Quantum Exceptional Group G2 and its Semisimple Conjugacy Classes

We construct quantization of semisimple conjugacy classes of the exceptional group G = G2 along with and by means of their representations on highest weight modules over the quantum group Uq(g)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$U_{q}(\mathfrak {g})$\end{document}. With every point t of a fixed maximal torus we associate a highest weight module Mt over Uq(g)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$U_{q}(\mathfrak {g})$\end{document} and realize the quantized polynomial algebra of the class of t by linear operators on Mt. Quantizations corresponding to points of the same orbit of the Weyl group are isomorphic.

on O that supports an action of the quantum group U q (g), where g is the Lie algebra of G. This action satisfies the quantum Leibnitz rule x.(ab) = (x (1) .a)(x (2) .b) for any a, b ∈ C q [O], x ∈ U q (g), where x (1) ⊗ x (2) is the Heyneman-Sweedler notation for the coproduct of x. In the classical limit q → 1, this action goes over to the classical action of U(g) on C [O].
In the present work, we consider G = G 2 as an algebraic subgroup of GL(C 7 ), where C 7 has the structure of the minimal fundamental representation. The triangular decomposition of GL(C 7 ) induces a triangular decomposition of G with a maximal torus T ⊂ G represented by diagonal matrices. We denote by O t ⊂ G the conjugacy class of an element t ∈ T .
It is known that stabilizers of semisimple conjugacy classes can be described with the help of the affine Dynkin diagram of the group, by erasing one or more nodes. They are divided in two types: the ones whose stabilizer K is a Levi subgroup, and the others which we call pseudo-Levi subgroups. In our case (G = G 2 ), the affine Dynkin diagram for g is It turns out that a Levi K is described with at most one node, while all pseudo-Levi K correspond to exactly two nodes of the affine diagram.
We get our quantization C q [O t ] of the coordinate algebra C[O t ] from a quantization C q [G] of the coordinate algebra C[G] of the group G, as a quotient by an invariant ideal N q (O t ). Thus we 'deform' the classical situation, where C[O t ] is the polynomial ring of an affine variety, i.e. is a quotient of the algebra C [G]. The quantization C q [G] is realized as a subalgebra in U q (g) generated by entries of a matrix Q. This matrix is constructed from the universal R-matrix R of U q (g) in the following way. Denote the above mentioned minimal representations of U q (g) on C 7 by π and set Q := (π ⊗ id)(R 21 R). This is an element of End(C 7 ) ⊗ U q (g) and is considered as a 7 × 7-matrix with entries in U q (g). Those entries are the generators of C q [G] ⊂ U q (g), This subalgebra is ad-invariant due to transformation properties of Q. In particular, the matrix Q satisfies the Reflection Equation Q 2 R 21 Q 1 R 12 = R 12 Q 1 R 21 Q 2 [6,13], as well as some more relations, whose exact form is not important for this study.
Besides presenting the algebra C q [O t ] as a quotient of C q [G], we realize it by linear operators on a vector space M. As C q [O t ] is a subalgebra in U q (g), we take for M a U q (g)module associated with a point in T ∩O t . Then the ideal N q (O t ) is the annihilator of M, and the image of C q [G] in End(M) is the quantization C q [O t ]. It is automatically equivariant by the construction.
The ideal N q (O t ) is generated by the matrix entries of the minimal polynomial of Q as an operator on C 7 ⊗ M, and the kernel of a central character of C q [G] determined by the highest weight of M. It is expressed through q-traces of the powers of Q. To study the minimal polynomial of Q, we analyse the module structure of C 7 ⊗ M. Namely, we prove that C 7 ⊗ M splits into a direct sum of submodules of highest weight. That guarantees semisimplicity of the matrix Q and helps to write down its minimal polynomial explicitly, which is crucial for our approach.
Our analysis of C 7 ⊗ M consists of two parts. First we construct singular vectors in C 7 ⊗ M. For given k, the subset of points whose centralizer is k forms an algebraic subset in T . That subset determines a subset of k-admissible weights parameterizing the module M of type k. We construct singular vectors in C 7 ⊗ M as regular functions of weight that never turn zero. We do it in several steps. First we process the Verma module M =M and then get singular vectors in C 7 ⊗ M for all other M. We do it by projection from C 7 ⊗M with a subsequent regularization, because some singular vectors acquire scalar factors to be canceled.
The second part of our analysis identifies the conditions when the sum of submodules in C 7 ⊗ M generated by the singular vectors exhausts all of C 7 ⊗ M and when the sum is direct. This study makes use of ideas close to [19] and is based on the concept of extremal twist operator and its determinant.
The setup of the paper is as follows. In Section 2 we present a classification of semisimple conjugacy classes of G 2 . It is followed by the basic information about the quantum group U q (g) and its representation on C 7 in Section 3. The quantization theorem is stated in Section 4, and is proved modulo direct sum decomposition of C 7 ⊗ M, which is completed in the subsequent sections. In Section 5, we establish some properties of generalized parabolic modules. The last four sections are more technical. In Section 6, we do regularization of singular vectors in C 7 ⊗ M for a general Verma module M and decompose it into a direct sum of Verma modules. The subsequent sections are devoted to a similar study for all other types of M. Some useful formulas including the entries of a matrix participating in the Shapovalov inverse form can be found in Appendix.
Throughout the paper we adopt the following general convention: • For better readability of formulas, we denote a scalar inverse by the bar, e.g.q = q −1 .
• The notation a b means that a is proportional to b with a non-zero scalar factor. If the coefficient is a scalar function, we thereby assume that it never turns zero.
The symbol also stands for isomorphism, which is always clear from the context and causes no confusion.
• Divisibility by a regular scalar function φ is denoted by φ .
• We use the notation [z] q = q z − q −z q − q −1 whenever q z makes sense. • We assume that q ∈ C is not a root of unity.

Semisimple Conjugacy Classes of G 2
In this section we describe semisimple conjugacy classes of the complex algebraic group G = G 2 . Let g denote the Lie algebra of G with a fixed Cartan subalgebra h. Its root system R is displayed on the figure below.
The half-sum of positive roots 5α + 3β is denoted by ρ.
The group G 2 has a faithful representation in C 7 . The corresponding representation of the quantum group is given in Section 3.1.
The affine Dynkin diagram of g suggests the following stabilizers k ⊂ g of semisimple conjugacy classes labelled by the their root bases k ⊂ R + g : The subscripts indicate the lengths of roots. There are three Levi types with # k 1 and two pseudo-Levi types with # k = 2.
Different although isomorphic k give rise to the same conjugacy class G/K, where the subgroup K with the Lie algebra k is the centralizer of the initial point. Still we make this distinction because we associate with them different representations of quantized G/K.
Let T denote the maximal torus of G corresponding to h and fix t ∈ T such that k is the centralizer of t. We parameterize T with a pair of non-zero complex coordinates x, y ∈ C . In the matrix realization that gives t = diag(xy, x, y, 1,ȳ,x,ȳx) ∈ End(C 7 ). (2.1) Regarding the roots as characters on T , we have α(t) = y, β(t) = xy −1 .
Define T k ⊂ T as the subset of points whose centralizer Lie algebra is k. We will also use the notation T k = T k . We select the subset T k reg ⊂ T k of regular points, whose minimal polynomial in the representation on C 7 has maximal degree. The complementary subset in T k is denoted by T k bord and called borderline, see [1]. Such points are present only for k = h and k = k l . They are a sort of "transitional" from Levi to pseudo-Levi type, hence the name.
The set T ∅ = T h is determined by the conditions x = 1, y = 1, x = y, xy 2 = 1, x 2 y = 1, xy = 1. Clearly, it is enough to use the first three diagonal matrix entries for description of t, so one can write: We have listed all possible k ⊂ g, so that the sets T s,l = T α,3α+2β ∪ T α+β,3α+β ∪ T 2α+β,β , T l,l = T β,3α+β , along with T ∅ and the group identity exhaust all of T . They consist of points whose conjugacy classes are isomorphic as homogeneous spaces. Denote byG the group SL (7) and byW its Weyl group. The Weyl group of G is denoted by W . It is an elementary fact that the intersection of aG-conjugacy class with G consists of a finite number of G-classes.

Proposition 2.1
The conjugacy class of each semisimple element t ∈ G is the intersection of itsG-conjugacy class with G.
Proof A semisimple conjugacy class ofG is determined by the set of eigenvalues and their multiplicities. It is sufficient to check thatW t ∩ T = W t for each t ∈ T .
Fix t = (xy, x, y) ∈ T ∅ such that y = −1. One can check that the multiplicity of the eigenvalue y is 1. PresentW t ∩ T as a union Y ∪Ȳ of subsets whose elements have either y orȳ among their first three coordinates. This union is disjoint for t ∈ T ∅ .
One can check that #|Y | = #|Ȳ | = 6. Then Y ⊂ W t since The setȲ is obtained from Y by inverting the coordinates. One hasȲ ⊂ W t as Y (xy, x, y) 3 ). This proves thatW t = W t for each t ∈ T .
A semisimpleG-class is determined by the set of eigenvalues and their multiplicities. The eigenvalues are fixed by the minimal polynomial while the multiplicities by the character of the subalgebra of invariants under conjugation. The subalgebra of invariants when restricted to maximal torusT ⊂G is generated by the functions t → Tr(t m ), m = 1, . . . , 7, and the character is evaluation at t. All possible minimal polynomials of t ∈ T are listed here: 3 ) = 3. Remark that regular points, contrary to borderline, separate irreducible k-submodules in C 7 . The two bottom lines correspond to the two pseudo-Levi classes. Proof Fix a semisimple point t ∈ G and consider its conjugacy classesÕ t ⊂G and O t ⊂ G. Let F 1 and F 2 be G-submodules in C End(C 7 ) generating the ideals N(Õ t ) and N(G), respectively. Put f i : End(C 7 ) → F * i to be the corresponding G-invariant maps and set f = f 1 ⊕f 2 . By Proposition 2.1, it suffices to prove that ker df (t) = ker df 1 (t)∩ker df 2 (t) has the same dimension as O t , cf. [17], Prop. 2.1.
Identify the tangent space at t withg via the left translation by t, then ker df 2 (t) = g. For the general linear group, ker df 1 (t) =m t =g k , wherem t is the sum of Ad teigenspaces ing corresponding to eigenvalues disctinct from 1 andk is the centralizer of t. Then ker df (t) = ker df 1 (t) ∩ ker df 2 (t) =m t ∩ g = g k since t ∈ G and it is semi-simple.

Quantized Universal Enveloping Algebra
Throughout the paper we assume that q ∈ C is a non-zero complex number that is not a root of unity. Denote by U q (g ± ) the C-algebra generated by e ±α , e ±β subject to the q-Serre relations [e ±α , e ∓β ] = 0.
Remark that the vector space h is not contained in U q (g), still it is convenient for calculations to keep reference to h. The comultiplication on the generators is defined as follows: for all μ ∈ . It is opposite to that in [4]. We will use the notation The subalgebras in U q (g) generated by U q (g ± ) over U q (h) are quantized universal enveloping algebras of the Borel subalgebras b ± = h + g ± ⊂ g denoted further by U q (b ± ).
The Chevalley generators can be supplemented with root vectors for compound μ ∈ R + . They participate in construction of a Poincaré-Birkhoff-Witt (PBW) basis in U q (g) and universal R-matrix, [4].
The universal R-matrix is an element of a certain extension of U q (g)⊗U q (g).
be an orthogonal basis in h * . The exact expression for R (up to the flip of tensor legs) is extracted from [4], Theorem 8.3.9: 2 , and the product is ordered in a certain way. Its reduction to the minimal representation can be found in [21,22].
We will also work with the C[[ ]]-extension of U q (g) completed in the -adic topology, which we denote by U (g). Note that the Cartan subalgebra in U (g) is isomorphic to the polynomial algebra on h while U q (h) is that on T .

Minimal Representation of U q (g)
In this section we describe a representation of U q (g) on the vector space C 7 . It is a deformation of the classical representation of g restricted from so (7). Our realization is close to [14].
Let I denote the set of integers from 1, . . . , 7. Fix a basis {w i } i∈I ∈ C 7 = V and let e ij ∈ End(V ) denote the standard matrix units, e ij w k = δ jk w i , i, j, k ∈ I . One can check that the assignment Up to scalar multiplies, the action of U q (g − ) can be depicted by the graph The representation of U q (g + ) is obtained by reversing the arrows. Let ν i ∈ h * denote the weight of w i , then For all ν ∈ Z we denote by P (ν) the set of pairs (i, j ) ∈ I × I such that ν i − ν j = ν. For each pair with i < j there is a unique monomial ψ in f α , f β such that w j = ψw i . Let ψ ij ∈ U q (g − ) be the monomial obtained from it by reversing the order of factors.

Quantum Conjugacy Classes
In this section we describe quantum semisimple conjugacy classes using ideas of [16] and [2]. The construction is based on certain facts from representation theory to be established in the subsequent sections. We regard roots as multiplicative characters of T and elements of T as spectral points of U q (h) via the correspondence t : q h α → α(t), for all t ∈ T . Fix t ∈ T and its stabilizer subalgebra k. Put t q = tq 2ρ k −2ρ ∈ T and choose the weight λ from the condition q 2λ = t q regarded as an equality in T upon the identification h h * via the inner product. Consider Due to the special choice of λ, it satisfies the conditions q 2(λ+ρ,α) = q (α,α) for all α ∈ k , and we denote by k the set of all such weights. For each α ∈ k there exists a singular vector v λ−α ∈ M λ of weight λ − α annihilated by e β for all β ∈ . Up to a scalar multiplier, they can be written explicitly asf ij v λ with (i, j ) ∈ P (α), wheref ij are matrix elements of the (reduced) Shapovalov inverse [18]. For our case, they are defined in Eq. 5.12.
This should not be confusing as the ring of scalars is always explicitly stated.
The key role in our approach to quantization of conjugacy classes belongs to an element which is a matrix with entries in the quantum group. It commutes with the coproduct of all elements from U q (g) and plays a key role in the theory. Its entries generate an ad-invariant subalgebra C q [G] ⊂ U q (g), which is a quantization of the U(g)-algebra C[G] equipped with the conjugation action. So Q is the matrix of "quantum coordinate" functions on G. It satisfies the so called "reflection equation" [6,13] rather than the RTT-relations of the Hopf dual to U q (g), [10].
The operator Q is scalar on every submodule of highest weight in V ⊗ M k λ as well as a quotient module. In the classical limit, the module V is completely reducible over k. Let I k ⊂ I denote the subset of indices of k-highest vectors w i . Then the eigenvalues of Q on V ⊗ M k λ are x j = q 2(λ+ρ,ν j )+(ν j ,ν j )−(ν 1 ,ν 1 )−2(ρ,ν 1 ) , j ∈ I k , and they obviously deform the eigenvalues ν j (t) of t ∈ O t . A proof for the case k = h can be found in [16] (valid for all simple groups), while for general k it readily follows from Proposition 5.4. We prove in the subsequent sections that Q is semisimple on V ⊗ M k λ , so its minimal polynomial is where the prime means that only distinct eigenvalues count (coincidences occur only for borderline t).
When t varies in a W -orbit, the highest weight of M k λ varies in an orbit under the shifted action. It follows from Proposition 5.4 that the eigenvalues of Q are determined by weights Since the ordinary W -action relates the sets {ν j } j ∈I k , the shifted action relates the sets {ν j + λ} j ∈I k . On the other hand, Q can be written as (z)(z −1 ⊗ z −1 ) for a central element z ∈ U q (g), [8]. Therefore the set of eigenvalues {x j } j ∈I k and the minimal polynomial (4.10) depend only on the class of the point t.
Put Tr q (X) = Tr π(q 2h ρ )X for a 7 × 7 matrix X with arbitrary entries. Then the elements τ m = Tr q (Q m ), m ∈ I , generate the center of C q [G]. Note that τ m are not independent, as the rank of G is 2.
Let χ λ denote the central character of C q [G] returning τ m v λ = χ λ (τ m )v λ . It is expressed through the eigenvalues {x i } 7 i=1 of Q on the tensor product of V with the Verma module of highest weight λ. The operator Q is a scalar multiple and returns x i on the submodule in V ⊗ M λ of the highest weight ν i + λ, i = 1, . . . , 7. Explicitly [16], It is invariant under the shifted action of the Weyl group W on h * . Clearly ker χ λ belongs to the annihilator of M k λ . In the subsequent sections we prove that the minimal polynomial of Q regarded as an operator on V ⊗ M k λ is a deformation of the minimal polynomial of O t , so the relations (4.10) and (4.11) deform the defining relations of O t . Based on this fact, we formulate the main result of this work. Every isotypic component of C [G] is finitely generated over its centre (a q-version of the Richardson theorem [15]). Then the quotient S of C [G] by the ideal generated by ker χ λ is a direct sum of finite isotypic components, and its image in T is C[[ ]]-free since multiplication by is injective on T . Then the kernel J ⊂ S is a direct summand, and one has an g-invariant embedding J 0 ⊂ S 0 of the zero fibers modulo .
Let J ⊂ J denote the ideal generated by the relations (4.10) and (4.11). In the classical limit modulo we have a map J 0 → J 0 . Its image coincides with the image of N(O t ) in S 0 , because the relations (4.10) and (4.11) go over to the defining relations of N(O t ). Since the image of N(O t ) in S 0 is a maximal g-invariant ideal in S 0 , the map J 0 → J 0 is surjective. By the Nakayama lemma the embedding J → J is an isomorphism, and the annihilator of M k λ in C q [G] is generated by Eqs. 4.10 and 4.11. The relations (4.10) and (4.11) are invariant under the shifted action of W on the weight λ. Therefore J is W -invariant, and C q [O t ] depends only on the class of t.
The key fact underlying the above reasoning is semisimplicity of Q. To complete the proof, it is sufficient to show that the sum of highest weight submodules exhausts all of V ⊗ M k λ . We solve a stronger problem: we establish exact criteria when V ⊗ M k λ splits into a direct sum of highest weight modules. The rest of the paper is devoted to this analysis.

Generalized Parabolic Verma Modules
Fix λ ∈ h * and consider the Verma module M λ of highest weight λ. Let * M λ denote the opposite Verma module of lowest weight −λ. There is a unique, up to a scalar multiple, U q (g)-invariant form M λ ⊗ * M λ → C (equivalent to the contravariant Shapovalov on M λ ), which is non-degenerate if and only if M λ is irreducible. As that is the case for generic weight, [5], there is a unique liftF ∈ U q (g + )⊗U q (b − ) of the inverse form, where the Borel subalgebra is extended over the ring of fractions of U q (h) and the tensor product is completed with series. The matrixF = (π ⊗ id)(F ) ∈ End(V ) ⊗ U q (b − ) is expressed through another matrix F ∈ End(V ) ⊗ U q (g − ) whose entries f ij are certain polynomial in the Chevalley generators. They are "extracted" from an R-matrix of U q (g), and their explicit expressions are presented in Appendix A.2.
Put ρ i = (ρ, ν i ),ρ i = ρ i + 1 2 ||ν i || 2 , i ∈ I , and define For an ascending sequence of integers m 1 , . . . , m k , j, put f m 1 ,..., where the summation is taken over all sequences m = (m 1 , . . . , m k ) such that i < m 1 , m k < j, including m = ∅. Finally, we setf ii = 1 for all i andf ij = 0 for i > j. We regard U q (h) as the algebra of trigonometric polynomials on h * . The linear iso- A singular vector v λ−α ∈ M λ of weight λ − α can be constructed as follows. It is known to be unique, up to a scalar factor. Therefore it is proportional tof ij v λ =f ij (λ)v λ with (i, j ) ∈ P (α), upon an appropriate regularization off ij (λ) if needed (the elementf ij can be also constructed using reduction algebras, [12]).

Lemma 5.1 Suppose that t ∈ T k and
The proof is based on the fact that in any presentation of α as a sum of positive roots the summands do not belong to R + k , see e.g. [2], Lemma 2.2.

Standard Filtration of V ⊗ M λ
Define V j ⊂ V ⊗ M λ , j ∈ I , to be the submodule generated by {w i ⊗ v λ } j i=1 . They form an ascending filtration V • of V ⊗ M λ , which we call standard. Its graded module gr V • is the direct sum ⊕ j i=1 gr V j , where gr V j = V j /V j −1 is isomorphic to M λ+ν j for all λ (the proof is similar to [3] for classical U(g)). It is generated by the image w j λ of w j ⊗ v λ in gr V j .
Assuming λ ∈ k denote by V k j the image of V j under the projection V ⊗M λ → V ⊗M k λ . Clearly the sequence V k • = (V k j ) forms an ascending filtration of V ⊗ M k λ . Denote byĪ k the complement of I k in I . Then j ∈Ī k if and only if there is i < j such that ν i − ν j ∈ k .

Proposition 5.4 The graded module gr
Proof Fix j and put and β = ν 1 −ν j . The module grV k j is a quotient of V j by the submodule Let M j ⊂ V ⊗M λ denote the submodule of highest weight λ+ν j and let u j be its highest weight generator. Furthermore, consider M k λ for λ ∈ k and let π k λ denote the projection . . , 7, of submodules is also invariant under the action of Q, which is semisimple on W k 7 . Semi-simplicity of Q is important for our studies, so the question is

Proposition 5.5
Suppose that k is Levi and fix j ∈ I k . Then the following statements are Proof It can be proved that, for Levi k, both M k j and gr V k j are parabolically induced from the same U q (k)-module. Hence the map M k j → gr V k j is epimorphism and isomorphism simultaneously unless it is zero.
Assuming ii) we find that all maps M k i → gr V k i are surjective and therefore injective; hence iv). Conversely, iv) implies that all maps W k i → gr V k i are surjective. Since, W k 1 = V k 1 , induction on i then proves ii). Furthermore, iv) implies that M k i → gr V k i are isomorphisms, and then For any k, a direct sum decomposition That is the case if all maps ℘ k j : M k j → gr V k j are onto, i.e. the generators of M k j are not killed by ℘ k j .
In this section, M λ is the Verma module of highest weight λ. We work out exact criteria for decomposition of V ⊗ M λ into a direct sum of submodules of highest weight. To that end, we undertake a detailed study of singular vectorsû j =F (w j ⊗ v λ ) ∈ V ⊗ M λ as rational trigonometric functions h * → V ⊗ U q (g − ), upon the natural identification of M λ with U q (g − ) as vector spaces. Although they cannot be evaluated at poles, singular vectors are defined up to a scalar multiplier and can be regularized. We end up with rescaled singular vectors u j , j ∈ I , that are regular on h * and never turn zero.

Singular Vectors in V ⊗ M λ
The vectorsû j =F (w j ⊗ v λ ), j ∈ I , are expanded aŝ They are singular for all λ where defined and generate submodules M j ⊂ V ⊗M λ of highest weight λ + ν j . They have rational trigonometric dependence on λ and may have zeros and poles. As singular vectors matter up to scalar factors, it is convenient to pass fromû j tǒ u j =Ā j 1,...,j −1 (λ)û j , which are regular in λ. Theň Eachǔ j generates a submodule M j ⊂ V ⊗ M λ if does not turn zero, otherwiseǔ j needs rescaling. That is the subject of our further study. Remark that, for any U q (g)-module Z, a singular vector u = i∈I w i ⊗ z i ∈ V ⊗ Z, defines a U q (g + )-equivariant map V * → Z. Since the U q (g + )-module V * is cyclicly generated by z 1 , we call it generating coefficient of u. Lemma 6.1 Suppose that Z is generated by the highest weight vector v λ . Suppose thať f mj v λ = 0 for some m < j. Thenf ij v λ = 0 for all i m.
Proof It follows now from Eq. 5.12 thať where a i (λ) are numerical factors. The second term disappears by the hypothesis. Let us show that the other terms are zero too. Recall that for all k m and μ = ν k − ν k+1 , e μfkj =f k+1,jĀ j k mod U q (g)g + , cf. [18]. Thenf mj v λ = 0 impliesĀ Proof "Only if" in i) and ii) follow from the equalitiesǔ jj =Ā j 1,...,j −1 v λ and, respectively, u 1j =f 1j v λ . "If" in ii) is due to Lemma 6.1 because then the generating coefficientǔ 1j = f 1j v λ vanishes.

Remark 6.3
Suppose that Z is a family of U q (g)-modules of highest weight λ ranging in an algebraic set ⊂ h * . Then Lemma 6.1 admits an obvious modification if one replaces equality to zero with divisibility by some φ ∈ C q [ ]. Assuming it indecomposable, Corollary 6.2 can be appropriately reworded if C q [ ] is a unique factorization domain and Z has no zero divisors. In what follows, we apply this modification to = k and Z = M k λ .
In the next statement we essentially assume that k = h.
Proof It is sufficient to consider the case k = {μ}. Choose λ ∈ k so thatǔ j = 0. By Proposition 6.8 below, such weights are dense in k . Let i ∈ I be such that ν i − ν j = μ. AsĀ On the other hand,ǔ kj =Ā j 1,...,k−1 (λ)f kj v λ = 0 for all i < k because the numerical factor contains vanishingĀ j i (λ). Thereforeǔ j ∈ V ⊗ M λ−μ for all λ ∈ k and vanishes in V ⊗ M k λ , along with M k j .

Projection to gr V •
It follows from Proposition 5.3 that the image ofû j in gr V • lies in gr V j , and thusû j = D j w j λ mod V j −1 with someD j ∈ C. Up to a scalar factor, 7 j =1D j coincides with the determinant of the extremal twist operator θ V ,M h λ , [19]. Its inverse was calculated in [9] in connection with dynamical Weyl group. ObviouslyD 1 = 1. For higher j , applying the results of [9], one haŝ where the quantities are related to eigenvalues of the operator Q by q 2ξ ij = x ixj . For reader's convenience, ξ ij are listed in Appendix A.1. Let j ⊂ h * , j > 4, denote the set of weights such that q 2η 4j (λ) = −q 2 .

Lemma 6.5 The module M j is not contained in
Proof Observe that q 2ξ 4j = −q 2 = 1 and q 2ξ j j = q 8 = 1. For all other i < j, the functions q 2ξ ij are not constant on j . Therefore all q 2ξ ij with i < j are distinct from 1, off an algebraic subset in j . For such weights, M j cannot be in V j −1 , since the Q-eigenvalue

Corollary 6.6
For all j > 4,f 1j identically vanishes on j .
Proof The productĎ j = j −1 k=1Ā j kD j is polynomial in q ±h μ and turns zero on j for j > 4 thanks to the factorĀ j j and the equality That is possible only in the following two cases: eitherǔ j turns zero or M j ⊂ V j −1 . By Lemma 6.5,ǔ j (λ) = 0 on • j and hence on j . This yieldsǔ 1j =f 1j v λ = 0 on j for the generating coefficient.

Regularization of Singular Vectors in V ⊗ M λ
In this section, we evaluate a scalar function δ j ǔ j and show that renormalized singular vectors δ −1 jǔ j do not turn zero at all λ. Denote by J the two-sided ideal in U q (g − ) generated by the relation f α f β =q 3 f β f α . The non-zero elements f ij modulo J read Introduceǧ ij as polynomials in y ±1 1 , . . . , y ±1 7 with coefficients in U q (g − ) by similar formulas asf ij with allĀ j k in Eqs. 5.12 and 6.15 replaced byĀ k = 1−y k q−q . Proof All calculations will be done modulo J . First of all, Computation of the coefficient in the brackets completes the proof.
Corollary 6.6 assures thatf 1j /δ j is a polynomial in q ±h μ via identification y i = q 2η ij , i < j. We also havě For each j 5 and all i < j we assign y i = q 2η ij and use Lemma 6.7: the key point is that δ j cancels the factorqy 4 + q in all cases. Modulo J , we havef 15 (λ) f 12ǧ25 anď f 16 (λ) f 12ǧ25 f 56 . This implies the statement for j = 5, 6. Finally, which proves it for j = 7.
Conversely, let λ be such that M j ⊂ M i . Then [ξ ij ] q = 0 and hence φ ij (λ) = 0 if i = j . If i = j , we can assume that M j ⊂ M k for k = j since λ is in the closure of such weights, as follows from Eq. 6.16. Proposition 5.5 then suggests that D j (λ) = 0, and the only vanishing factor can be φ j j . Then it is true for all λ.
As an application of Lemma 6.9, we describe direct sum decomposition of the module V ⊗ M λ . This corresponds to maximal conjugacy classes, with the stabilizer subalgebra k = h. Fix t = diag(t i ) ∈ T and choose λ to fulfill the condition tq −2h ρ = q 2h λ . This fixes the relation between the entries of t and the Q-eigenvalues as t i = q 2(λ+ρ,ν i ) = x i q 2δ i4 +2(ρ,ν 1 ) .

Proposition 6.10
for t ∈ T h . This implies the stated conditions on q guaranteeing D h = 0 (mind that q is not a root of unity).
Next we essentially assume that k = h and define φ k j = i∈Ī k j φ ij , j ∈ I k .

Lemma 6.11
For every j ∈ I k , the vector π k λ (u j ) ∈ V ⊗ M k λ vanishes once φ k j (λ) = 0. If k is of type k s or k l , then π k λ (u j ) is divisible by φ k j .
Proof By Lemma 6.9, u j ∈ M i for i ∈Ī k j once φ k j (λ) = 0. On the other hand, π k λ (u i ) = 0 if λ ∈ k , by Proposition 6.4. Therefore π k λ (u j ) vanishes in V ⊗ M k λ . If the semisimple part of k has rank 1, trigonometric polynomials on k form a principal ideal domain as dim k = 1. Therefore π k λ (u j ) is divisible by φ k j (λ).
Throughout this section ν ∈ R + is a short root and k = k s is the reductive subalgebra of maximal rank with the root system {±ν}. We aim to prove that the vectors u k λ are regular functions of λ and do not vanish at all weights. Their projections to gr V k • are equal to D k j w λ j with D k j i∈I k j φ ij .

Regularization of Singular Vectors in V ⊗ M k λ
Assuming j ∈ I k , denote by c k j the coefficient in the expansion u k j w j ⊗c k j v λ +. . ., where the suppressed terms belong to i<j w i ⊗M k λ . Up to a non-vanishing factor, c k  Proof We parameterize k ⊂ h * with the complex variable θ as in Table 1.
The case ν = α follows from the fact that the nil-sets of D k j and c k j do not intersect, cf. Table 1.
In the case of ν = α + β, the statement is trivial for j = 1 and immediate for u k α is equal to [η 25 ] q = [θ + 1] q , up to a non-vanishing factor. Hence u k 5 does not turn zero unless [θ + 1] q = 0. However, D k 5 = 0 at such λ. Therefore u k 5 = 0 at all weights from k .
Finally, consider the case ν = 2α + β. The projection M λ → M k λ is an isomorphism on subspaces of weights λ − μ with μ < 2α + β. Such are the weights of the generating coefficients in u k j = π k λ (u j ). They do not turn zero as u j = 0 at all weights. This completes the proof.

Decomposition of V ⊗ M k s λ
Fix t ∈ T ν and let k k s denote the stabilizer of t. Choose λ ∈ k from the equality q 2λ+2ρ = tq ν . We use x ∈ C , x 2 = 1, to parameterise the spectrum {x ±1 , 1} of t as in Eq. 2.4.
Proof Table 1 gives λ if and only if D k does not vanish. It follows that the eigenvalues qx ±1 , q 2 of the operator q 10 Q on V ⊗ M k λ are pairwise distinct, hence the sum is direct.
In this section, ν ∈ R + is one of the three long roots, and k k l ⊂ g is the reductive subalgebra of maximal rank with the root system {±ν}. Now k is the set of weights λ that satisfy the condition q 2(λ+ρ,ν)−6 = 1. We parameterize it with the complex variable θ = (λ, α). There are two pairs (l, k) ∈ P (ν), so #Ī k = 2 and #I k = 5.

Regularization of Singular Vectors for Levi k
From Table 2 we conclude that u j /φ k j may be divisible by the following factors:

Proposition 8.1
For all j ∈ I k , the singular vector π k λ (u j )/φ k j is divisible by ψ k j , and u k j = π k λ (u 4 )/(φ k j ψ k j ) = 0 at all λ.

Decomposition of V ⊗ M k λ for Pseudo-Levi k
Let μ and ν denote, respectively, the minimal and maximal roots in k , relative to the partial order induced from ZR. Put m ⊂ k to be the reductive subalgebra of maximal rank such that m = {μ}. For λ ∈ k ⊂ m , the homomorphism M λ → M k λ factors through the projection M m λ → M k λ . Its restriction M m λ [λ−ξ ] → M k λ [λ−ξ ] is an isomorphism for ξ < ν. It follows from here that the map ν i <ν w i ⊗ M m λ [λ − ν i ] → ν i <ν w i ⊗ M k λ [λ − ν i ] sends non-vanishing singular vectors u m j with j ∈ I k ⊂ I m over to singular vectors, u k j ∈ V ⊗M k λ . It follows that D k j (λ) = D m j (λ) = 0, for all j ∈ I k .
Proposition 9.1 For all pseudo-Levi k ∈ g, V ⊗ M k λ = ⊕ j ∈I k M k j .
Proof Since D k = 0, the sum j ∈I k M k j gives all V ⊗M k λ . It is direct as the Q-eigenvalues are distinct. Indeed, for k = k l,l we have q 2ξ 12 = e ∓ 2πi 3 , q 2ξ 14 = e ± 2πi 3 q 8 , q 2ξ 24 = e ∓ 2πi 3 q 6 , where the upper sign corresponds to the first row in Table 3. For the three k s,l -points we have q 2ξ 13 = −q 2 , q 2ξ 12 = −q 2 , q 2ξ 12 = −q −2 , respectively, from the top downward.