PUSH-PULL OPERATORS ON THE FORMAL AFFINE DEMAZURE ALGEBRA AND ITS DUAL

Transcription

1 PUSH-PULL OPERATORS ON THE FORMAL AFFINE DEMAZURE ALGEBRA AND ITS DUAL BAPTISTE CALMÈS, KIRILL ZAINOULLINE, AND CHANGLONG ZHONG Contents. Introduction 2. Formal Demazure and push-pull operators 3 3. Two bases of the formal twisted group algebra 6 4. The Weyl and the Hecke actions 8 5. Push-pull operators and elements 0 6. The push-pull operators on the dual 2 7. Another basis of the W Ξ -invariant subring 4 8. The formal affine Demazure algebra 6 9. The algebraic restriction to the fixed locus 8 0. The push-pull operators on D F 9. The non-degenerate pairing on the W Ξ -invariant subring 2 References 24. Introduction In a series of papers [KK86], [KK90] Kostant and Kumar introduced and successfully applied the techniques of nil (or 0-) Hecke algebras to study equivariant cohomology and K-theory of flag varieties. In particular, they showed that the dual of the nil Hecke algebra serves as an algebraic model for the T-equivariant singular cohomology of G/B (here G is a split semisimple linear algebrac group with a chosen split maximal torus T and G/B is the variety of Borel subgroups). In [HMSZ] and [CZZ], this formalism has been generalized using an arbitrary formal group law associated to an algebraic oriented cohomology theory in the sense of Levine-Morel [LM07], via the Quillen formula. Namely, given a formal group law F and a finite root system with a set of simple roots Π, one defines the formal affine Demazure algebra D F and its dual D F provides an algebraic model for the T-equivariant oriented cohomology h T (G/B). Specializing to the additive and the multiplicative formal group laws, one recovers Chow groups (or singular cohomology) and K-theory respectively. 200 Mathematics Subject Classification. 20C08, 4F43, 57T5. The first author acknowledges the support of the French Agence Nationale de la Recherche (ANR) under reference ANR-2-BL The second author was supported by the NSERC Discovery grant , NSERC DAS grant and the Early Researcher Award (Ontario). We appreciate the support of the Fields Institute; part of this work has been done while authors were attending the Thematic Program on Torsors, Nonassociative algebras and Cohomological Invariants at the Fields Institute.

2 2 BAPTISTE CALMÈS, KIRILL ZAINOULLINE, AND CHANGLONG ZHONG AnothermotivationforstudyingthealgebraD F comesfromitscloserelationship to Hecke algebras. Indeed, for the additive (resp. multiplicative) F it coincides with the completion of the nil (resp. 0-) affine Hecke algebra (see [HMSZ]). Moreover, in section 8, we show that for some elliptic formal group law F and a root system of Dynkin type A the non-affine part of D F is isomorphic to the classical Iwahori- Hecke algebra, hence, relating it to equivariant elliptic cohomology. In the present paper we pursue the algebraization program for oriented cohomology theories started in [CPZ] and continued in [HMSZ] and [CZZ]; the general idea is to match cohomology rings of flag varieties and elements of classical interest in them (such as classes of Schubert varieties) with algebraic and combinatorial objects that can be introduced simply and algebraically, in the spirit of [De73] or [KK86]. This approach is useful to study the structure of these rings, and to perform various computations. We focus here on algebraic constructions pertaining to T-equivariant oriented cohomology groups. The precise proofs and details of how our algebraic objects match cohomology groups will be treated in a forthcoming paper; however, for the convenience of the reader, we now give a brief description of the geometric setting. Given an equivariant oriented cohomology theory h over a base field whose spectrum is denoted by pt, the formal group algebra S will correspond to h T (pt). It is an algebra over R = h(pt). The T-fixed points of G/B are naturally in bijection with the Weyl group W. This gives a pull-back to the fixed locus map h T (G/B) h T (W) w W h T(pt). This map happens to be injective. We do not know a direct geometric reason for that, but it follows from our algebraic description, in which it appears as the map D F S W w W S of Definition 9.. It is then convenient to enlarge S to its localization Q at a multiplicative subset generated by Chern class of line bundles corresponding canonically to roots, which gives injections S Q, S W Q W and SW Q W. Although we do not know good geometric interpretations of Q, Q W or Q W, all the formulas and operators we are interested in are easily defined at that localized level, because they involve denominators. The main technical difficulties then lie in proving that these operators actually restrict to S, SW, D F etc., or so to speak, that the denominators cancel out. Our central object of study is a push-pull operator on D F, which is an algebraic version of the composition h T (G/P) p h T (G/Q) p h T (G/P) of the push-forward followed by the pull-back along the quotient map p: G/P G/Q, where P Q are two parabolic subgroups of G. Again p happens to be injective, and it identifies h T (G/Q) to a subring of h T (G/P), namely the subring of invariants under the action of the parabolic subgroup W Q of the Weyl group W. Thisdoesnotseemtobestraightforwardfromthegeometryeither, anditoncemore follows from our algebraic description: given subsets Ξ Ξ of a given set of simple roots Π (each giving rise to a parabolic subgroup), we define an element Y Ξ/Ξ in Q W (see 5.3). We define an action of the Demazure algebra D F on its S-dual D F, by precomposition by multiplication on the right. The action of Y Ξ/Ξ thus defines the desired push-pull operator A Ξ/Ξ : (D F )W Ξ (D F )WΞ. The formula for the We will require that the cohomology rings are complete in some precise sense, but this is a technical point, that we prefer to hide here for simplicity.

3 PUSH-PULL OPERATORS ON THE FORMAL AFFINE DEMAZURE ALGEBRA AND ITS DUAL3 element Y Ξ/Ξ with Ξ = had already appeared in related contexts, namely, in discussions around the Becker-Gottlieb transfer for topological complex-oriented theories (see [BE90, (2.)] and [GR2, 4.]). Finally, we define the algebraic counterpart of the natural pairing h T (G/B) h T (G/B) h T (pt) obtained by multiplication and push-forward to the point. It is a pairing D F D F S. We show that it is non-degenerate, and that algebraic classes corresponding to (chosen) desingularization of Schubert varieties form a basis of D F, with a very simple dual basis with respect to the pairing. We provide the same kind of description forh T (G/P). This generalizes(to parabolic subgroups and to equivariant cohomology groups) and simplifies several statements from[cpz, 4], as well as results from [KK86] and [KK90] (to arbitrary oriented cohomology theories). The paper is organized as follows. In sections 2 and 3, we recall definitions and basic properties from [CPZ, 2,3], [HMSZ, 6] and [CZZ, 4,5]: the formal group algebra S, the Demazure and push-pull operators α and C α for every root α, the formal twisted group algebra Q W and its Demazure and push-pull elements X α and Y α. In section 4, we introduce a left Q W -action on the dual Q W. It induces both an action of the Weyl group W on Q W (the Weyl-action) and an action of X α and Y α on Q W (the Hecke-action). In sections 5 and 6, we introduce and study the more general push-pull elements in Q W and operators on Q W with respect to given coset representatives of parabolic quotients of the Weyl group. In section 7, we construct a basis of the subring of invariants of Q W, which generalizes [KK90, Lemma 2.27]. In section 8, we recall the definition and basic properties of the formal (affine) DemazurealgebraD F following[hmsz, 6],[CZZ, 5]and[Zh3]. Weshowthatfor the special elliptic formal group law, the formal Demazure algebra is related to the classical Iwahori-Hecke algebra. In section 9, we define the algebraic restriction to the fixed locusmap which is usedin section 0to restrictallourpush-pull operators andelementstod F anditsduald F aswellastorestrictthenon-degeneratepairing on D F. At last, in section, we define and discuss the non-degenerate pairing on the subring of invariants of D F under a parabolic subgroup of the Weyl group. Acknowledgments We would like to mention that one of the ingredients of this paper, the push-pull formulas in the context of Weyl group actions, arose in discussions between the first author and Victor Petrov, whose unapparent contribution we therefore gratefully acknowledge. 2. Formal Demazure and push-pull operators In this section we recall definitions of the formal group algebra and of the formal Demazure and push-pull operators, following [CPZ] and [CZZ]. Let R be a commutative ring with unit, and let F be a one-dimensional commutative formal group law (FGL) over R, i.e. F(x,y) R[[x,y]] satisfies F(x,0) = 0, F(x,y) = F(y,x), F(x,F(y,z)) = F(F(x,y),z). Example 2.. Theadditive FGLisdefined byf a (x,y) = x+y, andamultiplicative FGL is defined by F m (x,y) = x+y βxy with β R. The coefficient ring of the universal FGL is generated by the coefficients a ij modulo relations induced by the above properties and is called the Lazard ring.

4 4 BAPTISTE CALMÈS, KIRILL ZAINOULLINE, AND CHANGLONG ZHONG Example 2.2. Consider an elliptic curve given in Tate coordinates by ( µ t µ 2 t 2 )s = t 3. The corresponding FGL over the coefficient ring R = Z[µ,µ 2 ] is given by, e.g. see [BB0, Example 63], F e (x,y) := x+y µxy +µ 2xy and will be called a special elliptic FGL. Observe that F e (x,y) = x+y xy(µ +µ 2 F e (x,y)), and thus that the formal inverse of F e is identical to the one of F m, i.e. F e (x,x) = 2x µx2 +µ 2x 2. x µ x, and Let Λ be an Abelian group and let R[[x Λ ]] be the ring of formal power series with variables x λ for all λ Λ. Define the formal group algebra R[[Λ]] F to be the quotient of R[[x Λ ]] by the closure of the ideal generated by elements x 0 and x λ+λ 2 F(x λ,x λ2 ) for any λ,λ 2 Λ. Here 0 is the identity element in Λ. Let I F denote the kernel of the augmentation map ǫ: R[[Λ]] F R, 0. Assume that Λ is a free Abelian group of finite rank and let Σ be a finite subset of Λ. A root datum is an embedding Σ Λ, α α into the dual of Λ satisfying certain conditions [SGA, Exp. XXI, Def...]. The rank of the root datum is the Q-rank of Λ Z Q. The root lattice Λ r is the subgroup of Λ generated by Σ, and the weight lattice Λ w is the Abelian group defined by Λ w := {ω Λ Z Q α (ω) Z for all α Σ}. We always assume that the root datum is reduced and semisimple (Q-ranks of Λ r, Λ w and Λ arethe sameand no root is twice another one). We saythat a rootdatum is simply connected (resp. adjoint) if Λ = Λ w (resp. Λ = Λ r ), and then use the notation Dn sc (resp. Dad n ) for irreducible root data where D = A,B,C,D,E,F,G is one of the Dynkin types and n is the rank. The Weyl group W of a root datum (Λ,Σ) is a subgroup of Aut Z (Λ) generated by simple reflections s α for all α Σ defined by s α (λ) := λ α (λ)α, λ Λ. We fix a set of simple roots Π = {α,...,α n } Σ that is a basis of the root datum: each element of Σ is an integral linear combination of simple roots with either all positive coefficients or negative. This partitions Σ into Σ + and Σ the subsets of positive roots, resp. negative roots. Let l denote the length function on W with respect to the set of simple roots Π. Let w 0 be the longest element of W with respect to l and let N := l(w 0 ). Following [CZZ, Def. 4.4] we say that the formal group algebra R[[Λ]] F is Σ- regular if is not a zero-divisor in R[[Λ]] F for all roots α Σ. We will always assume that: The formal group algebra R[[Λ]] F is Σ-regular. By [CZZ, Lemma 2.2] this holds if 2 is not a zero-divisor in R, or if the root datum does not contain any C sc as an irreducible component.

5 PUSH-PULL OPERATORS ON THE FORMAL AFFINE DEMAZURE ALGEBRA AND ITS DUAL5 Following [CPZ, Def. 3.5 and 3.2] for each α Σ we define two R-linear operators α and C α on R[[Λ]] F as follows: (2.) α (y) := y sα(y), C α (y) := κ α y α (y) = y x α + sα(y), y R[[Λ]] F, where κ α := + x α (note that κ α R[[Λ]] F ). The operator α is called the Demazure operator and the operator C α is called the push-pull operator or the BGG operator. Example 2.3. For the special elliptic formal group law F e we have κ α = µ + µ 2 F e (x α, ) = µ for each α Σ. If the root datum is of type A sc, we have Σ = {±α}, Λ = ω with simple root α = 2ω and C α ( ) = xα x α + x α = µ + µ, C α(x ω ) = xω x α + x ω = µ x ω +µ2x2 ω µ x ω. If it is of type A sc 2 we have Σ = {±α,±α 2,±(α +α 2 )}, Λ = ω,ω 2 with simple roots α = 2ω ω 2, α 2 = 2ω 2 ω and = 2x µx2 x2 µ2x2 x2 +µ 2x 2 µx2 2µ2xx2, C α2 (x ) = µ x, C α (x ) = µ x +µ2x2 µx2 2µ2xx2 µ x µ 2x x 2, where x := x ω and x 2 := x ω2. According to [CPZ, 3] the operators α satisfy the twisted Leibniz rule (2.2) α (xy) = α (x)y +s α (x) α (y), x,y R[[Λ]] F, i.e. α is a twisted derivation. Moreover, they are R[[Λ]] Wα F -linear, where W α = {e,s α }, and (2.3) s α (x) = x if and only if α (x) = 0. Remark 2.4. Properties (2.2) and (2.3) suggest that the Demazure operators can be effectively studied using the theory of twisted derivations and the invariant theory of W. On the other hand, push-pull operatorsdo not satisfy properties (2.2) and (2.3) but according to [CPZ, Theorem 2.4] they correspond to the push-pull maps between flag varieties and, hence, are of geometric origin. For the i-th simple root α i, let i := αi and s i := s αi. Given a sequence I = (i,...,i m ) with i j {,...,n}, denote I = m and define I := i im, C I := C i C im and set = C = id. We say that a sequence I is reduced in W if s i s i2...s im is a reduced expression in W. In this case we also say that I = I w is a reduced sequence of the element w(i) := s i s i2...s im. We set I e = (here e is the identity element of W). Remark 2.5. It is well-known that for a nontrivial root datum, all w W, Iw and C Iw are independent of the choice of a reduced sequence I w of w W if and only if F is ofthe form F(x,y) = x+y+βxy, β R. The if part ofthe statement is due to Demazure [De73, Th. ] and the only if part to Bressler and Evens [BE90, Theorem 3.7]. So for such F we can define w := Iw and C w := C Iw for each w W. The operators w and C w play a crucial role in Schubert calculus and computations of the singular cohomology (F = F a ) and the K-theory (F = F m ) rings of flag varieties. For a general F, e.g. for F = F e, the situation becomes much more intricate as we have to rely on choices of reduced decomposition I w.

6 6 BAPTISTE CALMÈS, KIRILL ZAINOULLINE, AND CHANGLONG ZHONG 3. Two bases of the formal twisted group algebra We now recall definitions and basic properties of the formal twisted group algebra Q W, Demazure elements X α and push-pull elements Y α, following [HMSZ] and [CZZ]. For a chosen set of reduced sequences {I w } w W we introduce two bases {X Iw } and {Y Iw } of Q W and describe the matrices (a X v,w ) and (ay v,w ) by expressing them on the canonical basis {δ w } of Q W. We also relate the coefficients a X v,w and the corresponding coefficients a X v,w of the reversed elements X Iw rev. For simplicity we write S := R[[Λ]] F. Since the formal group algebra S is Σ- regular, it embeds into the localization Q = S[ α Σ]. Let Q W be the twisted group algebra of Q and of the group ring R[W] over R, i.e. Q W = Q R R[W] as an R-module and the product in Q W is given by (q δ w )(q δ w ) = qw(q ) δ ww, q,q Q, w,w W. where δ w is the canonical element corresponding to w in R[W]. Note that Q W is not a Q-algebra since the embedding Q Q W,q q δ e is not central. Inside Q W, we use the notation q := q δ e and δ w := δ w, := δ e and δ α := δ sα for a root α Σ. Thus qδ w = q δ w and δ w q = w(q) δ w. Clearly, {δ w } w W is a basis of Q W as a left Q-module. For each α Σ we define the following elements of Q W (corresponding to the operators α and C α, respectively, by the action of (4.3)): X α := δ α, Y α := κ α X α = x α + δ α called the Demazure elements and the push-pull elements, respectively. By straightforward computations, for each α Σ we have (3.) X 2 α = κ αx α = X α κ α and Y 2 α = κ αy α = Y α κ α, X α q = s α (q)x α + α (q) and Y α q = s α (q)y α + α (q), q Q, X α Y α = Y α X α = 0. Let δ i := δ si, X i := X αi and Y i := Y αi be the i-th simple root α i. Given a sequence I = (i,i 2,...,i m ) with i j {,...,n}, let the product X i X i2...x im be denoted by X I and Y i Y i2...y im by Y I. Set X = Y =. By [Bo68, Ch. VI,, No 6, Cor. 2] if v W has a reduced decomposition v = s i s i2 s im, then (3.2) vσ Σ + = {α i,s i (α i2 ),...,s i (s i2 ( s im (α im ) ))}. In particular, s i Σ Σ + = {α i }. Lemma 3.. Let I v be a reduced sequence of an element v W. Then X Iv = w v ax v,wδ w for some a X v,w Q, where the sum is taken over all elements of W less or equal to v with respect to the Bruhat order and a X v,v = ( ) l(v) α vσ Σ + x α. Moreover, we have δ v = w v bx v,wx Iw for some b X v,w S such that b X v,e = and b X v,v = ( )l(v) α vσ Σ x + α. Proof. It follows from [CZZ, Lemma 5.4, Corollary 5.6] and the fact that δ α = X α. Similar for Y I s we have

7 PUSH-PULL OPERATORS ON THE FORMAL AFFINE DEMAZURE ALGEBRA AND ITS DUAL7 Lemma 3.2. Let I v be a reduced sequence for an element v W. Then Y Iv = w v ay v,wδ w for some a Y v,w Q and a Y v,v = ( ) l(w) a X v,v. Moreover, we have δ v = w v by v,w Y I w for some b Y v,w S and by v,v = ( )l(v) b X v,v. Proof. We follow the proof of [CZZ, Lemma 5.4] replacing X by Y. By induction we have Y Iv = (x β +x β δ β) w v a Y v,wδ w = x β s β(a Y v,v )δ v + w<va Y v,wδ w, where I v = (i,...,i m ) is a reduced sequence of v, β = α i and v = s β v. This implies the formulas for Y Iv and for a Y v,v. Remaining statements involving by v,w follow by the same arguments as in the proof of [CZZ, Corollary 5.6] using the fact that δ α = Y α xα x α and xα x α S. As in the proof of [CZZ, Corollary 5.6], Lemmas 3. and 3.2 immediately imply: Corollary 3.3. The elements {X Iv } v W (resp. {Y Iv } v W ) form a basis of Q W as a left Q-module. Example 3.4. For the root data A ad or A sc and the formal group law F e we have x Π = x α and ( ) (a Y v,w ) 0 v,w W = µ where the first row and column correspond to e W and the second to s α W. Given a sequence I = (i,...,i m ), let I rev := (i m,...,i ). We set x Π := α Σ S. Observe that by (3.2) we have s α (x Π )x α = x Π for α Σ +. Lemma 3.5. Let I = (i,...,i m ) be a sequence in {,...,n}. Let X I = v W a X I,v δ v and X I rev = v W a X I,v δ v for some a X I,v, a X I,v Q, then v(x Π )a X I,v = v(ax I,v )x Π. Similarly, let Y I = v W a Y I,vδ v and Y I rev = v W a Y I,vδ v for some a Y I,v, a Y I,v Q, then v(x Π )a Y I,v = v(ay I,v )x Π. Proof. If I = (i), then Y I = Y I rev. So a I,v = a I,v for all v W (we omit the superscripts Y ). For v / {e,s i } we have a I,v = a I,v = 0. For v = e we have x Π a I,e = x Πa I,e. For v = s i we have s i (x Π )a I,s i = x Π s i (a I,si ), since a I,si = i. The proof is analogous for X i. Now the conclusion immediately follows by induction on the length I using Lemma 3.6 below with x = x Π, f = Y J, f = Y J rev, g = Y K and g = Y K rev for any splitting of I in smaller sequences J followed by K.

8 8 BAPTISTE CALMÈS, KIRILL ZAINOULLINE, AND CHANGLONG ZHONG Lemma 3.6. Given x Q, assume that elements f = v W a v δ v and f = v W a v δ v with a v,a v Q g = v W b v δ v and g = v W b v δ v with b v,b v Q of Q W satisfy v(x)a v = v(a v )x and v(x)b v = v(b v )x for all v W. Then the product fg = c v δ v and the element (fg) := g f = c vδ v bear the same relation: v(x)c v = v(c v )x. Proof. By definition of the product, we have c v = b v v (a v 2 ) and c v = v v 2=v v v 2=v a v v2 (b 2 v ). We then compute (all sums still being over pairs (v,v 2 ) W 2 such that v v 2 = v) v(c v )x = ( v v 2 av v 2 = v v 2 (a v 2 2 (b v ) ) x = v v 2 (a v 2 )v (b v )x )v (x)b v = ( v v2 (a v )x ) b v 2 = ( v v2 (x)a ) v 2 b v = v(x) b v v (a v 2 ) = v(x)c v. 4. The Weyl and the Hecke actions In the present section we recall several basic facts concerning the Q-linear dual Q W following [HMSZ] and [CZZ]. We introduce a left Q W-action on Q W. The latter induces an action of the Weyl group W on Q W (the Weyl-action) and the action by means of X α and Y α on Q W (the Hecke-action). These two actions will play an important role in the sequel. Let Q W := Hom Q(Q W,Q) denote the Q-linear dual of the left Q-module Q W. By definition, Q W is a left Q-module via (qf)(z) := qf(z) for any z Q W, f Q W andq Q. Moreover,thereisaQ-basis{f w } w W ofq W dualtothebasis{δ w} w W defined by f w (δ v ) := δ w,v (the Kronecker symbol), w,v W. Definition 4.. We define a left action of Q W on Q W as follows: (z f)(z ) := f(z z), z,z Q W, f Q W. By definition, this action is left Q-linear, i.e. z (qf) = q(z f) and it induces a different left Q-module structure on Q W via the embedding q qδ e, i.e. (q f)(z) := f(zq). It also induces a Q-linear action of W on Q W via w(f) := δ w f. Lemma 4.2. We have q f w = w(q)f w and w(f v ) = f vw for any q Q and w,v W. Proof. We have (q f w )(δ v ) = f w (v(q)δ v ) = v(q)δ w,v which shows that q f v = v(q)f v. As for the second we have [w(f v )](δ u ) = f v (δ u δ w ) = δ v,uw, so w(f v ) = f vw.

9 PUSH-PULL OPERATORS ON THE FORMAL AFFINE DEMAZURE ALGEBRA AND ITS DUAL9 There is a coproduct on Q W defined by [CZZ, Def. 8.9]: : Q W Q W Q Q W, qδ w qδ w δ w. Here Q is the tensor product of left Q-modules. It is cocommutative with co-unit ε : Q W Q, qδ w q [CZZ, Prop. 8.0]. The coproduct structure on Q W induces a product structure on Q W, which is Q-bilinear for the natural action of Q on Q W (not the one using ). In terms of the basis {f w } w W the product is given by component-wise multiplication: (4.) ( v W q v f v )( w W q w f w) = w W q w q w f w, q w,q w Q. In other words, if we identify the dual Q W with the Q-module of maps Hom(W,Q) via Q W Hom(W,Q), f f, f (w) := f(δ w ), then the product is the classical multiplication of functions with values in a ring. The multiplicative identity of this product corresponds to the counit ε and equals = w W f w. We also have (4.2) q (ff ) = (q f)f = f(q f ) for q Q and f,f Q W. Lemma 4.3. For any α Σ and f,f Q W we have s α(ff ) = s α (f)s α (f ), i.e. W acts on the algebra Q W by Q-linear automorphisms. Proof. By Q-linearity of the action of W and of the product, it suffices to check the formula on basis elements f = f w and f = f v, for which it is straightforward. Observe that the ring Q can be viewed as a left Q W -module via the following action: (4.3) (qδ w ) q := qw(q ), q,q Q, w W. Then by definition we have (4.4) (q )(z) = z q, z Q W. Definition 4.4. For α Σ we define two Q-linear operators on Q W by A α (f) := Y α f and B α (f) := X α f, f Q W. An action by means of A α or B α will be called a Hecke-action on Q W. Remark 4.5. If F = F m (resp. F = F a ) one obtains actions introduced by Kostant Kumar in [KK90, I 8 ] (resp. in [KK86, I 5 ]). As in (2.2) and (2.3) we have (4.5) B α (ff ) = B α (f)f +s α (f)b α (f ) and B α s α = B α, for f,f Q W, (4.6) B α (f) = 0 if and only if f (Q W) Wα. Indeed, using (4.2) and Lemma 4.3 we obtain B α (f)f +s α (f)b α (f ) = [ ( δ α ) f]f +s α (f)[ ( δ α ) f ] = [ (f s α (f))]f +s α (f)[ (f s α (f ))] = (ff s α (f)s α (f )) = B α (ff ) and B α (s α (f)) = ( δ α ) s α (f) = (s α (f) f) = B α (f). As for (4.6) we have 0 = B α (f) = X α f = [( δ α ) f] which is equivalent to f = s α (f).

10 0 BAPTISTE CALMÈS, KIRILL ZAINOULLINE, AND CHANGLONG ZHONG And similarly to (3.) we obtain (4.7) A 2 α (f) = κ α A α (f) = A α (κ α f), B 2 α (f) = κ α B α (f) = B α (κ α f), A α B α = B α A α = 0. We set A i = A αi and B i := B αi for the i-th simple root α i. We set A I = A i... A im and B I = B i... B im for a sequence I = (i,...,i m ) with i j {,...,n}. The operators A I and B I are key ingredients in the proof that the natural pairing of Theorem 0.7 on the dual of the formal affine Demazure algebra is non-degenerate. Lemma 4.6. For any sequence I, we have A I rev(x Π f e ) = v W v(x Π )a Y I,vf v and B I rev(x Π f e ) = v W v(x Π )a X I,vf v. Proof. We prove the first formula only. The second one is obtained using similar arguments. Let Y I rev = v W a Y I,v δ v and Y I = v W ay I,v δ v as in Lemma 3.5. Then A I rev(x Π f e ) = Y I rev x Π f e = x Π (a Y I,v δ v f e ) = v W v W x Π (a Y I,v f v ) = x Π v (a Y I,v )f v = x Π v(a Y I,v )f v. v W v W The formula then follows by Lemma Push-pull operators and elements Let us now introduce and study a key notion of the present paper, the notion of push-pull operators (resp. elements) on Q (resp. in Q W ) with respect to given coset representatives in parabolic quotients of the Weyl group. Let (Σ,Λ) be a root datum with a chosen set of simple roots Π. Let Ξ Π and let W Ξ denote the subgroup of the Weyl group W of the root datum generated by simple reflections s α, α Ξ. We thus have W = {e} and W Π = W. Let Σ Ξ := {α Σ s α W Ξ } and let Σ + Ξ := Σ Ξ Σ +, Σ Ξ := Σ Ξ Σ be subsets of positive and negative roots respectively. Given subsets Ξ Ξ of Π, let Σ + Ξ/Ξ := Σ + Ξ \Σ+ Ξ and Σ Ξ/Ξ := Σ Ξ \Σ Ξ. We define x Ξ/Ξ := and set x Ξ := x Ξ/. α Σ Ξ/Ξ Lemma 5.. Given subsets Ξ Ξ of Π we have v(σ Ξ/Ξ ) = Σ Ξ/Ξ and v(σ + Ξ/Ξ ) = Σ + Ξ/Ξ for any v W Ξ. Proof. We prove the first statement only, the second one can be proven similarly. Since v acts faithfully on Σ Ξ, it suffices to show that for any α Σ Ξ/Ξ, the root β := v(α) Σ Ξ and is negative. Indeed, if β Σ Ξ, then so is α = v (β) (as v W Ξ ), which is impossible. On the other hand, if β is positive, then β = v(α) vσ Ξ Σ+ Ξ = vσ Ξ Σ+ Ξ, where the latter equality follows from (3.2) and the fact that v W Ξ. So α = v (β) Σ Ξ, a contradiction.

11 PUSH-PULL OPERATORS ON THE FORMAL AFFINE DEMAZURE ALGEBRA AND ITS DUAL Corollary 5.2. For any v W Ξ, we have v(x Ξ/Ξ ) = x Ξ/Ξ. Definition 5.3. Given a set of left coset representatives W Ξ/Ξ of W Ξ /W Ξ we define a push-pull operator on Q with respect to W Ξ/Ξ by C Ξ/Ξ (q) := w ( q x Ξ/Ξ ). and a push-pull element with respect to W Ξ/Ξ by Y Ξ/Ξ := ( ) δ w x. Ξ/Ξ We set C Ξ := C Ξ/ and Y Ξ := Y Ξ/ (so they do not depend on the choice of W Ξ/ = W Ξ in these two special cases). By definition, we have C Ξ/Ξ (q) = Y Ξ/Ξ q, where Y Ξ/Ξ acts on q Q by (4.3). Also in the trivial case where Ξ = Ξ, then x Ξ/Ξ =, while C Ξ/Ξ = id Q and Y Ξ/Ξ = if we choose e as representative of the only coset. Observe that for Ξ = {α i } we have W Ξ = {e,s i } and C Ξ = C i (resp. Y Ξ = Y i ) is the push-pull operator (resp. element) introduced before and preserves S. Example 5.4. For the formal group law F e and the root datum A 2, we have x Π = x α x α2 x α α 2 and C Π () = w W w( x Π ) = µ ( x α2 x α α 2 + x α 2 + +α 2 ) = µ 3 +µ µ 2. Lemma 5.5. The operator C Ξ/Ξ restricted to Q W Ξ is independent of the choices of representatives W Ξ/Ξ and it maps Q W Ξ to Q WΞ. Proof. Since /x Ξ/Ξ Q W Ξ by corollary 5.2, the independence statement is clear. The second part follows, since for any v W Ξ, and for any set of coset representatives W Ξ/Ξ, the set vw Ξ/Ξ is again a set of coset representatives. Actually, we will see in Corollary 0.4 that the operator C Ξ sends S to S WΞ. Remark 5.6. The formula for the operator C Ξ (with Ξ = ) had appeared before in related contexts, namely, in discussions around the Becker-Gottlieb transfer for topological complex-oriented theories (see [BE90, (2.)] and [GR2, 4.]). The definition of the element Y Ξ/Ξ can be viewed as a generalized algebraic analogue of this formula. Lemma 5.7 (Composition rule). Given subsets Ξ Ξ Ξ of Π and given sets of representatives W Ξ/Ξ and W Ξ /Ξ, take W Ξ/Ξ := {wv w W Ξ/Ξ, v W Ξ /Ξ } as the set of representatives of W Ξ /W Ξ. Then C Ξ/Ξ C Ξ /Ξ = C Ξ/Ξ and Y Ξ/Ξ Y Ξ /Ξ = Y Ξ/Ξ. Proof. We prove the formula for Y s, the one for C s follows since C acts as Y, and the composition of actions corresponds to multiplication. We have Y Ξ/Ξ Y Ξ /Ξ = ( δ w x )( δ Ξ/Ξ v x ) = δ Ξ /Ξ wv v (x Ξ/Ξ )x. Ξ /Ξ v W Ξ /Ξ,v W Ξ /Ξ By Corollary 5.2, we have v (x Ξ/Ξ ) = x Ξ/Ξ. Therefore, v (x Ξ/Ξ )x Ξ /Ξ = x Ξ/Ξ x Ξ /Ξ = x Ξ/Ξ. We conclude by definition of W Ξ/Ξ.

12 2 BAPTISTE CALMÈS, KIRILL ZAINOULLINE, AND CHANGLONG ZHONG The following lemma is clear from the defining formula of C Ξ/Ξ. Lemma 5.8 (Projection formula). We have C Ξ/Ξ (qq ) = qc Ξ/Ξ (q ) for any q (Q W ) WΞ and q (Q W ) W Ξ. Lemma 5.9. Given a subset Ξ of Π and α Ξ we have (a) Y Ξ = Y Y α = Y α Y for some Y and Y Q W, (b) Y Ξ X α = X α Y Ξ = 0, Y α Y Ξ = κ α Y Ξ and Y Ξ Y α = Y Ξ κ α. Proof. (a) The first identity follows from Lemma 5.7 applied to Ξ = {α} (in this case Y = Y Ξ/Ξ ). For the second identity, let α W Ξ be set of right coset representatives of W α \W Ξ, thus each w W Ξ can be written uniquely either as w = s α u or as w = u with u α W Ξ. Then Y Ξ = (+δ α )δ u x Ξ = (+δ α ) x α x α δ u x Ξ u α W Ξ u α W Ξ = Y α x α δ u x Ξ = Y α u α W Ξ (b) then follows from (a) and (3.). u α W Ξ δ w w (x α) x Ξ. 6. The push-pull operators on the dual We now introduce and study the push-pull operators on the dual of the twisted formal group algebra Q W. For w W, we define fw Ξ := v ww Ξ f v. Observe that fw Ξ = fw Ξ if and only if ww Ξ = w W Ξ. Consider the subring of invariants (Q W by means of the )WΞ -action of W Ξ on Q W and fix a set of representatives W Π/Ξ. By Lemma 4.2, we then have the following lemma: Lemma 6.. The set {f Ξ w} w WΠ/Ξ forms a basis of (Q W )WΞ as a left Q-module, and f Ξ wf Ξ v = δ w,v f Ξ v for any w,v W Π/Ξ. In other words, {f w } w WΠ/Ξ is a set of pairwise orthogonal projectors, and the direct sum of their images is (Q W )WΞ. Definition 6.2. Given subsets Ξ Ξ of Π and a set of representatives W Ξ/Ξ we define a Q-linear operator on Q W by A Ξ/Ξ (f) := Y Ξ/Ξ f, f Q W, and call it the push-pull operator with respect to W Ξ/Ξ. It is Q-linear since so is the -action. We set A Ξ = A Ξ/. Lemma 5.7 immediately implies: Lemma 6.3 (Composition rule). Given subsets Ξ Ξ Ξ of Π and sets of representatives W Ξ/Ξ and W Ξ /Ξ, let W Ξ/Ξ = {wv w W Ξ/Ξ, v W Ξ /Ξ }, then we have A Ξ/Ξ A Ξ /Ξ = A Ξ/Ξ. Here is an analogue of Lemma 5.5. Lemma 6.4. The operator A Ξ/Ξ restricted to (Q W )W Ξ is independent of the choices of representatives W Ξ/Ξ and it maps (Q W )W Ξ to (Q W )WΞ.

13 PUSH-PULL OPERATORS ON THE FORMAL AFFINE DEMAZURE ALGEBRA AND ITS DUAL 3 Proof. Let f (Q W )W Ξ. For any w W and v W Ξ, by Corollary 5.2, we have ( ) ( ) ( ) δwv f = δw δ v f = δw δv f = ( ) δ w f. x Ξ/Ξ x Ξ/Ξ x Ξ/Ξ x Ξ/Ξ which proves that the action on f of any factor δ w ( x ) in Y Ξ/Ξ Ξ/Ξ is independent of the choice of the coset representative w. Now if v W Ξ, we have v(a Ξ/Ξ (f)) = δ v Y Ξ/Ξ f = (δ v Y Ξ/Ξ ) f = A Ξ/Ξ (f) where the last equalityholds since δ v Y Ξ/Ξ is again an operatory Ξ/Ξ corresponding to the set of coset representatives vw Ξ/Ξ instead of W Ξ/Ξ. This proves the second claim. Lemma 6.5. We have A Ξ/Ξ (f v ) = v(x Ξ/Ξ ) f vw = v(x Ξ/Ξ ) w(f v ). In particular, we have A Ξ/Ξ (f Ξ v ) = v(x Ξ/Ξ ) fξ v and A Π/Ξ(f Ξ v ) = v(x Π/Ξ ). Proof. By Lemma 4.2 we obtain A Ξ/Ξ (f v ) = = ( ) fv = δ w x Ξ/Ξ In particular A Ξ/Ξ (fv Ξ ) = w W Ξ vw(x Ξ/Ξ ) u W Ξ/Ξ δ w ( v(x Ξ/Ξ ) f v f vwu = v(x Ξ/Ξ ) = v(x Ξ/Ξ ) where the second equality follows from Corollary 5.2. Together with Lemma 6. we therefore obtain: Corollary 6.6. We have A Ξ/Ξ ((Q W )W Ξ ) = (Q W )WΞ. Lemma 6.7 (Projection formula). We have ) = v(x Ξ/Ξ ) w W Ξ u W Ξ/Ξ f vw. f vwu w vw Ξ f w = v(x Ξ/Ξ ) fξ v A Ξ/Ξ (ff ) = fa Ξ/Ξ (f ) for any f (Q W )WΞ and f (Q W )W Ξ. Proof. Using (4.2) and Lemma 4.3, we compute A Ξ/Ξ (ff ) = Y Ξ/Ξ (ff ) = ( ) (ff ) = = = f δ w ( f( x Ξ/Ξ f ) ) = δ w x Ξ/Ξ δ w x Ξ/Ξ f = fa Ξ/Ξ (f ) (δ w f)(δ w x Ξ/Ξ f ) δ w x Ξ/Ξ (ff ) Lemma 6.8. Given a sequence I in {,...,n}, for any x,y S and f,f Q W we have C Π ( I (x)y) = C Π (x I rev(y)) and A Π (B I (f)f ) = A Π (fb I rev(f )).

14 4 BAPTISTE CALMÈS, KIRILL ZAINOULLINE, AND CHANGLONG ZHONG Proof. We prove the second formula only. The first one is obtained similarly. By Lemma 5.9.(b) we have Y Π X α = 0 for any α Π. By (4.5) we obtain 0 = A Π ( Bα (s α (f)f ) ) = A Π ( fbα (f ) B α (f)f ). Hence, A Π (B α (f)f ) = A Π (fb α (f )). The formula then follows by iteration. Let {XI w } w W and {YI w } w W be the Q-linear basis of Q W dual to {X I w } w W and {Y Iw } w W, respectively, i.e. XI w (X Iv ) = δ w,v for w,v W. We have XI e =. Indeed, from Lemma 3., we have δ v = w v bx v,w X I w with b X v,e =. So for each v W we have XI e (δ v ) = b X v,e = = (δ v). Lemma 6.9. Let w 0 be the longest element in W, of length N. We have A Π (X I w0 ) = ( ) N and A Π (Y I w0 ) =. Proof. Consider the first formula. By Lemma 3. δ v = w v bx v,w X I w, therefore X I w = v w bx v,w f v. Lemma 6.5 yields A Π (X I w ) = v w b X v,w v(x Π). If w = w 0 is the longest element, then A Π (XI w0 ) = bx w 0,w 0 w 0(x Π). By Lemma 3. we have b X w 0,w 0 = ( ) N α Σ x + α which by (3.2) equals to ( ) N w 0 (x Π ). The second formula is obtained similarly using Lemma 3.2 instead. Definition 6.0. We define the characteristic map c: Q Q W by q q. By the definition of the action, c is an R-algebra homomorphism given by c(q) = w W w(q)f w. Note that c is Q W -equivariant with respect to the action (4.3) and the -action. Indeed, c(z q) = (z q) = z (q ) = z c(q). In particular, it is W-equivariant. The following lemma provides an analogue of the push-pull formula of [CPZ, Theorem. 2.4]. Lemma 6.. Given subsets Ξ Ξ of Π, we have A Ξ/Ξ c = c C Ξ/Ξ. Proof. By definition, we have A Ξ/Ξ (c(q)) = Y Ξ/Ξ c(q) = c(y Ξ/Ξ q) = c(c Ξ/Ξ (q)). 7. Another basis of the W Ξ -invariant subring Recall that {f Ξ w} w WΠ/Ξ is a basis of the invariant subring (Q W )WΞ. In the present section we construct another basis {X I u } u W Ξ of the subring (Q W )WΞ, which generalizes [KK90, Lemma 2.27]. Given a subset Ξ of Π we define W Ξ = {w W l(ws α ) > l(w) for any α Ξ}. Note that W Ξ is a set of left coset representativesof W/W Ξ such that each w W Ξ is the unique representative of minimal length. We will extensively use the following fact [Hu90,.0]: (7.) For any w W there exist unique u W Ξ and v W Ξ such that w = uv and l(w) = l(u)+l(v).

15 PUSH-PULL OPERATORS ON THE FORMAL AFFINE DEMAZURE ALGEBRA AND ITS DUAL 5 Definition 7.. Let Ξ be a subset of Π. We say that the set of reduced sequences {I w } w W is Ξ-compatible if for each w W and the unique factorization w = uv with u W Ξ and v W Ξ, l(w) = l(u)+l(v) of (7.) we have I w = I u I v, i.e. I w starts with I u and ends by I v. Observe that there always exists a Ξ-compatible set of reduced sequences. Indeed, we can take any reduced sequence for w W Ξ W Ξ and then define I w := I u I v for w = uv with u W Ξ and v W Ξ. Lemma 7.2. For any z Q W, we have X I e (z) = z, where e is the neutral element of W and z is defined in (4.3). In particular, for any sequence I with I, the coefficient of X I e (X I ) = 0. Proof. Since X α = 0, for any sequence I with I, we have X I = 0, and of course X Ie = =. Thus z = ( w W X I w (z)x Iw ) = X I e (z). It follows that if I and we express X I = v W q vx Iv, then q e = 0. Lemma 7.3. For any reduced sequence I of an element w and q Q we have X I q = v wφ I,v (q)x Iv for some φ I,Iv (q) Q. Proof. For any subsequence J of I (not necessarily reduced), we have w(j) w by [De77, Th..]. Thus, by developing all X i = i ( δ αi ), moving all coefficients to the left, and then using Lemma 3. and transitivity of the Bruhat order, X I q = w v φ I,w (q)δ w = w v φ I,w (q)x Iw for some coefficients φ I,w (q) and φ I,w (q) Q. Theorem 7.4. Assume that the set of reduced sequences {I w } w W is Ξ-compatible. For any u W Ξ, and for any sequence I of length at least and in W Ξ (i.e. α i Ξ for any i appearing in the sequence I), we have X I u (zx I ) = 0 for all z Q W. Proof. Since {X Iw } w W is a basis of Q W, we may assume that z = X Iw for some w W. We proceed by induction on the length of w. First decompose X I = e<v W Ξ q v X Iv with q v Q by Lemma 7.2. When l(w) = 0, we havex Iw = X Ie =. Since W Ξ W Ξ = {e}, for any v W Ξ, v e, we have XI u (X Iv ) = 0 so this case is clear. Then, the induction step goes as follows: since the sequences are Ξ-compatible, we have X Iw X I = X Iw X Iv X I = X Iw X I with w W Ξ, v W ξ and I W Ξ with l(i ) l(i). We can thus assume that w W Ξ. Then, by Lemma 7.3, X Iw X I = X Iw q v X Iv = φ Iw,w (q v )X Iw X Iv. v e w w,v e Now X I u (X Iw X Iv ) = X I u (X Iwv ) = 0 since wv is not minimal (v e) so wv u. Applying X I u to other terms in the above summation gives zero by induction.

16 6 BAPTISTE CALMÈS, KIRILL ZAINOULLINE, AND CHANGLONG ZHONG Remark 7.5. The proof will not work if we replace X s by Y s, because constant terms appear. Corollary 7.6. Assume that the set of reduced sequences {I w } w W is Ξ-compatible. The elements {X I u } u W Ξ form a Q-module basis of (Q W )WΞ. Proof. For every α i Ξ we have (δ i X I u )(z) = X I u (zδ i ) = X I u (z( x i X i )) = X I u (z), z Q W, where the last equality follows by Theorem 7.4. Therefore, X I u is W Ξ -invariant. Let σ (Q W )WΞ, i.e. for each α i Ξ we have σ = s i (σ) = δ i σ. Then σ(zx i ) = σ(z i ( δ αi )) = σ(z i ) (δ i σ)(z x i ) = (σ δ i σ)(z x i ) = 0 for any z Q W. Write σ = w W c wx I w for some c w S. If w / W Ξ, then I w ends by some i such that α i Ξ which implies that c w = σ(x Iw ) = σ(x Iw\iX i ) = 0, where I w \i is the sequence obtained by deleting the last i in I w. So σ is a linear combination of {XI u } u W Ξ. Corollary 7.7. Assume that the set of reduced sequences {I w } w W is Ξ-compatible. Then we have b X wv,u = b X w,u for any v W Ξ, u W Ξ and w W, where b X wv,u are the coefficients of Lemma 3.. Proof. From Lemma 3. we have XI u = w u bx w,uf w. By Lemma 4.2 we obtain that v(xi u ) = w u bx w,u f wv for any v W Ξ. Since XI u is W Ξ -invariant by Corollary 7.7 and {f w } is a basis of Q W, this implies that bx wv,u = bx w,u. 8. The formal affine Demazure algebra In the present section we recall the definition and basic properties of the formal (affine) Demazure algebra D F following [HMSZ], [CZZ] and [Zh3]. We show that for the special elliptic formal group law F e, the formal Demazure algebra is related to the classical Iwahori-Hecke algebra. Following [HMSZ], we define the formal affine Demazure algebra D F to be the R-subalgebra of the twisted formal group algebra Q W generated by elements of S and the Demazure elements X i for all i {,...,n}. By [CZZ, Lemma 5.8], D F is also generated by S and all X α for all α Σ. Since κ α S, the algebra D F is also generated by Y α s and elements of S. Finally, since δ α = X α, all elements δ w are in D F, and D F is a sub-s W -module of Q W, both on the left and on the right. Remark 8.. Since {X Iw } w W is a Q-linear basis of Q W, restricting the action (4.3) of Q W onto D F we obtain an isomorphism between the algebra D F and the R-subalgebra D(Λ) F of End R (S) generated by operators α (resp. C α ) for all α Σ, and multiplications by elements from S. This isomorphism maps X α α and Y α C α. Therefore, for any identity or statement involving elements X α or Y α there is an equivalent identity or statement involving operators α or C α. According to [HMSZ, Theorem 6.4] (or [CZZ, 7.9] when the ring R is not necessarily a domain), in type A n, the algebra D F is generated by the Demazure elements X i, i {,...,n}, and multiplications by elements from S subject to the folowing relations:

17 PUSH-PULL OPERATORS ON THE FORMAL AFFINE DEMAZURE ALGEBRA AND ITS DUAL 7 (a) X 2 i = X i (b) X i X j = X j X i for i j >, (c) X i X j X i X j X i X j = κ ij (X j X i ) for i j = and (d) X i q = s i (q)x i + i (q), Recall that the Iwahori-Hecke algebra H of the symmetric group S n+ is an Z[t,t ]-algebra with generators T i, i {,...,n}, subject to the following relations: (A) (T i +t)(t i t ) = 0 or, equivalently, T 2 i = (t t)t i +, (B) T i T j = T j T i for i j > and (C) T i T j T i = T j T i T j for i j =. (TheT i sappearinginthedefinitionoftheiwahori-heckealgebrain[cg0, Def.7..] correspond to tt i in our notation, where t = q /2.) Following [HMSZ, Def. 6.3] let D F denote the R-subalgebra of D F generated by the elements X i, i {,...,n}, only. By [HMSZ, Prop. 7.], over R = C, if F = F a (resp. F = F m ), then D F is isomorphic to the completion of the nil-hecke algebra (resp. the 0-Hecke algebra) of Kostant-Kumar. The following observation provides another motivation for the study of formal (affine) Demazure algebras. Let us consider the special elliptic formal group law of example 2.2 with coefficient µ =. Then its formal inverse is x/(x ), and since ( + µ 2 x i x j )x i+j = x i +x j x i x j, the coefficient κ ij of relation (c) is simply µ 2 : (8.) κ ij = x i+jx j x i+jx i x ix j = xi+xj xixj xi+j x ix jx i+j = (+µ2xixj)xi+j xi+j x ix jx i+j = µ 2 Proposition 8.2. Let F e be a normalized (i.e. µ = ) special elliptic formal group law over an integral domain R containing Z[t,t ], and let a,b R. Then the following are equivalent () The assignment T i ax i + b, i {,...,n}, defines an isomorphism of R-algebras H Z[t,t ] R D F. (2) We have a = t+t or t t and b = t or t respectively. Furthermore µ 2 (t+t ) 2 = ; in particular, the element t+t is invertible in R. Proof. Assume there is an isomorphism of R-algebras given by T i ax i +b. Then relations (b) and (B) are equivalent and relation (A) implies that 0 = (ax i +b) 2 +(t t )(ax i +b) = [a 2 +2ab+a(t t )]X i +b 2 +b(t t ). Therefore b = t or t and a = t t 2b = t +t or t t respectively, since and X i are S-linearly independent in D F D F. Relations (C) and (a) then imply 0 = (ax i +b)(ax j +b)(ax i +b) (ax j +b)(ax i +b)(ax j +b) = a 3 (X i X j X i X j X i X j )+(a 2 b+ab 2 )(X i X j ). Therefore, by relation (c) and (8.), we have a 3 µ 2 a 2 b ab 2 = 0 which implies that 0 = a 2 µ 2 ab b 2 = (t+t ) 2 µ 2 +. Conversely, by substituting the values of a and b, it is easy to check that the assignment is well defined, essentially by the same computations. It is an isomorphism since a = ±(t+t ) is invertible in R.

18 8 BAPTISTE CALMÈS, KIRILL ZAINOULLINE, AND CHANGLONG ZHONG Remark 8.3. The isomorphism of Theorem 8.2 provides a presentation of the Iwahori-Hecke algebra with t+t inverted in terms of the Demazure operators on the formal group algebra R[[Λ]] Fe. 9. The algebraic restriction to the fixed locus In the present section we define the algebraic counterpart of the restriction to T-fixed points of G/B. Let S W be the twisted group algebra of S and the group ring R[W], i.e. S W = S R R[W] as a R-module and the multiplication is defined by (xδ w )(x δ w ) = xw(x )δ ww, x,x S, w,w W. The algebra S W is a free S-module with basis {δ w } w W. Since S is Σ-regular, it injects into its localization Q. Therefore, S W injects into Q W via δ w δ w. Since δ α = X α for each α Σ, there is a natural inclusion of S-modules η: S W D F. By [CZZ, Prop. 7.7] the elements {X Iw } w W and, hence, {Y Iw } w W form a basis of D F as a left S-module. Tensoring η by Q we obtain an isomorphism η Q : Q W Q S D F, because both are free Q-modules and their bases {X Iw } w W are mapped to each other. Observe that by definition D F injects into Q S D F Q W. Consider the S-linear dual S W = Hom S(S W,S). Since {δ w } w W is a basis for both S W and Q W, S W can be identified with the free S-submodule of Q W with basis {f w } w W or, equivalently, with the subset {f Q W f(s W) S}. Consider the S-linear dual D F = Hom S(D F,S). Definition 9.. The induced map η : D F S W (composition with η) will be called the algebraic restriction to the fixed locus, because of its geometric interpretation, given in the introduction. Lemma 9.2. The map η is injective and its image in S W Q W = Q S S W coincides with the subset Proof. There is a commutative diagram {f Q W f(d F ) S}. D F η S W Q S D F η Q Q S S W where the vertical maps are injective by freeness of the modules and because S injects into Q. The description for the image then follows from the fact that {X Iw } w W is a basis for both D F and Q W. Remark 9.3. Observe that η is not surjective, unless the root datum is trivial. Indeed, since X α = δ α, we have f sα (X α ) = / S, so f sα is not in the image of η.

19 PUSH-PULL OPERATORS ON THE FORMAL AFFINE DEMAZURE ALGEBRA AND ITS DUAL 9 Since D F is a subring of Q W, the -action ofq W on Q W restrictsto an S-linear action of D F on D F given by the same formula: (z σ)(z) = σ(zz ) for σ D F and z,z in D F. Thus, the action of W on Q W restricts to an action on D F. By [CZZ, Theorem 9.2] the coproduct on Q W restricts to a coproduct on D F. Hence, the dual D F becomes a subring of Q W. 0. The push-pull operators on D F In this section we restrict the push-pull operators onto the dual of the formal affine Demazure algebra D F, and define a non-degenerate pairing on it. Lemma 0.. For any subset Ξ of Π we have Y Ξ D F. Proof. The ringq W is functorial in the rootdatum (i.e. alongmorphisms oflattices that send roots to roots) and in the formal group law. This functoriality sends elements X α (or Y α ) to themselves, so it restricts to a functoriality the subring D F. We can therefore assume that the root datum is adjoint, and that the formal group law is the universal one over the Lazard ring, in which all integers are regular, since it is a polynomial ring over Z. ConsidertheinvolutionιonQ W givenbyqδ w ( ) l(w) w (q)δ w. Itsatisfies ι(zz ) = ι(z )ι(z). Since ι(x α ) = Y α, it restricts to an anti-automorphism on D F. Hence, it suffices to show that ι(y Ξ ) D F. By definition we have ι(y Ξ ) = ( ) l(w) x Ξ δ w = x Ξ ( ) l(w) δ w. w W Ξ w W Ξ Since the root datum is adjoint, D F = {z Q W z S S} by [CZZ, Remark 7.8]. It therefore suffices to show that ι(y Ξ ) x S for any x S. We have ι(y Ξ ) x = x Ξ ( ) l(w) w(x). w W Ξ For any α Σ Ξ let α W Ξ = {w W Ξ l(s α w) > l(w)}. Then W Ξ = α W Ξ s α α W Ξ and the sum ( ) l(w) w(x) = ( ) l(w) (w(x) s α (w(x))) = ( ) l(w) α (w(x)) w W Ξ w α W Ξ w α W Ξ is divisible by. By using the next lemma recursively, we then conclude that x Ξ divides the sum and thus that (ι(y Ξ )) x D F. Lemma 0.2. Assume that all integers are regular in R, and that the root datum is adjoint. Let α and β be roots such that α ±β and let x S. Then if x β x, we have x. Proof. By [CZZ, Lemma 2.], the root α can be completed as a basis (e = α,e 2,...,e n )ofthelattice, givinganr-algebraisomorphismφ : S R[[x,...,x n [[, sending to x, by [CPZ, Cor 2.3]. Since β ±α, we have β = i n ie i with n i 0 for some i and φ(x β ) = i n ix i + z, where z I 2, the square of the augmentation ideal (generated by the variables). Since R[[x,...,x n ]]/(x ) R[[x 2,...,x n ]], the result follows from the regularity of the class of x β in that quotient, which in turn follows from [CZZ, Lemma 2.3].

20 20 BAPTISTE CALMÈS, KIRILL ZAINOULLINE, AND CHANGLONG ZHONG Corollary 0.3. The operator Y Ξ (resp. A Ξ ) restricted to S (resp. to D F ) defines an operator on S (resp. on D F ). Moreover, we have Y Ξ (S) S WΞ and A Ξ (D F) (D F) WΞ. Proof. Here Y Ξ acts on S Q via (4.3). Since Y Ξ D F {z Q W z S S} by [CZZ, Remark 7.8] and Y Ξ Q (Q) WΞ, the result follows. As for A Ξ, by Lemma 9.2 any f D F has the property that f(d F) S. Therefore, (A Ξ (f))(d F ) = (Y Ξ f)(d F ) = f(d F Y Ξ ) S, so A Ξ (f) D F. The result then follows by Lemma 6.7. Corollary 0.4. Suppose that the root datum has no irreducible component of type Cn sc or that 2 is invertible in R. Then if W Ξ is regular in R, for any Ξ Ξ Π, we have Y Ξ/Ξ (S W Ξ ) S WΞ. Proof. Let x (S) W Ξ, then W Ξ x = w W Ξ w(x). So we have W Ξ Y Ξ/Ξ (x) = Y Ξ/Ξ ( W Ξ x) = = u W Ξ/Ξ v W Ξ uv( x x Ξ/Ξ ) = w W Ξ w( u W Ξ/Ξ u( W Ξ x ) x Ξ/Ξ xx Ξ x Ξ ) S WΞ. Thus W Ξ Y Ξ/Ξ (x) S, which implies that Y Ξ/Ξ (x) S by [CZZ, Lemma 3.5]. Besides, it is fixed by W Ξ by 5.5. For each v W define f v := x Π f v S W, i.e. f v ( w W q w δ w ) = x Π q v. Lemma 0.5. For any v W we have f v D F. Proof. We first show that f w0 D F. By Lemma 3., we have X Iv = a X v,w δ w, where a X v,v = ( ). w v α (vσ ) Σ + Since w 0 Σ = Σ +, we have f w0 (X Iv ) = f w0 ( w v a X v,w δ w) = (x Π a X w 0,w 0 )δ v,w0 = α Σ + ( x α )δ v,w0 S. By Lemma 9.2, we have f w0 D F. For an arbitrary v W, by Lemma 4.2, we obtain f v = x Π f w0w 0 v = v w 0 (x Π f w0 ) = v w 0 ( f w0 ) D F. Corollary 0.6. For any X D F we have x Π X S W. Proof. It suffices to show that for any sequence I v, x Π X Iv S W. And indeed, x Π X Iv = x Π ( a X v,w δ w) = (x Π a X v,w )δ w = f w (X Iv )δ w S W. w v w v w v

21 PUSH-PULL OPERATORS ON THE FORMAL AFFINE DEMAZURE ALGEBRA AND ITS DUAL 2 Theorem 0.7. For any v,w W, we have Consequently, the pairing A Π (Y I v A I rev w (x Πf e )) = δ w,v = A Π (X I v B I rev w (x Πf e )). A Π : D F D F (D F )W = S, (σ,σ ) A Π (σσ ) is non-degenerate andsatisfies that {A I rev (x w Πf e )} w W is dual tothe basis {YI v } v W, and {B I rev (x w Πf e )} w W is dual to the basis {XI v } v W. Proof. We prove the first identity. The second identity is obtained similarly. Let Y I rev = w v W a w,v δ v and Y Iw = v W a w,vδ v. Let δ w = v W b w,vy Iv so that v W a w,vb v,u = δ w,u and YI u = v W b v,uf v. Combining the formula of Lemma 4.6 with the formula A Π (f v ) = v(x of Π) Lemma 6.5, we obtain A Π (YI u A I rev (x w Πf e )) = b v,u v(x Π )a w,v A Π (f v ) = b v,u a w,v = δ w,u. v W v W The characteristic map c introduced in 6.0 restricts to a homomorphism of R- algebras c S : S D F which maps x S to the evaluation at x using the action of D F on S, that is c S (x)(z) = z x for z D F. In particular, we have c S (x)(x I ) = I (x) and c S (x)(δ w ) = w(x), w W. Lemma 0.8. For any sequence I and x S, we have A Π (c S (x)a I rev(x Π f e )) = C I (x) and A Π (c S (x)b I rev(x Π f e )) = I (x). Proof. We prove the first formula only. The second formula is obtained similarly. Let Y I = v W ay I,v δ v. Since c S (x) = v W v(x)f v, by Lemma 4.6, we get A Π (c S (x)a I rev(x Π f e )) = A Π ( v W v(x)v(x Π )a Y I,vf v ) = v W a Y I,vv(x) = C I (x).. The non-degenerate pairing on the W Ξ -invariant subring In this last section, we construct a non-degenerate pairing on the subring of invariants(d F )WΞ. Using this pairingwe provideseverals-module bases of(d F )WΞ. Lemma.. Suppose that the set of reduced sequences {I w } w W is Ξ-compatible, then the set {X I u } u W Ξ is a basis of (D F )WΞ. Proof. This follows immediately from Lemma 6. and the identity (Q W )WΞ D F = (D F )WΞ. Given representatives u,u W Ξ we set d Y u,u := u (x Π/Ξ ) v W Ξ a Y u,u v, d X u,u := u (x Π/Ξ ) v W Ξ a X u,u v, η Ξ = where a X w,v and ay w,v are the coefficients introduced in Lemma 3. and 3.2. w W Ξ w(x Π/Ξ ) Lemma.2. For any u W Ξ we have A Ξ (A I rev (x u Πf e )) = d Y u,u fξ u, A Ξ(B I rev (x u Πf e )) = d X u,u fξ u. u W Ξ u W Ξ