma 6.20(c). □
Since \(\Delta _{w_\circ \omega _{i^\star }}^{-1} x_j \Delta _{w_\circ \omega _{i^\star }}=q^{-(w_\circ \omega _{i^\star }+\omega _{i^\star },\alpha _j)}x_j =q_j^{\delta _{i,j}-\delta _{i^\star ,j}}x_j\), we can write
$$\displaystyle \begin{aligned}\displaystyle (q_i-q_i^{-1})\widehat E_i(x_j)\Delta_{w_\circ \omega_{i^\star}}\\\displaystyle =q_i^{\frac 12(\delta_{i,i^\star}-1)} (q^{\frac 12(\alpha_i,\alpha_j)}x_j E_i(\Delta_{w_\circ\omega_{i^\star}})-q^{-\frac 12(\alpha_i,\alpha_j)} q_j^{\delta_{i,j}-\delta_{i^\star,j}} E_i(\Delta_{w_\circ\omega_{i^\star}}) x_j). \end{aligned} $$
Using Lemma 6.27 we conclude that \((q_i-q_i^{-1})\widehat E_i(x_j)\Delta _{w_\circ \omega _{i^\star }}=\delta _{i,j}(q_i-q_i^{-1})\Delta _{w_\circ \omega _{i^\star }}\) and so \(\widehat E_i(x_j)=\delta _{i,j}\). Part (a) of Proposition 6.25 is proven.
The first assertion in Proposition 6.25(b) is immediate since \(|\widehat \sigma (x)|=w_\circ |x|\) for \(x\in \mathcal B_q({\mathfrak {g}})\) homogeneous. Furthermore, by (6.14) we obtain for all \(x\in \mathcal B_q({\mathfrak {g}})\).
$$\displaystyle \begin{aligned}\displaystyle \widehat\sigma(F_i(x))=\frac{ \widehat\sigma(x_i \widehat K_{\frac 12\alpha_i}(x))-\widehat\sigma(\widehat K_{-\frac 12\alpha_i}(x)x_i)}{q_i-q_i^{-1}} =\frac{ \widehat K_{-\frac 12\alpha_{i^\star}}(\widehat\sigma(x)) z_{i^\star}-z_{i^\star} \widehat K_{\frac 12\alpha_{i^\star}}(\widehat\sigma(x))}{q_i-q_i^{-1}} \\\displaystyle =\widehat E_{i^\star}(\widehat\sigma(x)). \end{aligned} $$
It remains to prove that \(\widehat \sigma (E_i(x))=\widehat F_{i^\star }(\widehat \sigma (x))\) for all \(x\in \mathcal B_q({\mathfrak {g}})\). Let \(D_i=\widehat \sigma \circ E_i-\widehat F_{i^\star }\circ \widehat \sigma \). Since \(\widehat \sigma \circ K_{\pm \frac 12\alpha _i}= \widehat K_{\mp \frac 12\alpha _{i^\star }}\circ \widehat \sigma \) it follows that Di is a \(\widehat K_{\frac 12\alpha _{i^\star }}\circ \widehat \sigma \)-derivation from \(\mathcal B_q({\mathfrak {g}})\) to \(\widehat {\mathcal B}_q({\mathfrak {g}})^{op}\). We have
$$\displaystyle \begin{aligned} D_i(v_\lambda)=\widehat\sigma(E_i(v_\lambda))-\widehat F_{i^\star}(v_{w_\circ\lambda})=0, \end{aligned}$$
By Proposition 6.25(a) we have
$$\displaystyle \begin{aligned} \delta_{i,j}=\widehat E_i(x_j)=\frac{ q^{\frac 12(\alpha_i,\alpha_j)}x_j z_i-q^{-\frac 12(\alpha_i,\alpha_j)}z_i x_j}{q_i-q_i^{-1}}= \frac{q_j-q_j^{-1}}{q_i-q_i^{-1}}\,\widehat F_j(z_i) \end{aligned} $$
(6.15)
and so
$$\displaystyle \begin{aligned} D_i(x_j)=\widehat\sigma(E_j(x_i))-\widehat F_{i^\star}(z_{j^\star})=\delta_{i,j}-\delta_{i^\star,j^\star}=0. \end{aligned}$$
Thus Di = 0 on generators of \(\mathcal B_q({\mathfrak {g}})\). Then Di = 0 by Lemma 6.5. This completes the proof of Proposition 6.25(b). □

To prove part (b) of Theorem 6.21, we need to show that \(\widehat \sigma (V_\lambda )\subset V_\lambda \). The following lemma results from Proposition 6.25(a) by an obvious induction.

Lemma 6.28

For any \(b\in \mathcal B_q({\mathfrak {g}})\), r ≥ 1, (i1, …, ir) ∈ Ir, \(\widehat E_{i_1}\cdots \widehat E_{i_r}(b)=E_{i_1}\cdots E_{i_r}(b)\in \mathcal B_q({\mathfrak {g}})\). In particular, for any v  Vλ, λ  P+ we have \(\widehat E_{i_1}\cdots \widehat E_{i_r}(v)\in V_\lambda \).

Since Vλ is spanned by the \(F_{i_1}\cdots F_{i_r}(v_\lambda )\), r ≥ 0, (i1, …, ir) ∈ Ir, it suffices to show that \(\widehat \sigma ( F_{i_1}\cdots F_{i_r}(v_\lambda ))\in V_\lambda \). We have by Proposition 6.25(b)
$$\displaystyle \begin{aligned} \widehat \sigma( F_{i_1}\cdots F_{i_r}(v_\lambda))=\widehat E_{i_1^\star}\cdots \widehat E_{i_r^\star}(v_{w_\circ\lambda})\in V_\lambda\end{aligned} $$
by Lemma 6.28 applied with \(v=v_{w_\circ \lambda }\).

Consider the operator \(\sigma ^I_{V_\lambda }\circ \widehat \sigma \). Clearly, it maps vλ to itself and commutes with the \(U_q({\mathfrak {g}})\)-action by Proposition 6.25(b). Since Vλ is a simple \(U_q({\mathfrak {g}})\)-module generated by vλ, it follows that \(\widehat \sigma |{ }_{V_\lambda }= (\sigma ^I_{V_\lambda })^{-1}=\sigma ^I_{V_\lambda }\). In particular, \(\widehat \sigma \) is an involution on each Vλ and hence an anti-involution on \(\mathcal C_q(\lambda )\). □

Corollary 6.29

Let \({\mathfrak {g}}\) be finite-dimensional reductive. Then σI is an anti-involution on the algebra \(\mathcal C_q({\mathfrak {g}})\).

6.4 σI on Upper Global Crystal Basis

Denote Bup the dual canonical basis in \(\mathcal A_q({\mathfrak {g}})\) and denote Bλ the upper global crystal basis of Vλ. Let \(\mathbf B=\bigsqcup _{\lambda \in P^+} \mathbf B_\lambda \) be the upper global crystal basis of \(\mathcal C_q({\mathfrak {g}})\).

Theorem 6.30

For any finite dimensional reductive \({\mathfrak {g}}\) we have \(\widehat \sigma (\mathbf B)=\mathbf B\). In particular, \(\sigma ^I_{V_\lambda }(\mathbf B_\lambda )=\mathbf B_\lambda \).

Proof

We need the following □

Lemma 6.31 (See e.g. [27, Proposition 2.33])

For any λ  P+, j(Bλ) ⊂Bup.

In particular, since \(v_{w_\circ \lambda }\in \mathbf B_\lambda \), it follows that \(\Delta _{w_\circ \lambda }\in \mathbf B^{up}\).

Denote
$$\displaystyle \begin{aligned} \widehat{\mathbf B^{up}}&=\{ q^{\frac 12(w_\circ\lambda+\lambda,|b|)}b\Delta_{w_\circ\lambda}^{-1}\,:\, b\in\mathbf B^{up},\,\lambda\in P^+\}\\ &=\{ q^{-\frac 12(w_\circ\lambda+\lambda,|b|)}\Delta_{w_\circ\lambda}^{-1}b\,:\, b\in\mathbf B^{up},\,\lambda\in P^+\}. \end{aligned} $$
We need the following

Lemma 6.32

In the notation of Proposition 6.22, \(\sigma _0(\mathbf B^{up})\subset \widehat {\mathbf B^{up}}\).

Proof

Note that κ(Bup) = Bup since it is a composition of two involutions preserving Bup. In particular, \(\kappa (\Delta _{w_\circ \lambda })=\Delta _{w_\circ \lambda }\) for all λ ∈ P+.

Let b ∈Bup, λ ∈ P+. We have
$$\displaystyle \begin{aligned}\displaystyle \widehat\kappa( q^{\frac 12(w_\circ\lambda+\lambda,|b|)}b\Delta_{w_\circ\lambda}^{-1})= q^{\frac 12(w_\circ\lambda+\lambda,|b|)}(\kappa(\Delta_{w_\circ\lambda}))^{-1} \kappa(b) \\\displaystyle = q^{-\frac 12(w_\circ\lambda+\lambda,w_\circ|\kappa(b)|}\Delta_{w_\circ\lambda}^{-1}\kappa(b) \in \widehat{\mathbf B^{up}}. \end{aligned} $$
By [27, Theorem 5.4] we have \(\zeta _{\mathbf c_0}(\mathbf B^{up})\subset \widehat {\mathbf B^{up}}\) where \(\zeta _{\mathbf c_0}:\mathcal A_q({\mathfrak {g}})\to \widehat {\mathcal A}_q({\mathfrak {g}})\) is as in the proof of Proposition 6.22. Since \(\sigma _0=\widehat \kappa \circ \zeta _{\mathbf c_0}\), the assertion follows. □
Define
$$\displaystyle \begin{aligned} \widetilde{\mathbf B}=\{ q^{\frac 12(\lambda,|b|)} b v_{\lambda}\,:\, b\in\mathbf B^{up},\, \lambda\in P^+\}. \end{aligned}$$
It is immediate from the definition that \(\mathbf j(\widetilde {\mathbf B})=\mathbf B^{up}\) and that \(\widetilde {\mathbf B}\) is a basis in \(\mathcal B_q({\mathfrak {g}})\). Moreover, it follows from Lemma 6.31 that \(\mathbf B\subset \widetilde {\mathbf B}\) and
$$\displaystyle \begin{aligned}\mathbf B=\mathcal C_q({\mathfrak{g}})\cap \widetilde{\mathbf B}. \end{aligned} $$
(6.16)
Finally, define
$$\displaystyle \begin{aligned} \widehat{\mathbf B}=\{ q^{-\frac 12(w_\circ\lambda,|b|)} bv_{w_\circ\lambda}\,:\, b\in \widehat{\mathbf B^{up}},\, \lambda\in P^+\}\subset \widehat{\mathcal B}_q({\mathfrak{g}}). \end{aligned}$$

Proposition 6.33

\(\widehat {\mathbf B}=\{ q^{\frac 12(\lambda ,|b|)} b v_\lambda \,:\, b\in \widehat {\mathbf B^{up}},\,\lambda \in P^+\}\). In particular, \(\widehat {\mathbf B}\) is a basis of \(\widehat {\mathcal B}_q({\mathfrak {g}})\). Finally, \(\widetilde {\mathbf B}\subset \widehat {\mathbf B}\).

Proof

We need the following

Lemma 6.34

Let R be a Open image in new window -algebra and let S  R ∖{0} be a commutative Ore submonoid. Let B be a basis of R and suppose that \(\widehat B=\{ \tau _s(b)s^{-1}\,:\, b\in B,\, s\in S\}\) is a basis of R[S−1] where τs : R  R is some family of automorphisms satisfying \(\tau _{ss^{\prime }}=\tau _s\circ \tau _{s^{\prime }}\), \(\tau _s|{ }_S= \operatorname {\mathrm {id}}_S\). Then \(\hat \tau _s(\widehat B)s^{-1}=\widehat B\) for any s  S, where \(\hat \tau _s\) is the unique lifting of τs to R[S−1] provided by Lemma 6.4.

Proof Define fs : R[S−1] → R[S−1] by \(f_s(x)=\hat \tau _s(x)s^{-1}\), x ∈ R[S−1]. We claim that
$$\displaystyle \begin{aligned} f_s\circ f_{s^{\prime}}=f_{ss^{\prime}},\qquad s,s^{\prime}\in S \end{aligned} $$
(6.17)
and fs is invertible with \(f_s^{-1}(x)=\hat \tau _s^{-1}(x)s\). Indeed, for all x ∈ R[S−1] we have
$$\displaystyle \begin{aligned} f_s(f_{s^{\prime}}(x))=\hat\tau_s(\hat \tau_{s^{\prime}}(x)s^{\prime}{}^{-1})s^{-1} =\hat\tau_{ss^{\prime}}(x)s^{\prime}{}^{-1}s^{-1}=f_{ss^{\prime}}(x). \end{aligned}$$
and also \(f_s(\hat \tau _s^{-1}(x)s)=x\) and \(\hat \tau _{s^{-1}}(f_s(x))=x=f_s(\hat \tau _{s^{-1}}(x))\).
We have \(\widehat B=\{ f_s(b)\,:\, b\in B,\,s\in S\}\) and the assertion of the lemma is equivalent to \(f_s(\widehat B)=\widehat B\) for all s ∈ S. Clearly, (6.17) implies that \(f_s(\widehat B)\subset \widehat B\) for all s ∈ S. To prove the opposite inclusion, let \(\hat b\in \widehat B\). Write
$$\displaystyle \begin{aligned} f^{-1}_s(\hat b)=\sum_{\hat b^{\prime}\in \widehat B} \lambda_{\hat b^{\prime}} \hat b^{\prime}. \end{aligned}$$
Then
$$\displaystyle \begin{aligned} \hat b=\sum_{\hat b^{\prime}\in\widehat B} \lambda_{\hat b^{\prime}}f_s(\hat b^{\prime})=\sum_{\hat b^{\prime\prime}\in f_s(\widehat B)\subset \widehat B} \lambda_{f^{-1}_s(\hat b^{\prime\prime})} \hat b^{\prime\prime}, \end{aligned}$$
where we used that \(f_s(\widehat B)\subset \widehat B\). Since \(\widehat B\) is a basis, this implies that \(\lambda _{f^{-1}_s(\hat b^{\prime \prime })}=\delta _{\hat b,\hat b^{\prime \prime }}\) and so \(\hat b\in f_s(\widehat B)\). Therefore, \(\widehat B\subset f_s(\widehat B)\). □

Apply this lemma to \(R=\mathcal A_q({\mathfrak {g}})\), \(S=\mathcal S_{w_\circ }\), B = Bup and \(\widehat B=\widehat {\mathbf B^{up}}\). By [27, Proposition 3.9], \(\widehat {\mathbf B^{up}}\) is a basis of \(\widehat {\mathcal A}_q({\mathfrak {g}})\).1 We have \(\tau _{\Delta _{w_\circ \lambda }}(b)=q^{\frac 12(w_\circ \lambda +\lambda ,|b|)}b\). Then all assumptions of Lemma 6.34 are satisfied. Indeed, \(\tau _{\Delta _{w_\circ \lambda }}\tau _{\Delta _{w_\circ \mu }}(b)=q^{\frac 12(w_\circ (\lambda +\mu )+\lambda +\mu ,|b|)}b=\tau _{\Delta _{w_\circ (\lambda +\mu )}}(b)\) and \(\tau _{\Delta _{w_\circ \lambda }}(\Delta _{w_\circ \mu })=q^{\frac 12(w_\circ \lambda +\lambda ,w_\circ \mu -\mu )}\Delta _{w_\circ \mu }=\Delta _{w_\circ \mu }\), λ, μ ∈ P+. Thus by Lemma 6.34 we have, for any λ ∈ P+, \(\widehat {\mathbf B^{up}}=\hat \tau _{\Delta _{w_\circ \lambda }}^{-1}(\widehat {\mathbf B^{up}})\Delta _{w_\circ \lambda } =\{ q^{-\frac 12(w_\circ \lambda +\lambda ,|b|)}b\Delta _{w_\circ \lambda }\,:\, b\in \widehat {\mathbf B^{up}}\}\).

Since \(v_{w_\circ \lambda }=q^{\frac 12(w_\circ \lambda -\lambda ,\lambda )}\Delta _{w_\circ \lambda }v_\lambda \) we have
$$\displaystyle \begin{aligned} \widehat{\mathbf B}&=\{ q^{-\frac 12(w_\circ\lambda,|b|)}b v_{w_\circ\lambda}\,:\, b\in \widehat{\mathbf B^{up}},\,\lambda\in P^+\}\\ &=\{ q^{\frac 12(\lambda,|b|)} \tau_{\Delta_{w_\circ\lambda}}^{-1}(b) v_{w_\circ\lambda}\,:\, b\in \widehat{\mathbf B^{up}},\,\lambda\in P^+\}\\ &=\{ q^{\frac 12(\lambda,|b|+w_\circ\lambda-\lambda)}\tau^{-1}_{\Delta_{w_\circ\lambda}}(b)\Delta_{w_\circ\lambda}v_\lambda\,:\, b\in \widehat{\mathbf B^{up}},\,\lambda\in P^+\}\\&= \{ q^{\frac 12(\lambda,|b^{\prime}|)}b^{\prime}v_\lambda\,:\, b^{\prime}\in \widehat{\mathbf B^{up}},\,\lambda\in P^+\} \end{aligned} $$
where we denoted \(b^{\prime }=\tau _{\Delta _{w_\circ \lambda }}^{-1}(b)\Delta _{w_\circ \lambda }\) and observed that |b| = |b| + wλ − λ. This proves the first assertion of Proposition 6.33. The second and third assertions are now immediate. □
Now we can complete the proof of Theorem 6.30. It follows from Proposition 6.33 and Lemma 6.32 that for any b ∈Bup, λ ∈ P+ we have
$$\displaystyle \begin{aligned} \widehat\sigma(q^{\frac 12(\lambda,|b|)}b v_\lambda)=q^{\frac 12(\lambda,|b|)}v_{w_\circ\lambda}\sigma_0(b)=q^{-\frac 12(w_\circ\lambda,|\sigma_0(b)|)}\sigma_0(b)v_{w_\circ\lambda} \in \widehat{\mathbf B}. \end{aligned}$$
Thus, \(\widehat \sigma (\widetilde {\mathbf B})\subset \widehat {\mathbf B}\). Since \(\mathbf B\subset \widetilde {\mathbf B}\), it follows that \(\widehat \sigma (\mathbf B)\subset \widehat {\mathbf B}\). Then by Theorem 6.21(b) we conclude that \(\widehat \sigma (\mathbf B)\subset \widehat {\mathbf B}\cap \mathcal C_q({\mathfrak {g}})\). On the other hand, since \(\widehat {\mathbf B}\) is linearly independent by Proposition 6.33, its intersection with \(\mathcal C_q({\mathfrak {g}})\) is also linearly independent. Since \(\widetilde {\mathbf B}\subset \widehat {\mathbf B}\) by Proposition 6.33, it follows from (6.16) that \(\mathbf B=\widetilde {\mathbf B}\cap \mathcal C_q({\mathfrak {g}})\subset \widehat {\mathbf B}\cap \mathcal C_q({\mathfrak {g}})\). But B is a basis of \(\mathcal C_q({\mathfrak {g}})\) and so \(\mathbf B=\widehat {\mathbf B}\cap \mathcal C_q({\mathfrak {g}})\). Thus, \(\widehat \sigma (\mathbf B)\subset \mathbf B\). Since by Theorem 6.21(b) \(\widehat \sigma \) is an involution on \(\mathcal C_q({\mathfrak {g}})\), \(\widehat \sigma (\mathbf B)=\mathbf B\) which completes the proof of the first assertion of Theorem 6.30. The second assertion is immediate from the first and Theorem 6.21(b). □

6.5 Proof of Theorem 1.10

Proof

Let V  be any object in \(\mathscr O^{int}_q({\mathfrak {g}})\). Let (Lup, Bup) be an upper crystal basis of V  and let G(Bup) be the corresponding upper global crystal basis (see [25]). By [25, Theorem 3.3.1] there exists a direct sum decomposition V =∑jVj such that Bj := G(Bup) ∩ Vj is a basis of Vj and each \(V^j\cong V_{\lambda _j}\), λj ∈ P+. The latter isomorphism identifies Bj with \(\mathbf B_{\lambda _j}\). Since by Theorem 4.10, \(\sigma ^i_V\) is compatible with direct sum decompositions, the restriction of \(\sigma ^i_V\) to Vj coincides with \(\sigma ^I_{V_j}\) and under the above isomorphism it identifies with \(\sigma ^I_{V_{\lambda _j}}\) and thus preserves \(\mathbf B_{\lambda _j}\) by Theorem 6.30. □

7 Examples

7.1 Thin Modules

Let λ ∈ P+. We say that Vλ is quasi-miniscule if Vλ(β) ≠ 0 implies that β ∈  ∪{0}. For example, \(V_{\omega _i}\), i ∈ I for \({\mathfrak {g}}={\mathfrak {sl}}_n\) are (quasi)-miniscule, as well as the quantum analogue of the adjoint representation of \({\mathfrak {g}}\).

Lemma 7.1

Conjecture 1.2 holds for any quasi-miniscule V = Vλ.

Proof

Let v = v(λ) ∈ Vλ(λ). Then in the notation of (3.2) we have Open image in new window . As shown in Proposition 3.8, the action of W(V ) on the basis [v]W of Open image in new window is given by the Weyl group action on \(W/W_{J_\lambda }\). It remains to observe that \(\sigma ^i|{ }_{V_\lambda (0)}= \operatorname {\mathrm {id}}_{V_\lambda (0)}\), i ∈ I. □

This result can be extended to a larger class of modules. We say that \(V\in \mathscr O^{int}_q({\mathfrak {g}})\) is thin if \(\dim V(\beta )\le 1\) for all β ∈ P ∖{0}. By definition, every quasi-miniscule module is thin. Furthermore, all modules \(V_{m\omega _1}\), \(V_{m\omega _n}\), \(m\in {\mathbb Z}_{\ge 0}\) are thin for \({\mathfrak {g}}={\mathfrak {sl}}_{n+1}\).

Theorem 7.2

Conjecture 1.2 holds for thin modules.

Proof

Let (L, B) be an upper crystal basis of \(V\in \mathscr O^{int}_q({\mathfrak {g}})\) and let Gup(B) ⊂ V  be the corresponding global crystal basis. We say that b ∈ Gup(B) of weight β ∈ P is thin if either β = 0 or Open image in new window . Denote \(G_0^{up}(B)\) the set of thin elements in Gup(B). Clearly, V  is thin if and only if \(G_0^{up}(B)=G^{up}(B)\). We need the following

Proposition 7.3

For any b  Gup(B) ∩ V (β), β  P with \(\dim V(\beta )=1\) we have ΦV(b) ⊂ Gup(B).

Proof It suffices to prove that σJ(b) ∈ Gup(B) for all \(J\in \mathscr J\). Since \(\sigma ^J(V(\beta ))=V(w_\circ ^J\beta )\) and \(\dim V(w_\circ ^J\beta )= \dim V(\beta )=1\), it follows that σJ(b) = cb for some \(b^{\prime }\in G^{up}(B)\cap V(w_\circ ^J\beta )\). Let \( \underline b\) be the image of b in B under the quotient map LLqL. By Theorem 1.8, \(\tilde \sigma ^J( \underline b)= \underline b^{\prime }\) and so \(c\in 1+q \mathbb A\). It follows from Proposition 5.8 that \(\overline {c}=c\) and so c = 1. □

Let g : B → Gup(B) be Kashiwara’s bijection (cf. [25]) and let \(B_0=g^{-1}(G_0^{up}(B))\). Then by Theorem 1.8 and Proposition 7.3 we have \(g(\tilde \sigma ^i(b))=\sigma ^i(g(b))\) for all b ∈ B0. Since the action of \(\tilde \sigma ^i\) on B coincides with the action of W defined in [26], it follows from [26, Theorem 7.2.2] that W(V ) is a homomorphic image of W.

We may assume, without loss of generality, that V = Vλ and J(V ) = ∅. In view of Proposition 3.8(b), the action of W(V ) on the set [vλ]W, vλ ∈ Vλ(λ) is faithful and coincides with that of W. This implies that ψV from Theorem 1.1 is an isomorphism. □

7.2 Crystallizing Cactus Group Action for \({\mathfrak {g}}={\mathfrak {sl}}_3\)

We now describe combinatorial consequences of Theorem 1.8 for \({\mathfrak {g}}={\mathfrak {sl}}_3\). It turns out that the corresponding action of \( \operatorname {\mathrm {\mathsf {Cact}}}_{S_3}\) lifts to the ambient set
$$\displaystyle \begin{aligned} \widehat{\mathbf M}=\{ (m_1,m_2,m_{12},m_{21},m_{01},m_{02})\in{\mathbb Z}_{\ge 0}^2\times{\mathbb Z}^4\,:\, m_1m_2=0\} \end{aligned}$$
where the crystal basis for \(\mathcal C_q({\mathfrak {sl}}_3)\) identifies with \(\mathbf M=\widehat {\mathbf M}\cap {\mathbb Z}_{\ge 0}^6\).
We need some notation. Define \( \operatorname {\mathrm {wt}}_i:\widehat {\mathbf M}\to {\mathbb Z}\) by
$$\displaystyle \begin{aligned}\operatorname{\mathrm{wt}}_i({\mathbf{m}})=m_{0i}-m_i+m_j-m_{ij},\qquad \{i,j\}=\{1,2\}. \end{aligned}$$
Furthermore, define \(e_i^r:\widehat {\mathbf M}\to \widehat {\mathbf M}\), i ∈{1, 2}, \(r\in {\mathbb Z}\), by
$$\displaystyle \begin{aligned} e_i^r(m_1,m_2,m_{12},m_{21},m_{01},m_{02})=(m^{\prime}_1,m^{\prime}_2,m^{\prime}_{12},m^{\prime}_{21},m^{\prime}_{01},m^{\prime}_{02}) \end{aligned}$$
where, for \({\mathbf {m}}=(m_1,m_2,m_{12},m_{21},m_{01},m_{02})\in \widehat {\mathbf M}\) we set
$$\displaystyle \begin{gathered} m^{\prime}_i=[m_i-m_j-r]_+,\quad m^{\prime}_j=[m_j-m_i+r]_+,\quad m^{\prime}_{ji}=m_{ji},\quad m^{\prime}_{0j}=m_{0j}, \\ m^{\prime}_{ij}=m_{ij}+\min(m_i-r,m_j), \,\,\,\, m^{\prime}_{0i}=m_{0i}+r+\min(m_i-r,m_j),\,\,\,\,\,\, \{i,j\}=\{1,2\}, \end{gathered} $$

and \([x]_+:=\max (x,0)\), \(x\in {\mathbb Z}\). The following is well-known (cf. [5, Example 6.26]).

Lemma 7.4

The \(e_i^r\), i ∈{1, 2}, \(r\in {\mathbb Z}\) satisfy
$$\displaystyle \begin{aligned} e_1^r e_1^s=e_1^{r+s},\quad e_2^r e_2^s=e_2^{r+s},\quad e_1^r e_2^{r+s}e_1^s=e_2^s e_1^{r+s}e_2^r,\quad r,s\in{\mathbb Z}. \end{aligned} $$
(7.1)

In particular, \(e_i^0= \operatorname {\mathrm {id}}\) and \((e_i^r)^{-1}=e_i^{-r}\), \(r\in {\mathbb Z}\), i ∈{1, 2}.

Proof

Define a map \(\widehat {\mathbf k}:\widehat {\mathbf M}\to {\mathbb Z}^5\) by \(\widehat {\mathbf k}(m_1,m_2,m_{12},m_{21},m_{01},m_{02})\mapsto (a_1,a_2,a_3,l_1,l_2)\) where
$$\displaystyle \begin{aligned} a_1=m_1+m_{21},\quad a_2=m_2+m_{12}+m_{21},\quad a_3=m_{12},\quad l_i=m_i+m_{3-i,i}+m_{0i} \end{aligned}$$
with i ∈{1, 2}. It is easy to see that \(\widehat {\mathbf k}\) is a bijection with its inverse given by
$$\displaystyle \begin{aligned} (a_1,a_2,a_3,l_1,l_2)\mapsto (m_1,m_2,m_{12},m_{21},m_{01},m_{02}), \end{aligned}$$
where m1 = [a1 + a3a2]+, m2 = [a2a1a3]+, m12 = a3, \(m_{21}=\min (a_1,a_2-a_3)\), m01 = l1 − a1, \(m_{02}=l_2-a_2+\min (a_1,a_2-a_3)\).
The action of operators \(e_i^r\), i ∈{1, 2}, \(r\in {\mathbb Z}\) on \({\mathbb Z}^5\) induced by this bijection coincides with the action constructed in [5, Example 6.26]
$$\displaystyle \begin{aligned} &e_1^{r}(a_1,a_2,a_3,l_1,l_2)=(a_1+[\delta-r]_+-[\delta]_+,a_2,a_3+[\delta]_+-\max(\delta,r),l_1,l_2)\\ &e_2^r(a_1,a_2,a_3,l_1,l_2)=(a_1,a_2-r,a_3,l_1,l_2). \end{aligned} $$
where δ = a1 + a3 − a2. The identities from the Lemma are now easy to obtain by using tropicalized relations for the ei given after Definition 2.20 in [5] in the context of [5, Example 6.26]. □
Define \( \underline \sigma = \underline \sigma ^{\{1,2\}}:\widehat {\mathbf M}\to \widehat {\mathbf M}\) by
$$\displaystyle \begin{aligned} (m_1,m_2,m_{12},m_{21},m_{01},m_{02})\mapsto (m_1,m_2,m_{02},m_{01},m_{21},m_{12}). \end{aligned}$$
Clearly, \( \underline \sigma \) is an involution and \( \underline \sigma (\mathbf M)=\mathbf M\). Furthermore, define \( \underline \sigma ^i:\widehat {\mathbf M}\to \widehat {\mathbf M}\), i ∈{1, 2} by
$$\displaystyle \begin{aligned} \underline\sigma^i({\mathbf{m}})=e_i^{-\operatorname{\mathrm{wt}}_i({\mathbf{m}})}({\mathbf{m}}),\qquad {\mathbf{m}}\in\widehat{\mathbf M}. \end{aligned}$$

Proposition 7.5

The following identities hold in \(\operatorname {Bij}(\widehat {\mathbf M})\)
$$\displaystyle \begin{gathered} \underline\sigma^i\circ\underline\sigma^i=\operatorname{\mathrm{id}},\qquad \underline\sigma^i\circ e_i^r=e_i^{-r}\circ\underline\sigma^i,\qquad \underline\sigma\circ e_i^r=e_j^{-r}\circ\underline\sigma,\\ \underline\sigma^i\circ \underline\sigma^j\circ\underline\sigma^i=\underline\sigma^j\circ\underline\sigma^i\circ\underline\sigma^j, \qquad \underline\sigma^i\circ\underline\sigma=\underline\sigma\circ\underline\sigma^j, \end{gathered} $$

where {i, j} = {1, 2}. In particular, the assignments \(\tau _{i,i+1}\mapsto \underline \sigma ^i\), i ∈{1, 2}, \(\tau _{13}\mapsto \underline \sigma \) define an action of \( \operatorname {\mathrm {\mathsf {Cact}}}_{S_3}\) on \(\widehat {\mathbf M}\).

Proof

Since \( \operatorname {\mathrm {wt}}_i(e_i^r({\mathbf {m}}))= \operatorname {\mathrm {wt}}_i({\mathbf {m}})+2r\) for any \({\mathbf {m}}\in \widehat {\mathbf M}\), we have
$$\displaystyle \begin{aligned} \underline\sigma^i\circ\underline\sigma^i({\mathbf{m}})=e_i^{-(\operatorname{\mathrm{wt}}_i({\mathbf{m}})-2\operatorname{\mathrm{wt}}_i({\mathbf{m}}))-\operatorname{\mathrm{wt}}_i({\mathbf{m}})}({\mathbf{m}})={\mathbf{m}}, \end{aligned}$$
while
$$\displaystyle \begin{aligned} \underline\sigma^i\circ e_i^r({\mathbf{m}})=e_i^{-r-\operatorname{\mathrm{wt}}_i({\mathbf{m}})}({\mathbf{m}})=e_i^{-r}\circ\underline \sigma^i({\mathbf{m}}). \end{aligned}$$
To prove the third identity, note that \(e_i^r( \underline \sigma ({\mathbf {m}}))=(\tilde m_1,\tilde m_2,\tilde m_{12},\tilde m_{21}, \tilde m_{01},\tilde m_{02})\), where
$$\displaystyle \begin{gathered} \tilde m_i=[m_i-m_j-r]_+,\quad \tilde m_j=[m_j-m_i+r]_+,\quad \tilde m_{ji}=m_{0i},\quad \tilde m_{0j}=m_{ij}, \\ \tilde m_{ij}=m_{0j}+\min(m_i-r,m_j), \,\,\,\, \tilde m_{0i}=m_{ji}+\min(m_i,m_j+r),\,\,\,\,\,\,\,\, \{i,j\}=\{1,2\}, \end{gathered} $$
which is easily seen to coincide with \( \underline \sigma (e_j^{-r}({\mathbf {m}}))\). The braid identity follows from the last relation in (7.1) (known as Verma relations) and the identity \( \operatorname {\mathrm {wt}}_j(e_i^r({\mathbf {m}}))= \operatorname {\mathrm {wt}}_j({\mathbf {m}})-r\), {i, j} = {1, 2}. Finally,
$$\displaystyle \begin{aligned} \underline\sigma\circ\sigma^j({\mathbf{m}})=\underline\sigma\circ e_j^{-\operatorname{\mathrm{wt}}_j({\mathbf{m}})}({\mathbf{m}})=e_i^{\operatorname{\mathrm{wt}}_j({\mathbf{m}})}(\underline\sigma({\mathbf{m}})) =e_i^{-\operatorname{\mathrm{wt}}_i(\underline\sigma({\mathbf{m}}))}(\underline\sigma({\mathbf{m}}))=\underline\sigma^i\circ \underline\sigma({\mathbf{m}}), \end{aligned}$$
where we used the identity \( \operatorname {\mathrm {wt}}_i( \underline \sigma ({\mathbf {m}}))=m_{ji}-m_i+m_j-m_{0j}=- \operatorname {\mathrm {wt}}_j({\mathbf {m}})\). □

Remark 7.6

It would be interesting to define analogues of \(\widehat {\mathbf M}\) for other \({\mathfrak {g}}\) and study the action of the corresponding cactus groups on \(\widehat {\mathbf M}\). We plan to study this in a subsequent publication via the approach of [5].

Given \(l_1,l_2\in {\mathbb Z}\) define
$$\displaystyle \begin{aligned} \widehat{\mathbf M}_{l_1,l_2}=&\{ (m_1,m_2,m_{12},m_{21},m_{01},m_{02})\in\widehat{\mathbf M}\\ &:\,m_{01}+m_1+m_{21}=l_1,\, m_{02}+m_2+m_{12}=l_2\}. \end{aligned} $$
Clearly, \( \underline \sigma \), \( \underline \sigma ^i\), \(e_i^r\), i ∈{1, 2}, \(r\in {\mathbb Z}\) preserve \(\widehat {\mathbf M}_{l_1,l_2}\) for any \(l_1,l_2\in {\mathbb Z}\). Set \(\mathbf M_{l_1,l_2}=\widehat {\mathbf M}_{l_1,l_2}\cap \mathbf M\).

In view of [5, Example 6.26], \(\widehat {\mathbf k}(\mathbf M_{l_1,l_2})\), where \(\widehat {\mathbf k}\) is defined in the proof of Lemma 7.4, identifies with the upper crystal basis \(B^{up}(V_{l_1\omega _1+l_2\omega _2})\) of \(V_{l_1\omega _1+l_2\omega _2}\). In particular, \(\widehat {\mathbf k}(\mathbf M)\) identifies with the upper crystal basis \(B^{up}(\mathcal C_2)=\bigsqcup _{\lambda \in P^+} B^{up}(V_\lambda )\) of \(\mathcal C_2=\mathcal C_q({\mathfrak {sl}}_3)\). We use this identification throughout the rest of this chapter.

Proposition 7.7

Under the above identification, the restrictions of \( \underline \sigma \), \( \underline \sigma ^i\), i ∈{1, 2} to M coincide with the action of \( \operatorname {\mathrm {\mathsf {Cact}}}_{S_3}\) on \(B^{up}(\mathcal C_2)\) provided by Theorem 1.8 with \({\mathfrak {g}}={\mathfrak {sl}}_3\) and \(V=\mathcal C_2\).

Proof

It follows from Corollary 5.6 applied to \(f= \underline \sigma \) extended to Bλ ∪{0}, and Proposition 7.5 that \( \underline \sigma \) coincides with \(\tilde \sigma ^{\{1,2\}}_{V_\lambda }\) for all λ ∈ P+. On the other hand, by Remark 5.7 we have \( \underline \sigma ^i=\tilde \sigma ^{\{i\}}_{V_\lambda }\), i ∈{1, 2}, λ ∈ P+. □

7.3 Gelfand–Kirillov Model for \({\mathfrak {g}}={\mathfrak {sl}}_3\)

Our goal here is to illustrate results and constructions from Sect. 6 for \({\mathfrak {g}}={\mathfrak {sl}}_3\) and provide some evidence for Conjecture 1.2. We freely use the notation from Sects. 6 and 7.2. In this case the algebra \(\mathcal A_2=\mathcal A_q({\mathfrak {g}})\) is generated by the xi, i ∈{1, 2} subject to the relations
$$\displaystyle \begin{aligned} x_i^2 x_j-(q+q^{-1})x_i x_j x_i+x_j x_i^2=0,\qquad \{i,j\}=\{1,2\}. \end{aligned} $$
(7.2)
Define
$$\displaystyle \begin{aligned} x_{ij}=\frac{q^{\frac 12} x_i x_j-q^{-\frac 12} x_j x_i}{q-q^{-1}},\qquad \{i,j\}=\{1,2\}. \end{aligned}$$
Then \(x_i x_j=q^{\frac 12} x_{ij}+q^{-\frac 12}x_{ji}\), {i, j} = {1, 2} and (7.2) is equivalent to xixij = qxijxi or xixji = q−1xjixi, {i, j} = {1, 2}. The following is well-known1 (see e.g. [7]).

Lemma 7.8

The dual canonical basis B up in the algebra \(\mathcal A_2\) is
$$\displaystyle \begin{aligned} \mathbf B^{up}=&\{ q^{\frac 12(m_1-m_2)(m_{21}-m_{12})}x_1^{m_1} x_2^{m_2} x_{12}^{m_{12}}x_{21}^{m_{21}}\\ &:\,(m_1,m_2,m_{12},m_{21})\in{\mathbb Z}_{\ge0}^4,\, m_1m_2=0\}. \end{aligned} $$
We have \(\widehat {\mathcal A_2}=\mathcal A_2[\mathcal S_{w_\circ }^{-1}]=\mathcal A_2[x_{12}^{-1},x_{21}^{-1}]\). It follows from Lemma 7.8 that
$$\displaystyle \begin{aligned} \widehat{\mathbf B^{up}}=&\{ q^{\frac 12(m_1-m_2)(m_{21}-m_{12})}x_1^{m_1} x_2^{m_2} x_{12}^{m_{12}}x_{21}^{m_{21}}\\&:\, (m_1,m_2,m_{12},m_{21})\in{\mathbb Z}_{\ge0}^2\times {\mathbb Z}^2,\, m_1m_2=0\}. \end{aligned} $$
The following is immediate

Lemma 7.9

  1. (a)
    The algebra \(\mathcal B_2:=\mathcal B_q({\mathfrak {g}})\) is generated by \(\mathcal A_2\) and Open image in new window , where \(v_i=v_{\omega _i}\) , as subalgebras subject to the relations
    $$\displaystyle \begin{aligned} v_i x_j=q^{-\delta_{i,j}}x_j v_i,\qquad i,j\in\{1,2\}. \end{aligned}$$
     
  2. (b)

    \(\mathcal B_2\) is a \(U_q({\mathfrak {g}})\)-module algebra with the \(U_q({\mathfrak {g}})\)-action defined in Lemma 6.7.

     
Abbreviate \(z_i=F_i(v_i)=v_{s_i\omega _i}\) and \(z_{ij}=F_iF_j(v_j)=v_{s_is_j\omega _j}\). Clearly,
$$\displaystyle \begin{aligned} z_i=q^{-\frac 12} x_i v_i,\qquad z_{ij}=q^{-\frac 12} x_{ij}v_j,\qquad \{i,j\}=\{1,2\}. \end{aligned} $$
(7.3)
The following Lemma is an immediate consequence of Lemma 7.9

Lemma 7.10

  1. (a)
    The algebra \(\mathcal C_2=\mathcal C_q({\mathfrak {sl}}_3)\) is generated by v 1 , v 2 , z 1 , z 2 , z 12 and z 21 subject to the relations
    $$\displaystyle \begin{aligned} v_1 v_2&=v_2 v_1,\quad v_i z_j=q^{-\delta_{i,j}} z_j v_i,\quad v_i z_{12}=q^{-1}z_{12} v_i,\quad v_i z_{21}=q^{-1}z_{21}v_i,\\ &\quad i,j\in\{1,2\}.\\ z_i z_j&=q v_i z_{ij}+q^{-1}z_{ji}v_j,\quad z_k z_{ij}=q^{-\delta_{j,k}}z_{ij}z_k,\quad \{i,j\}=\{1,2\},\, k\in \{1,2\} \end{aligned} $$

    and z12z21 = z21z12.

     
  2. (b)
    \(\mathcal C_2\) is a \(U_q({\mathfrak {g}})\) -module algebra via
     
  3. (c)

    The P-grading on \(\mathcal C_q({\mathfrak {g}})\) is given by |vi| = ωi, |zi| = siωi = ωi − αi, |zij| = sjsiωj = ωj − αi − αj, {i, j} = {1, 2}.

     

The following is an immediate corollary of Theorem 6.21.

Corollary 7.11

The assignments
$$\displaystyle \begin{aligned} v_i\mapsto z_{ji},\quad z_i\mapsto z_i,\quad z_{ij}\mapsto v_j,\qquad \{i,j\}=\{1,2\}. \end{aligned}$$

define an anti-involution of \(\mathcal C_2\) which coincides with \(\sigma =\sigma ^{\{1,2\}}_{\mathcal C_2}\).

Given \({\mathbf {m}}\in \widehat {\mathbf M}\), define \(\mathbf b_{{\mathbf {m}}}\in \mathcal C_2\) as
$$\displaystyle \begin{aligned} \mathbf b_{{\mathbf{m}}}= q^{\frac 12(m_1(m_{21}-m_{01})+m_2(m_{12}-m_{02})-(m_{12}+m_{21})(m_{01}+m_{02}))} z_1^{m_1} z_2^{m_2} z_{12}^{m_{12}}z_{21}^{m_{21}} v_1^{m_{01}}v_2^{m_{02}} \end{aligned} $$
(7.4)
if m ∈M and bm = 0 if \({\mathbf {m}}\in \widehat {\mathbf M}\setminus \mathbf M\). Thus, \(|\mathbf b_{{\mathbf {m}}}|=(m_{01}+m_1+m_{21})\omega _1+(m_{02}+m_2+m_{12})\omega _2-(m_1+m_{12}+m_{21})\alpha _1-(m_2+m_{12}+m_{21})\alpha _2 = \operatorname {\mathrm {wt}}_1({\mathbf {m}})\omega _1+ \operatorname {\mathrm {wt}}_2({\mathbf {m}})\omega _2\), m ∈ M.

The following is a consequence of Lemma 7.10, (6.16) and Proposition 6.33

Lemma 7.12

The upper global crystal basis B of \(\mathcal C_2\) is B = {bm  :  m ∈M}. Furthermore, for each λ = l1ω1 + l2ω2 ∈ P+ we have \(\mathbf B_\lambda =\{ \mathbf b_{{\mathbf {m}}}\,:\, {\mathbf {m}}\in \mathbf M_{l_1,l_2}\}\). Moreover, under the identification of M with the upper crystal basis of \(\mathcal C_q({\mathfrak {g}})\) (cf. Sect.7.2), the map M →B defined by mbm, m ∈ M is Kashiwara’s bijection G [25] between an upper crystal basis of \(\mathcal C_2\) and its upper global crystal basis.

The following is an explicit form of Theorem 6.30 for \({\mathfrak {g}}={\mathfrak {sl}}_3\) and is immediate from (7.4) and Corollary 7.11.

Lemma 7.13

We have \( \sigma (\mathbf b_{{\mathbf {m}}})=\mathbf b_{ \underline \sigma ({\mathbf {m}})}\) for all m ∈M.

Remark 7.14

It is easy to check that
$$\displaystyle \begin{aligned} |\mathbf B_\lambda(0)|&=|\{{\mathbf{m}}\in \mathbf M_{l_1,l_2}\,:\, \operatorname{\mathrm{wt}}_1({\mathbf{m}})\\&=\operatorname{\mathrm{wt}}_2({\mathbf{m}})=0\}|= \begin{cases} \min(l_1,l_2)+1,& l_1\equiv l_2\pmod 3\\ 0,&\text{otherwise}. \end{cases} \end{aligned} $$
It follows from the definition of \( \underline \sigma \) that σ is trivial on Bλ(0) if and only if \(\dim V_\lambda (0)=1\) (that is, if and only if \(\min (l_1,l_2)=0\) and \(\max (l_1,l_2)\in 3{\mathbb Z}_{>0}\)). Thus, \(\tau _{1,3}\notin \mathsf K_{{\mathfrak {sl}}_3}\) in the notation introduced after Problem 1.7. On the other hand, it is immediate from the definitions that the σi, i ∈{1, 2} act trivially on Vλ(0) for any \(V\in \mathscr O^{int}_q({\mathfrak {g}})\). In particular, σ is not contained in W(Vλ) if \(\dim V_\lambda (0)>1\).

In order to calculate σi, i ∈{1, 2} we need the following result.

Lemma 7.15

For any m ∈M, r ≥ 0 and i ∈{1, 2} we have
$$\displaystyle \begin{aligned} &E_i^{(r)}(\mathbf b_{{\mathbf{m}}})=\binom{m_i+m_{ij}}{r}_q \mathbf b_{e_i^r({\mathbf{m}})}+ \sum_{1\le t\le r} C^{(r)}_t(m_j+m_{ij},m_i+m_{ij}) \mathbf b_{e_i^{r}({\mathbf{m}})+t\mathbf a^+_i} \\ &F_i^{(r)}(\mathbf b_{{\mathbf{m}}})=\binom{m_j+m_{0i}}{r}_q \mathbf b_{e_i^{-r}({\mathbf{m}})}+ \sum_{1\le t\le r} C^{(r)}_t(m_i+m_{0i},m_j+m_{0i}) \mathbf b_{e_i^{-r}({\mathbf{m}})-t\mathbf a^+_i} \end{aligned} $$
where \(\mathbf a^+_1=(0,0,-1,1,-1,1)\) , \(\mathbf a^+_2=-\mathbf a^+_1\) and
$$\displaystyle \begin{aligned} C^{(r)}_t(c,d)=\begin{cases} \displaystyle \binom{c}{t}_q\binom{d-t}{r-t}_q,& d-c\ge r,\\ \displaystyle \binom{d-c}{t}_q \binom{d-t}{r}_q,& d-c<r, \end{cases} \end{aligned} $$
(7.5)

with the convention that \(\binom {k}{l}_q=0\) if k < l.

Proof

Both identities can be proven by induction on r using Lemma 7.10 and the fact that the Ei, Fi act on \(\mathcal C_2\) by \(K_{\frac 12\alpha _i}\)-derivations. □

Denote
$$\displaystyle \begin{aligned} \mathbf b^{(i)}_{{\mathbf{m}}}=E_i^{(r_i)}(\mathbf b_{e_i^{-r_i}({\mathbf{m}})}),\qquad r_i=m_j+m_{0i},\, i\in \{1,2\},\, {\mathbf{m}}\in\mathbf M. \end{aligned}$$
The following is immediate from Lemma 7.15.

Lemma 7.16

For each i ∈{1, 2}, \(\mathbf B^{(i)}=\{ \mathbf b^{(i)}_{{\mathbf {m}}}\,:\, {\mathbf {m}}\in \mathbf M\}\) is a basis of \(\mathcal C_2\). Moreover, for each i ∈{1, 2}, λ = l1ω1 + l2ω2 ∈ P+, \( \mathbf B^{(i)}_\lambda :=\{ \mathbf b^{(i)}_{{\mathbf {m}}},\,:\, {\mathbf {m}}\in \mathbf M_{l_1,l_2}\} \) is a basis of Vλ. Finally,
$$\displaystyle \begin{aligned} \sigma^i(\mathbf b^{(i)}_{{\mathbf{m}}})=\mathbf b^{(i)}_{\underline\sigma^i({\mathbf{m}})},\qquad i\in \{1,2\},\,{\mathbf{m}}\in \mathbf M. \end{aligned}$$

In particular, \(\sigma ^i(\mathbf B^{(i)}_\lambda )=\mathbf B^{(i)}_\lambda \).

Thus, σi, i ∈{1, 2} are easy to calculate in respective bases B(i). To attack Conjecture 1.2 we need to find the matrix of both of them in a same basis. Note the following consequence of Lemma 7.15.

Corollary 7.17

For each λ = l1ω1 + l2ω2 ∈ P+ we have
$$\displaystyle \begin{aligned} \mathbf b^{(i)}_{{\mathbf{m}}}=\sum_{ {\mathbf{m}}^{\prime}\in \mathbf M_{l_1,l_2}} C^{i;\lambda}_{{\mathbf{m}}^{\prime},\mathbf{m}}\mathbf b_{{\mathbf{m}}^{\prime}} \end{aligned}$$
where C i; λ is an \(\mathbf M_{l_1,l_2}\times \mathbf M_{l_1,l_2}\) -matrix given by
$$\displaystyle \begin{aligned} C^{i;\lambda}_{{\mathbf{m}}^{\prime},{\mathbf{m}}}=\begin{cases} \displaystyle\binom{m_i+m_{0i}+m_j+m_{ij}}{m_i+m_{ij}}_q,& {\mathbf{m}}^{\prime}={\mathbf{m}},\\ C^{(m_j+m_{0i})}_{t}(m_j+m_{ij},m_i+m_{0i}+m_j+m_{ij}),& {\mathbf{m}}^{\prime}-{\mathbf{m}}=t\mathbf a^+_i,\,t\in{\mathbb Z}_{>0},\\ 0,& \mathit{\text{otherwise}}. \end{cases} \end{aligned}$$

Remark 7.18

The bases \(\mathbf B^{(i)}_\lambda \), i ∈{1, 2}, λ ∈ P+ are in fact Gelfand-Tsetlin bases. The matrices Ci;λ appeared first in the classical limit (q = 1) in [17]. According to [17, Theorem 10] their entries are closely related to Clebsch-Gordan coefficients, and so one should expect that our matrices are related to quantum Clebsch-Gordan coefficients. It is easy to see that σi(Bλ) = Bλ if and only if Vλ is thin.

Theorem 7.19

For each λ = l1ω1 + l2ω2 ∈ P+ and i ∈{1, 2} the matrix Ni;λ of σi with respect to the basis Bλ of Vλ is given by
$$\displaystyle \begin{aligned} N^{i;\lambda}=C^{i;\lambda}P^{i;\lambda} (C^{i;\lambda}){}^{-1} \end{aligned}$$
where \(P^{i;\lambda }=(P^{i;\lambda }_{{\mathbf {m}}^{\prime },{\mathbf {m}}})_{{\mathbf {m}},{\mathbf {m}}^{\prime }\in \mathbf M_{l_1,l_2}}\) with
$$\displaystyle \begin{aligned} P^{i;\lambda}_{{\mathbf{m}}^{\prime},{\mathbf{m}}}=\delta_{{\mathbf{m}}^{\prime},\underline\sigma^i({\mathbf{m}})},\qquad {\mathbf{m}},{\mathbf{m}}^{\prime}\in \mathbf M_{l_1,l_2}. \end{aligned}$$

Conjecture 7.20 (Conjecture 1.2 for \({\mathfrak {g}}={\mathfrak {sl}}_3\))

For each λ = l1ω1 + l2ω2 ∈ P+ we have
$$\displaystyle \begin{aligned} (N^{1;\lambda}N^{2;\lambda})^3=1. \end{aligned}$$

This was verified using Mathematica® for all \(l_1,l_2\in {\mathbb Z}_{\ge 0}\) such that l1 + l2 ≤ 14.

Footnotes

  1. 1.

    The difference between our notation and that of [7, 27] is in the linear automorphism of \(\mathcal A_q({\mathfrak {g}})\) defined on homogeneous elements x by \(x\mapsto q^{\frac 12(|x|,|x|)-(|x|,\rho )}x\).

Notes

Acknowledgements

The first two authors are grateful to Anton Alekseev and Université de Genève, Switzerland, for their hospitality. An important part of this work was done during the second author’s stay at the Weizmann Institute of Science, Israel and during the conference in honor of Anthony Joseph’s 75th birthday, at the Weizmann Institute and at the University of Haifa. The authors would like to use this opportunity to thank Maria Gorelik and Anna Melnikov for organizing that wonderful event.

This work was partially supported by a BSF grant no. 2016363 (A. Berenstein), the Simons foundation collaboration grant no. 245735 (J. Greenstein) and by the Minerva foundation with funding from the Federal German Ministry for Education and Research (J.-R. Li).

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Arkady Berenstein
    • 1
    Email author
  • Jacob Greenstein
    • 2
  • Jian-Rong Li
    • 3
  1. 1.Department of MathematicsUniversity of OregonEugeneUSA
  2. 2.Department of MathematicsUniversity of CaliforniaRiversideUSA
  3. 3.Department of MathematicsThe Weizmann Institute of ScienceRehovotIsrael

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