Equivariant K-theory of the semi-infinite flag manifold as a nil-DAHA module

The equivariant K-theory of the semi-infinite flag manifold, as developed recently by Kato, Naito, and Sagaki, carries commuting actions of the nil-double affine Hecke algebra (nil-DAHA) and a q-Heisenberg algebra. The action of the latter generates a free submodule of rank |W|, where W is the (finite) Weyl group. We show that this submodule is stable under the nil-DAHA, which enables one to express the nil-DAHA action in terms of |W|×|W|\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|W|\times |W|$$\end{document} matrices over the q-Heisenberg algebra. Our main result gives an explicit algebraic construction of these matrices as a limit from the (non-nil) DAHA in simply-laced type. This construction reveals that multiplication by equivariant scalars, when expressed in terms of the Heisenberg algebra, is given by the nonsymmetric q-Toda system introduced by Cherednik and the author.


Introduction
Let G be a connected, simply-connected complex simple algebraic group, with Borel subgroup and maximal torus B ⊃ T .The semi-infinite flag manifold [FF, FM] associated with G is the homogeneous space Q rat = G(K)/(T (C) • U (K)) where K = C((z)) and U is the unipotent radical of B. This variant of the affine flag manifold captures the level-zero representation theory of the untwisted affine Lie algebra associated with G [BF2, K1,KNS,MRY].Furthermore, the space of quasi-maps P 1 → G/B into the finite-dimensional flag variety admits a closed embedding into Q rat , and thus semi-infinite flag manifolds are intimately related to quantum Ktheory of G/B [BF1].
In [KNS], an equivariant K-group K I⋊C * (Q rat ) is introduced, where I ⊂ G(R), R = C [[z]], is the Iwahori subgroup and C × acts by loop rotation.There is a major difficulty in applying the usual construction of equivariant algebraic K-theory-namely, the Grothendieck group of equivariant coherent sheaves-to Q rat , as this space is an ind-infinite scheme which is not Noetherian.Thus K I⋊C * (Q rat ) is not the Grothendieck group of a category of coherent sheaves, but is constructed to behave as if it were.We review its definition and basic properties in §2.One may object to working with this formal substitute for algebraic K-theory, but as a demonstration of its value we mention that K I⋊C * (Q rat ) has already found applications in the quantum K-theory of G/B [K2,LNS], and in particular to the K-theoretic version of Peterson's isomorphism [LLMS].
The K-group K I⋊C * (Q rat ) has the structure of an (H 0 , H)-bimodule, where H 0 is the nil-double affine Hecke algebra (nil-DAHA) and H is a q-Heisenberg algebra.Acting on the subset of Schubert classes in K I⋊C * (Q rat ) indexed by the Weyl group W of G, the q-Heisenberg algebra H generates a free submodule of finite rank |W |.We prove (Theorem 5.1(1)) that this free H-module is stable under the nil-DAHA H 0 , giving rise to a homomorphism ̺ 0 from H 0 to the algebra Mat W (H) of W × W matrices over H. 0.1.Algebraic construction.Assuming that G is simply-laced, our main result (Theorem 5.1) gives a different construction of the homomorphism ̺ 0 : H 0 → Mat W (H) which is purely algebraic, starting from the polynomial representation of the double affine Hecke algebra.The nonsymmetric q-Whittaker function and its symmetries [CO] are ultimately responsible for linking the geometry and with our algebra construction (see (5.4) in proof of Theorem 5.1).In particular, our result shows that the action of H 0 on K I⋊C * (Q rat ), expressed through ̺ 0 , is by the nonsymmetric q-Toda operators of [CO].
We also define the "spherical part" of K I⋊C * (Q rat ) (see §5.1), which should be regarded as the (G(R) ⋊ C × )-equivariant K-theory of Q rat .By taking the diagonal entry of ̺ 0 at the identity element of W , we obtain a homomorphism ̺ sph 0 : Z[X] W → H corresponding to the spherical nil-affine Hecke algebra action on the spherical part of K I⋊C * (Q rat ) (Corollary 5.3).
Here Z[X] W ⊂ H 0 is a copy of the representation ring R(G × C × ) inside H 0 .
A consequence of our construction is that homomorphism ̺ sph 0 coincides with the q-Toda system of difference operators [C2,CO].In particular, our Corollary 5.3 is closely related to results of Braverman and Finkelberg [BF2].The role of q-Toda systems in quantum K-theory and related geometries goes back to the influential works [GL] and [BF1].More recently, [Ko, KoZ] give geometric incarnations of type A (q, t)-Macdonald difference operators using the equivariant K-theory of quasimaps into the cotangent bundle of the flag variety.
As mentioned above, our results assume that G is simply-laced.While we expect that certain adjustments can be made to extend them to general G, our method of proof of Theorem 5.1 applies only in the simply-laced case.(See the remarks at the beginning of Section 3 for more about this.)Thus we do not attempt to formulate the most general result in this paper.0.2.Inverse Pieri-Chevalley formula.For any G-weight λ ∈ P one has in the usual way an equivariant line bundle O(λ) on Q rat .The Pieri-Chevalley formula in K I⋊C * (Q rat ) expresses the action of multiplication by O(λ) on Schubert classes {[O w ]} w∈W aff , where W aff is the affine Weyl group, as a combinatorial (in general, infinite) sum: where c λ w,ṽ ∈ R(T × C × ) are equivariant scalars.The first Pieri-Chevalley formula in K I⋊C * (Q rat ) was given by [KNS] for dominant λ ∈ P + , as an infinite sum over semi-infinite Lakshmibai-Seshadri paths.A finite Pieri-Chevalley formula in K I⋊C * (Q rat ) for antidominant λ ∈ P − was stated and proved in [NOS]. 1 A Pieri-Chevalley formula in K I⋊C * (Q rat ) for arbitrary 1 By an isomorphism from [K2], this implies the finiteness of multiplication in the small equivariant quantum K-theory of G/B.In this setting, the anti-dominant Pieri-Chevalley rule also has a close connection to the shift operator of [IMT].
λ ∈ P , interpolating between the dominant and anti-dominant cases, was found recently by Lenart, Naito, and Sagaki [LNS].
Our main result directly pertains to the inverse Pieri-Chevalley formula, namely to the expansion: ] and e λ ∈ R(I ⋊ C × ) is an equivariant scalar.Theorem 5.1 gives an algebraic construction of the inverse Pieri-Chevalley formula for arbitrary λ ∈ P , simply because the multiplication by equivariant scalars e λ is part of the nil-DAHA action.A consequence of Theorem 5.1 is the finiteness of the inverse Pieri-Chevalley formula in K I⋊C * (Q rat ) for arbitrary λ ∈ P ; in particular, an immediate coarse observation is that the right-hand side of (0.2) can be expressed as a sum over ũ ∈ W aff less than or equal to the translation element y λ in the usual Bruhat order (see §1.2 for the notation used here).In future work, we plan to use our construction give a complete and explicit description of the inverse Chevalley rule in type A and to derive implications for the structure of the coefficients d λ,µ w,ṽ in general.

Notation
1.1.Root data.Let G ⊃ B ⊃ T be as in the introduction.Denote by R, Q, P (respectively, R ∨ , Q ∨ , P ∨ ) the (co)roots, (co)root lattice, and (co)weight lattice of G.For any root α ∈ P let α ∨ ∈ R ∨ be its associated coroot.
Let R = R + ⊔ R − be the decomposition of R into positive and negative roots determined by B. We write α > 0 (resp., α < 0) to indicate that α ∈ R + (resp., α ∈ R − ).Let I be a Dynkin index set for G and let {α i } i∈I , {ω i } i∈I (respectively, {α ∨ i } i∈I , {ω ∨ i } i∈I ) be the simple (co)roots and fundamental (co)weights, respectively.Let W = s i : i ∈ I be the Weyl group of G, where s i = s α i is the simple reflection through α i .Let ℓ(w) be the length of w ∈ W with respect to {s i } i∈I and let w 0 be the longest element of W .
Let Q ∨ + = ⊕ i∈I Z + α ∨ i and P + = ⊕ i∈I Z + ω i be the cones of effective coweights and dominant weights, respectively, where Z + = Z ≥0 .Let ≤ be the partial order on Q ∨ given by α ≤ β if and only if β − α ∈ Q ∨ + .For λ ∈ P + , let V (λ) be the irreducible G-module with highest weight λ and let V (λ) µ ⊂ V (λ) be its µ-weight space for any µ ∈ P .
We say that a statement depending on λ ∈ P holds for sufficiently dominant λ if there exists an M ∈ Z + such that the statement is true whenever λ = i∈I m i ω i with m i ≥ M for all i ∈ I.
Let Z[P ] = R(T ) be the group algebra of P , with basis elements e µ (µ ∈ P ) such that e λ+µ = e λ e µ and e 0 = 1.For any finite-dimensional T -module V , define its character as ch V = µ∈P m µ e µ ∈ Z[P ] where m µ is the dimension of µ-eigenspace of T in V .
1.2.Affine Weyl groups.Let W aff = W ⋉ Q ∨ and W ext = W ⋉ P ∨ be the affine and extended affine Weyl groups.We denote elements (w, β) of these groups by wy β , i.e., w = (w, 0) and y β = (e, β) where e ∈ W is the identity element.The group W aff = s i : i ∈ I aff is a Coxeter group where I aff = I ⊔ {0} and with θ the highest (long) root of G.
The group Π = P ∨ /Q ∨ acts on W aff by diagram automorphisms.This can be realized as a subgroup of W ext by the elements where r ∈ I is an index of a minuscule fundamental coweight (i.e., α r appears with coefficient 1 in θ) and u r is the shortest element of W sending ω ∨ r to the antidominant chamber.
Let Q aff = Q ⊕ Zδ be the affine root lattice, which has basis {α i } i∈I aff where α 0 = −θ + δ.Let P aff = P ⊕ Zδ be the level-zero affine weight lattice.The affine Weyl group W aff acts on P aff as follows: where , : Q ∨ × P → Z is the canonical pairing.
The semi-infinite Bruhat order [L] (see also [INS,§2.4 and §A.3]) is the partial order ≺ on W aff generated by relations s α+kδ w ≺ w if and only if w−1 (α + kδ) ≺ 0. The resulting poset is graded by the length function ℓ ∞ 2 (wy β ) = ℓ(w) + β, 2ρ , where 2ρ = α∈R + α. 1.3.Smash products.Suppose S is a commutative ring with 1.For any S-algebra S ′ and any group Γ acting by S-algebra automorphisms on S ′ , we write S ′ ⋊ Γ for the smash product S ′ ⊗ S S[Γ], which is an S-algebra with multiplication ( In the case when Γ is an abelian group, written additively, we use exponential notation {x γ } γ∈Γ for the standard basis elements of the group algebra S[Γ], so that x γ x γ ′ = x γ+γ ′ and x 0 = 1.As we will encounter several instances of such group algebras, and sometimes the same algebra will appear in different contexts, we will use various letters for the base of exponentials (e.g., x, y, e, X, Y ) 1.4.q-Heisenberg algebras.The following special case of smash products will arise frequently.Let S = Z[q ±1 ] and suppose A and B are abelian groups, written additively, together with a bilinear form 1.5.Matrices.Given a commutative ring S with 1 and any S-algebra S ′ (not necessarily commutative), let Mat W (S ′ ) denote the S-algebra of W ×W matrices with entries in S ′ .

Semi-infinite flag manifolds
In this section we recall the construction of the semi-infinite flag manifold Q rat due to [FM] and the equivariant K-group K I⋊C * (Q rat ) of [KNS].Our presentation follows [KNS], but we elaborate further on some points which our crucial for the present work.Near the end of this section we give a detailed example for G = SL(2).
Consider a (T × C × )-module V with the following properties: if V = ⊕ i∈Z V i where C × acts in V i by q i , then each V i is a finite-dimensional Tmodule and V i = 0 for i sufficiently large (or small).For such V , we define ch V = i∈Z q i ch V i as an element of Z[P ]((q −1 )) (or Z[P ]((q))).We also define the graded dual Let C × act on K by loop rotation, i.e., a • p(z) = p(a −1 z) for a ∈ C × and p(z) ∈ K. Let q ∈ R(C * ) stand for the weight of z under this actionnamely, the class of the representation q(a) = a −1 .
Let I ⊂ G(R) be the Iwahori subgroup, which is the pre-image of B under the evaluation map G(R) → G at z = 0.Both I and G(R) are C × -stable.
2.1.Semi-infinite flags.Let Q be the infinite-type scheme of [FM] which parametrizes tuples (ℓ λ ) λ∈P + of C-lines in λ∈P + P(V (λ) [[z]]) satisfying the Plücker equations.Such a collection is uniquely determined by the lines (ℓ ω i ) i∈I , and the map (ℓ λ ) λ∈P + → (ℓ ω i ) i∈I is the Drinfeld-Plücker embedding The semi-infinite flag manifold Q rat is the direct limit of the family For any given by the restriction of ⊠ i∈I O(m i ) on P (resp.its limit).Here the C × -action on all objects is induced by loop rotation on K.
Two basic features of K I⋊C * (Q) are: induced by tensor product of line bundles is well-defined on K I⋊C * (Q) for any ν ∈ P .• For suitable quasicoherent sheaves E on Q (see [KNS,Theorem 5.4 as equivariant sheaves, for all w ∈ W aff and λ ∈ P , and correspondingly ] for all λ ∈ P .One easily checks that this map is: (i) well-defined, i.e., it respects convergence (2.1) and equivalence (2.2), and (ii) injective.Moreover, one has (i β ) * [O w(λ)] = q β,λ [O wy β (λ)] for any w ∈ W aff,+ and λ ∈ P .
The equivariant K-theory of Q rat is defined as where the direct limit of ( One obtains classes [O w(λ)] ∈ K I⋊C * (Q rat ) for w ∈ W aff and λ ∈ P which are well-defined by ] and the sum is finite.Without loss of generality we may assume that w ∈ W aff,+ , so that this equation holds in K I⋊C * (Q), and that λ ∈ P + .This is achieved via (2.6) for sufficiently large α ∈ Q ∨ + and then by tensoring with [O(ν)] sufficiently dominant ν ∈ P + .We may also assume that c w,λ ∈ Z[q −1 ] after multiplying by a power of q −1 .
2.4.Functional realization.Let F P be the Z ]], with pointwise addition and scalar multiplication.Let F P be its quotient by the Z[P ][[q −1 ]]-submodule of functions vanishing on sufficiently dominant µ.
Define the q-Heisenberg algebra ).The spaces F rat P and F rat P are right H-modules under P is an H-module monomorphism, and Proof.The image Ψ(K I⋊C * (Q rat )) ⊂ F rat P is stable under H, due to: where we choose γ ∈ Q ∨ + so that α−β +γ ∈ Q ∨ + .These computations extend to convergent infinite sums.Hence K I⋊C * (Q rat ) can be made uniquely into an H-module such that Ψ is an H-homomorphism.
To obtain (2.11) and (2.12), we simply apply Ψ to both sides and check that they agree using (2.4) and (2.8).Finally, the freeness assertion is Lemma 2.2.2.6.nil-DAHA.The nil-DAHA H 0 is the ring defined by generators T i (i ∈ I aff ), X ν (ν ∈ P ), X ±δ (2.13) and relations We set D i = 1 + T i (i ∈ I aff ).These elements satisfy the braid relations and ] of H 0 is given multiplication operators X ν+kδ → q −k e −ν and Demazure operators In this representation we have: By [KNS,Prop. 6.4] (attributed to unpublished work of Braverman and Finkelberg), By comparison of these formulas with the Heisenberg action, we observe that K I⋊C * (Q rat ) is an (H 0 , H)-bimodule.
The nil-DAHA H 0 acts on F rat P and F rat P via its P -pointwise action in the polynomial representation.
More generally, we have by [K1, Theorem A(3)] that for any w ∈ W aff , i ∈ I aff , and λ ∈ P .We have e ≺ s i for i = 0 (which gives another way to see the above) and s 0 ≺ e.
For i = 0 we therefore have Lemma 2.4 and (2.27) together give for all w ∈ W aff and λ ∈ P .
Below we will show (see Theorem 5.1): The main content of (2.29) is therefore that K is stable under X ν (ν ∈ P ), which is not immediate.
Granting (2.29) for now, we have that K is an (H 0 , H)-bimodule which is free as a right H-module with basis {[O w ]} w∈W .Hence for any H ∈ H there exists a unique W × W matrix A H with entries in H such that for all w ∈ W .We obtain an algebra homomorphism given by ̺ 0 (H) = A H .
Our goal in the next sections is to give an algebraic construction of the homomorphism ̺ 0 .
The matrices ̺ 0 (D i ) can be directly computed from (2.28).For i = 0 we have s i w ≺ w if and only if s i w < w in the usual Bruhat order on W .For i = 0 we have s 0 w ≺ w if and only if w −1 (−θ) < 0 if and only if w < s θ w in the usual Bruhat order on W . Hence (2.33) 2.7.Example: G = SL(2).Let W = {e, s} be the Weyl group, α the simple root, and ω the fundamental weight, with respect to the standard upper triangular Borel subgroup and diagonal torus.
Let C 2 = V (ω) be the standard representation of SL(2).In this case there are no Plücker equations and we have The semi-infinite Schubert varieties are given by the following equations Hence we have the standard exact sequence Tensoring by O(ω) gives: By a similar exact sequence, we obtain and hence Thus we find that 2.8.Dual versions.To prepare for the algebraic construction of ̺ 0 , it is convenient to apply a dual twist to all preceding constructions.Let * : Z[P ]((q −1 )) → Z[P ]((q)) denote the dual map on characters given by (e ν ) * = e −ν , q * = q −1 .
Correspondingly, let F rat * P be the Z[P ]((q))-module of all functions ψ : P → Z[P ]((q)).Extend the definition of * P -pointwise to * : F rat P → F rat * P .Define F rat * P as the quotient of F rat * P by functions vanishing on sufficiently dominant weights.We have an induced map * : F rat P → F rat * P .
We also denote the inverse of any of these maps by * .We make F rat * P into a right H-module as follows: Then * is compatible with the involutive ring automorphism of H, also denoted * , which is given by: q * = q −1 , (x λ ) * = x −λ , (y β ) * = y β .Then: for all ψ ∈ F rat P and h ∈ H.We make F rat * P into a left H 0 -module by the P -pointwise action of the following operators on Z[P ]((q)): Then * is a left H 0 -module homomorphism.

DAHA
We assume for the rest of this paper that G is simply-laced.(3.1) We will at times identify Q with Q ∨ and P with P ∨ by imposing α = α ∨ for α ∈ R. We extend the canonical pairing to , : P × P → Q.We choose e ≥ 1 minimal so that e P, P ⊂ 2Z.We have α, α = 2 for all α ∈ R and β, β ∈ 2Z for all β ∈ Q.
The constructions of this section and the next can be applied more generally in the setting of twisted affine root data.We have opted for less generality in an effort to keep our notation relatively simple.Moreover, the overlap with the preceding constructions-which were based on an untwisted affine root system-is precisely the case simply-laced root data.
Further details on the constructions of this section in the twisted case can be found in [CO].
3.2.Duality.The extended DAHA H carries, among other symmetries, a Z[t ±1/2 ]-linear algebra involutive anti-automorphism ϕ which is uniquely determined by ϕ(X δ/e ) = X δ/e = Y −δ/e , ϕ(X ν ) = Y −ν , and ϕ(T i ) = T i for all i ∈ I (see [CO] or [C1] and the references therein).We will make use of the algebra automorphism τ + of H [C1], which fixes T i (i = 0), X δ/e , and X ν (ν ∈ P ) and is given on the remaining generators by ]-subalgebra of H generated by T ′ i (i ∈ I aff ) and Y ν (ν ∈ P ) and Y δ/e = X −δ/e , where T ′ i = T i for i ∈ I and T ′ 0 = X −α 0 tT −1 0 .We deduce that H ′ can be presented as the Z[t ±1/2 ]-algebra with generators T ′ i (i ∈ I aff ) and Y ν (ν ∈ P ) and Y δ/e and relations for all i, j ∈ I aff and ν, µ ∈ P .
The map ϕ restricts to an anti-isomorphism ϕ : H → H ′ given by 3.3.Polynomial representation.Let k = Q(q 1/e , t 1/2 ).The group algebra k[P ], with k-basis {x λ } λ∈P , is a left H-module such that and Π acts by its group action on k[P ]: π(x λ ) = x π(λ) .Thus a general element H ∈ H acts on k[P ] by an operator given by a finite sum (a differencereflection operator): where h w ∈ k(P ) = Frac(k[P ]), viewed as a multiplication operator.This gives an embedding whose image leaves k[P ] stable.
3.4.Functions on W .Let k(P ) W be the k-algebra functions φ : W → k(P ) under pointwise addition and multiplication.We make k(P ) W a W extmodule as follows: where u, v ∈ W and µ ∈ P .This action is by k-algebra automorphisms, thus making k(P ) W a module over k(P ) W ⋊ W ext (letting k(P ) W act on itself by multiplication).
For any D ∈ k(P ) W ⋊ W ext there exist unique A vw ∈ k(P ) ⋊ P (v, w ∈ W ) such that and hence the corresponding matrix is given by be the induced map.Definition 3.2.Let ̺ ′ : H ′ ֒→ Mat W (k(P ) ⋊ P ) be the restriction to H ′ of the following composite map: Example 3.4.For any i = 0 and any ν ∈ P we have 2) and label rows and columns of matrices by (e, s 1 ).We have: Here we use that Y ω 1 = t −1/2 πT 1 , where π = y ω 1 s 1 , so that in the polynomial representation The corresponding nil-DAHA's H 0 and H ′ 0 are obtained by specializing t = 0, i.e., H 0 = H/t H and H ′ 0 = H ′ /t H ′ .Thus H 0 admits the presentation of §2.6 and H ′ 0 can be presented as follows: H ′ 0 is the ring with generators and relations for all i, j ∈ I aff and ν, µ ∈ P .
We set D ′ i = T ′ i + 1 for i ∈ I aff .We obtain a ring anti-automorphism ϕ 0 : H 0 → H ′ 0 as the specialization of ϕ.
Example 4.2.For G = SL(2), we have: Example 4.3.In general, for i = 0 we have and all other entries of ̺ ′ 0 (T ′ i ) vanish.For i = 0 we have and hence and all other entries of ̺ ′ 0 (T ′ 0 ) vanish.
For any (left or right) H ′ -module M , let M W be the set of all functions φ : W → M , made into a (left or right) Mat W (H ′ )-module as follows: thinking of φ as a column or row vector, respectively.Theorem 5.1.
(2) The following diagram is commutative: varieties in Q rat which are G(R)-stable, it is reasonable to regard K sph as (part of) the (G(R) ⋊ C × )-equivariant K-theory of Q rat .Observe that K sph is generated freely as a right H-module by [O Q ].
It is well known from the theory of affine Hecke algebras that the subalgebra Z[X] W := Z[X ν+kδ : ν ∈ P, k ∈ Z] W ⊂ H 0 commutes with T i for i = 0. Thus Z[X] W acts on K sph and hence for any H = f (X) ∈ Z[X] W we can write for a unique ̺ sph 0 (f (X)) ∈ H, giving a homomorphism ̺ sph 0 : Z[X] W → H. Let π vw (A) = A vw for any W ×W -matrix A. We deduce that ̺ sph 0 (f (X)) = π ee (̺ 0 (f (X))) and π we (̺ 0 (f (X))) = 0 for all w = e.