Affine Gindikin-Karpelevich formula via Uhlenbeck spaces
aa r X i v : . [ m a t h . R T ] D ec Dedicated to S. Patterson on the occasion of his 60th birthday
AFFINE GINDIKIN-KARPELEVICH FORMULA VIA UHLENBECKSPACES
ALEXANDER BRAVERMAN, MICHAEL FINKELBERG AND DAVID KAZHDAN
Abstract.
We prove a version of the Gindikin-Karpelevich formula for untwisted affineKac-Moody groups over a local field of positive characteristic. The proof is geometric andit is based on the results of [1] about intersection cohomology of certain Uhlenbeck-typemoduli spaces (in fact, our proof is conditioned upon the assumption that the results of [1]are valid in positive characteristic; we believe that generalizing [1] to the case of positivecharacteristic should be essentially straightforward but we have not checked the details).In particular, we give a geometric explanation of certain combinatorial differences betweenfinite-dimensional and affine case (observed earlier by Macdonald and Cherednik), whichhere manifest themselves by the fact that the affine Gindikin-Karpelevich formula hasan additional term compared to the finite-dimensional case. Very roughy speaking, thatadditional term is related to the fact that the loop group of an affine Kac-Moody group(which should be thought of as some kind of “double loop group”) does not behave wellfrom algebro-geometric point of view; however it has a better behaved version which hassomething to do with algebraic surfaces.A uniform (i.e. valid for all local fields) and unconditional (but not geometric) proof ofthe affine Gindikin-Karpelevich formula is going to appear in [2]. The problem
Classical Gindikin-Karpelevich formula.
Let K be a non-archimedian local fieldwith ring of integers O and let G be a split semi-simple group over O . The classicalGindikin-Karpelevich formula describes explicitly how a certain intertwining operator actson the spherical vector in a principal series representation of G ( K ). In more explicit termsit can be formulated as follows.Let us choose a Borel subgroup B of G and an opposite Borel subgroup B − ; let U, U − be their unipotent radicals. In addition, let Λ denote the coroot lattice of G , R + ⊂ Λ – theset of positive coroots, Λ + – the subsemigroup of Λ generated by R + . Thus any γ ∈ Λ + can be written as P a i α i where α i are the simple roots. We shall denote by | γ | the sum ofall the a i .Set now Gr G = G ( K ) /G ( O ). Then it is known that U ( K )-orbits on Gr are in one-to-onecorrespondence with elements of Λ (this correspondence will be reviewed in Section 2); forany µ ∈ Λ we shall denote by S µ the corresponding orbit. The same thing is true for U − ( K )-orbits. For each γ ∈ Λ we shall denote by T γ the corresponding orbit. It is well-known that T γ ∩ S µ is non-empty iff µ − γ ∈ Λ + and in that case the above intersection is finite. TheGindikin-Karpelevich formula allows one to compute the number of points in T − γ ∩ S for More precisely, the Gindikin-Karpelevich formula answers the analogous question for real groups; itsanalog for p -adic groups (usually also referred to as Gindikin-Karpelevich formula) is proved e.g. in Chapter4 of [6]. ∈ Λ + (it is easy to see that the above intersection is naturally isomorphic to T − γ + µ ∩ S µ for any µ ∈ Λ). The answer is most easily stated in terms of the corresponding generatingfunction:
Theorem 1.2. (Gindikin-Karpelevich formula) X γ ∈ Λ + T − γ ∩ S ) q −| γ | e − γ = Y α ∈ R + − q − e − α − e − α . Formulation of the problem in the general case.
Let now G be a split sym-metrizable Kac-Moody group functor in the sense of [8] and let g be the corresponding Liealgebra. We also let b G denote the corresponding ”formal” version of G (cf. page 198 in [8]).The notations Λ , Λ + , R + , Gr G , S µ , T γ make sense for b G without any changes (cf. Section 2for more detail). Conjecture 1.4.
For any γ ∈ Λ + the intersection T − γ ∩ S is finite. This conjecture will be proved in [2] when G is of affine type. In this paper we are goingto prove the following result: Theorem 1.5.
Assume that K = k (( t )) where k is finite. Then Conjecture 1.4 holds. So now (at least when K is as above) we can ask the following Question:
Compute the generating function I g ( q ) = X γ ∈ Λ + T − γ ∩ S ) q −| γ | e − γ . One possible motivation for the above question is as follows: when G is finite-dimensional,Langlands [6] has observed that the usual Gindikin-Karpelevich formula (more precisely,some generalization of it) is responsible for the fact that the constant term of Eisensteinseries induced from a parabolic subgroup of G is related to some automorphic L -function.Thus we expect that generalizing the Gindikin-Karpelevich formula to general Kac-Moodygroup will eventially become useful for studying Eisenstein series for those groups. Thiswill be pursued in further publications.We don’t know the answer for general G . In the case when G is finite-dimensional theanswer is given by Theorem 1.2. In this paper we are going to reprove that formula bygeometric means and give a generalization to the case when G is untwisted affine.1.6. The affine case.
Let us now assume that g = g ′ aff where g ′ is a simple finite-dimensional Lie algebra. The Dynkin diagram of g has a canonical (”affine”) vertex andwe let p be the corresponding maximal parabolic subalgebra of g . Let g ∨ denote the Lang-lands dual algebra and let p ∨ be the corresponding dual parabolic. We denote by n ( p ∨ ) its(pro)nilpotent radical.Let ( e, h, f ) be a principal sl (2)-triple in ( g ′ ) ∨ . Since the Levi subalgebra of p ∨ is C ⊕ g ′ ⊕ C (where the first multiple is central in g ∨ and the second is responsible for the “looprotation”), this triple acts on n ( p ∨ ) and we let W = ( n ( p ∨ )) f (the centralizer of f in n ( p ∨ )). The reason that we use the notation I g rather than I G is that it is clear that this generating functiondepends only on g and not on G . e are going to regard W as a complex (with zero differential) and with grading comingfrom the action of h (thus W is negatively graded). In addition W is endowed with anaction of G m , coming from the loop rotation in g ∨ . In the case when g ′ is simply lacedwe have ( g ′ ) ∨ ≃ g ′ and n ( p ∨ ) = t · g ′ [ t ] (i.e. g ′ -valued polynomials, which vanish at 0).Hence W = t · ( g ′ ) f [ t ] and the above G m -action just acts by rotating t . Let d , · · · , d r bethe exponents of g ′ (here r = rank( g ′ )). Then ( g ′ ) f has a basis ( x , · · · , x r ) where each x i is placed in the degree − d i . We let Fr act on W by requiring that it acts by q i/ onelements of degree i . Also for any n ∈ Z let W ( n ) be the same graded vector space butwith Frobenius action multiplied by q − n .Consider now Sym ∗ ( W ). We can again consider it as a complex concentrated in degrees ≤ G m . For each n ∈ Z we let Sym ∗ ( W ) n be the part ofSym ∗ ( W ) on which G m acts by the character z z n . This is a finite-dimensional complexwith zero differential, concentrated in degrees ≤ δ denote the minimal positiveimaginary coroot of g . Set ∆ W ( z ) = ∞ X n =0 Tr(Fr , Sym ∗ ( W ) n ) z n . In particular, when g ′ is simply laced we have∆( z ) = r Y i =1 ∞ Y j =0 (1 − q − d i z j ) − . Theorem 1.7. (Affine Gindikin-Karpelevich formula)Assume that the results of [1] are valid over k and let K = k (( t )) . Then I g ( q ) = ∆ W (1) ( e − δ )∆ W ( e − δ ) Y α ∈ R + (cid:18) − q − e − α − e − α (cid:19) m α . Here m α denotes the multiplicity of the coroot α . Remark.
Although formally the paper [1] is written under the assumption that char k = 0,we believe that adapting all the constructions of [1] to the case char k = p should be moreor less straightforward. We plan to discuss it in a separate publication.Let us make two remarks about the above formula: first, we see that it is very similar tothe finite-dimensional case (of course in that case m α = 1 for any α ) with the exception ofa “correction term” (which is equal to ∆ W ( e − δ )∆ W (1) ( e − δ ) ). Roughly speaking this correction termhas to do with imaginary coroots of g . The second remark is that the same correctionterm appeared in the work of Macdonald [7] from purely combinatorial point of view (cf.also [3] for a more detailed study). The main purpose of this note is to explain how theterm ∆ W ( e − δ )∆ W (1) ( e − δ ) appears naturally from geometric point of view (very roughly speaking it isrelated to the fact that affine Kac- Moody groups over a local field of positive characteristiccan be studied using various moduli spaces of bundles on an algebraic surface). The relationbetween the present work and the constructions of [3] and [7] will be discussed in [2]. .8. Acknowledgments.
We thank I. Cherednik, P. Etingof and M. Patnaik for veryhelpful discussions. A. B. was partially supported by the NSF grant DMS-0901274. M. F.was partially supported by the RFBR grant 09-01-00242 and the Science Foundation of theSU-HSE awards No.T3-62.0 and 10-09-0015. D. K. was partially supported by the BSFgrant 037.8389. 2.
Interpretation via maps from P to B Generalities on Kac-Moody groups.
In what follows all schemes will be consideredover a field k which at some point will be assumed to be finite. Our main reference for Kac-Moody groups is [8]. Assume that we are given a symmetrizable Kac-Moody root data andwe denote by G (resp. b G ) the corresponding minimal (resp. formal) Kac-Moody groupfunctor (cf. [8], page 198); we have the natural embedding G ֒ → b G . We also let W denotethe corresponding Weyl group and we let ℓ : W → Z ≥ be the corresponding length function.The group G is endowed with closed subgroup functors U ⊂ B, U − ⊂ B − such that thequotients B/U and B − /U − are naturally isomorphic to the Cartan group H of G ; also H is isomorphic to the intersection B ∩ B − . Moreover, both U − and B − are still closed assubgroup functors of b G . On the other hand, B and U are not closed in b G and we denoteby b B and b U their closures.The quotient G/B has a natural structure of an ind-scheme which is ind-proper; thesame is true for the quotient b G/ b B and the natural map G/B → b G/ b B is an isomorphism.This quotient is often called the thin flag variety of G . Similarly, one can consider thequotient B = b G/B − ; it is called the thick flag variety of G or Kashiwara flag scheme . As issuggested by the latter name, B has a natural scheme structure. The orbits of B on B arein one-to-one correspondence with the elements of the Weyl group W ; for each w ∈ W wedenote by B w the corresponding orbit. The codimension of B w is ℓ ( w ); in particular, B e isopen. There is a unique H -invariant point y ∈ B e . The complement to B e is a divisor in B whose components are in one-to-one correspondence with the simple roots of G .In what follows Λ will denote the coroot lattice of G , R + ⊂ Λ – the set of positive coroots,Λ + – the subsemigroup of Λ generated by R + . Thus γ ∈ Λ + can be written as P a i α i where α i are the simple coroots. We shall denote by | γ | the sum of all the a i .In what follows we shall assume that G is ”simply connected”, which means that Λ isequal to the full cocharacter lattice of H .2.2. Some further notations.
For any variety X and any γ ∈ Λ + we shall denote bySym γ X the variety parametrizing all unordered collections ( x , γ ) , ... ( x n , γ n ) where x j ∈ X, γ j ∈ Λ + such that P γ j = γ .Assume that k is finite and let S be a complex of ℓ -adic sheaves on a variety X over k .We set χ k ( S ) = X i ∈ Z ( − i Tr(Fr , H i ( X, S )) , where X = X × Spec k Spec k . e shall denote by ( Q l ) X the constant sheaf with fiber Q l . According to the Grothendieck-Lefschetz fixed point formula we have χ k (( Q l ) X ) = X ( k ) . Semi-infinite orbits.
As in the introduction we set K = k (( t )), O = k [[ t ]]. We letGr = b G ( K ) / b G ( O ), which we are just going to consider as a set with no structure. Each λ ∈ Λ is a homomorphism G m → H ; in particular, it defines a homomorphism K ∗ → H ( K ).We shall denote the image of t under the latter homomorphism by t λ . Abusing the notation,we shall denote its image in Gr by the same symbol. Set S λ = b U ( K ) · t λ ⊂ Gr; T λ = U − ( K ) · t λ ⊂ Gr . Lemma 2.4. Gr is equal to the disjoint union of all the S λ . Proof.
This follows from the
Iwasawa decomposition for G of [5]; we include a different prooffor completeness. Since Λ ≃ b U ( K ) \ b B ( K ) / b B ( O ), the statement of the lemma is equivalentto the assertion that the natural map b B ( K ) / b B ( O ) → b G ( K ) / b G ( O ) is an isomorphism; inother words, we need to show that b B ( K ) acts transitively on Gr. But this is equivalentto saying that b G ( O ) acts transitively on b G ( K ) / b B ( K ), which means that the natural map b G ( O ) / b B ( O ) → b G ( K ) / b B ( K ) is an isomorphism. However, the left hand side is ( b G/ b B )( O )and the right hand side is ( b G/ b B )( K ) and the assertion follows from the fact that the ind-scheme b G/ b B satisfies the valuative criterion of properness. (cid:3) The statement of the lemma is definitely false if we use T µ ’s instead of S λ ’s since the scheme b G/B − does not satisfy the valuative ctiterion of properness. Let us say that an element g ( t ) ∈ b G ( K ) is good if its projection to B ( K ) = B − ( K ) \ b G ( K ) comes from a point of B ( O ).Since B ( O ) = B − ( O ) \ b G ( O ), it follows that the set of good elements of b G ( K ) is just equalto B − ( K ) · G ( O ), which immediately proves the following result: Lemma 2.5.
The preimage of S γ ∈ Λ T γ in b G ( K ) is equal to the set of good elements of b G ( K ) . Spaces of maps.
Recall that the Picard group of B can be naturally identified withΛ ∨ (the dual lattice to Λ). Thus for any map f : P → B we can talk about the degree of f as an element γ ∈ Λ. The space of such maps is non-empty iff γ ∈ Λ + . We say that amap f : P → B is based if f ( ∞ ) = y . Let M γ be the space of based maps f : P → B ofdegree γ . It is shown in the Appendix to [1] that this is a smooth scheme of finite type over k of dimension 2 | γ | . We have a natural (“factorization”) map π γ : M γ → Sym γ A which isrelated to how the image of a map P → B intersects the complement to B e . In particular,if we set F γ = ( π γ ) − ( γ · , then F γ consists of all the based maps f : P → B of degree γ such that f ( x ) ∈ B e for any x = 0. Theorem 2.7.
There is a natural identification F γ ( k ) ≃ T − γ ∩ S . ince F γ is a scheme of finite type over k , it follows that F γ ( k ) is finite and thus Theo-rem 2.7 implies Theorem 1.5.The proof of Theorem 2.7 is essentially a repetition of a similar proof in the finite-dimensional case, which we include here for completeness. Proof.
First of all, let us construct an embedding of the union of all the F γ ( k ) into S = b U ( K ) / b U ( O ). Indeed an element of S γ ∈ Λ + F γ is uniquely determined by its restriction to G m ⊂ P ; this restriction is a map f : G m → B e such that lim x →∞ f ( x ) = y . We mayidentify B e with b U (by acting on y ). Thus we get [ γ ∈ Λ + F γ ⊂ { u : P \{ } → b U | u ( ∞ ) = e } . (2.1)We have a natural map from the set of k -points of the right hand side of (2.1) to b U ( K );this map sends every u as above to its restriction to the formal punctured neighbourhoodof 0. We claim that after projecting b U ( K ) to S = b U ( K ) / b U ( O ), this map becomes anisomorphism. Recall that b U is a group-scheme, which can be written as a projective limit offinite-dimensional unipotent group-schemes U i ; moreover, each U i has a filtration by normalsubgroups with successive quotients isomorphic to G a . Hence it is enough to prove that theabove map is an isomorphism when U = G a . In this case we just need to check that anyelement of the quotient k (( t )) / k [[ t ]] has unique lift to a polynomial u ( t ) ∈ k [ t, t − ] such that u ( ∞ ) = 0, which is obvious.Now Lemma 2.5 implies that a map u ( t ) as above extends to a map P → B if and onlyif the corresponding element of S lies in the intersection with some T − γ .It remains to show that F γ ( k ) is exactly equal to S ∩ T − γ as a subset of S . Let Λ ∨ be the weight lattice of G and let Λ ∨ + denote the set of dominant weights of G . For each λ ∨ ∈ Λ ∨ + we can consider the Weyl module L ( λ ∨ ), defined over Z ; in particular, L ( λ ∨ )( K )and L ( λ ∨ )( O ) make sense. By the definition L ( λ ∨ ) is the module of global sections of aline bundle L ( λ ∨ ) on B . Moreover we have a weight decomposition L ( λ ∨ ) = M µ ∨ ∈ Λ ∨ L ( λ ∨ ) µ ∨ where each L ( λ ∨ ) µ ∨ is a finitely generated free Z -module and L ( λ ∨ ) λ ∨ := l λ ∨ has rank one.Geometrically, l λ ∨ is the fiber of L ( λ ∨ ) at y and the corresponding projection map from L ( λ ∨ ) = Γ( B , L ( λ ∨ )) to l λ ∨ is the restriction to y .Let η λ ∨ denote the projection of L ( λ ∨ ) to l λ ∨ . This map is U − -equivariant (where U − acts trivially on l λ ∨ ). Lemma 2.8.
The projection of a good element g ∈ G ( K ) lies in T ν (for some ν ∈ Λ ) if andonly if for any λ ∨ ∈ Λ ∨ we have: η λ ∨ ( g ( L ( λ ∨ )( O ))) ⊂ t h ν,λ ∨ i l λ ∨ ( O ); η λ ∨ ( g ( L ( λ ∨ )( O ))) t h ν,λ ∨ i− l λ ∨ ( O ) . (2.2) Proof.
First of all, we claim that if the projection of g lies in T ν then the above conditionis satisfied. Indeed, it is clearly satisfied by t ν ; moreover (2.2) is clearly invariant underleft multiplication by U − ( K ) and under right multiplication by G ( O ). Hence any g ∈ U − ( K ) · t ν · G ( O ) satisfies (2.2). n the other hand, assume that a good element g ∈ G ( K ) satisfies (2.2). Since g liesin U − ( K ) · t ν ′ · G ( O ) for some ν ′ , it follows that g satisfies (2.2) when ν is replaced by ν ′ .However, it is clear that this is possible only if ν = ν ′ . (cid:3) It is clear that in (2.2) one can replace g ( L ( λ ∨ )( O )) with g ( L ( λ ∨ )( k )) (since the lattergenerates the former as an O -module).Let now f be an element of F γ . Then f ∗ L ( λ ∨ ) is isomorphic to the line bundle L ( h γ, λ ∨ i )on P . On the other hand, the bundle L ( λ ∨ ) is trivialized on B e by means of the action of U ; more precisely, the restriction of L ( λ ∨ ) is canonically identified with the trivial bundlewith fiber l λ ∨ . Let now s ∈ L ( λ ∨ )( k ); we are going to think of it as a section of L ( λ ∨ ) on B . In particular, it gives rise to a function e s : B e → l λ ∨ . Let also u ( t ) be the element of U ( K ), corresponding to f . Then η λ ∨ ( u ( t )( s )) can be described as follows: we consider thecomposition e s ◦ f and restrict it to the formal neighbourhood of 0 ∈ P (we get an elementof l λ ∨ ( K )). On the other hand, since f ∈ F γ , it follows that f ∗ L ( λ ∨ ) is trivialized awayfrom 0 and any section of it can be thought of as a function P \{ } with pole of order ≤ h γ, λ ∨ i at 0. Hence e s ◦ f has pole of order ≤ h γ, λ ∨ i at 0.To finish the proof it is enough to show that for some s the function e s ◦ f has pole oforder exactly h γ, λ ∨ i at 0 (indeed if f ∈ T − γ ′ for some γ ′ ∈ Λ, then by (2.2) e s ◦ f has poleof order ≤ h γ ′ , λ ∨ i at 0 and for some s , it has pole of order exactly h γ ′ , λ ∨ i which impliesthat γ = γ ′ ). To prove this, let us note that since L ( λ ∨ ) is generated by global sections, theline bundle f ∗ L ( λ ∨ ) is generated by sections of the form f ∗ s , where s is a global section of L ( λ ∨ ). This implies that for any s ∈ Γ( P , f ∗ L ( λ ∨ )) there exists a section s ∈ Γ( B , L ( λ ∨ ))such that the ratio s/s is a rational function on P , which is invertible at 0. Taking s suchthat its pole with respect to the above trivialization of f ∗ L ( λ ∨ ) is exactly equal to h γ ′ , λ ∨ i and taking s as above, we see that the pole of f ∗ s with respect to the above trivializationof f ∗ L ( λ ∨ ) is exactly equal to h γ ′ , λ ∨ i . (cid:3) Proof of Theorem 1.2 via quasi-maps
Quasi-maps.
We shall denote by QM γ the space of based quasi-maps P → B . Ac-cording to [4] we have the stratification QM γ = [ γ ′ ≤ γ M γ ′ × Sym γ − γ ′ A . The factorization morphism π γ extends to the similar morphism π γ : QM γ → Sym γ andwe set F γ = ( π γ ) − (0). Thus we have F γ = [ γ ′ ≤ γ F γ ′ . (3.1)There is a natural section i γ : Sym γ A → QM γ . According to [4] we have Theorem 3.2. (1)
The restriction of IC QM γ to F γ ′ is isomorphic to ( Q l ) F γ ′ [2](1) ⊗| γ ′ | ⊗ Sym ∗ ( n ∨ + [2](1)) γ − γ ′ . (2) There exists a G m -action on QM γ which contracts it to the image of i γ . In partic-ular, it contracts F γ to one point (corresponding to γ ′ = 0 in (3.1) ). Let s γ denote the embedding of γ · into Sym γ A . Then s ∗ γ i ! γ IC QM γ = Sym ∗ ( n + ) γ (here the right hand is a vector space concentrated in cohomological degree 0 andwith trivial action of Fr ). The assertion 2) implies that π γ ! IC QM γ = i ! γ IC QM γ and hence H ∗ c ( F , IC QM γ | F γ ) = s ∗ γ π γ ! IC QM γ = s ∗ γ i ! γ IC QM γ = Sym ∗ ( n + ) γ . Thus, setting, S γ = IC QM γ | F γ we get X γ ∈ Λ + χ k ( S γ ) e − γ = Y α ∈ R + − e − α . (3.2)On the other hand, according to 1) we have χ k ( S γ ) = X γ ′ ≤ γ ( F γ ′ ) q −| γ ′ | Tr(Fr , Sym ∗ ( n ∨ + [2](1)) γ − γ ′ )which implies that X γ ∈ Λ + χ k ( S γ ) e − γ = P γ ∈ Λ + F γ ( k ) q −| γ | e − γ Q α ∈ R + − q − e − α = I g ( q ) Q α ∈ R + − q − e − α . (3.3)Hence I g ( q ) = Y α ∈ R + − q − e − α − e − α . Proof of Theorem 1.7
Flag Uhlenbeck spaces.
We now assume that G = ( G ′ ) aff where G ′ is some semi-simple simply connected group. We want to follow the pattern of Section 3. Let γ ∈ Λ + . Asis discussed in [1] the corresponding space of quasi-maps behaves badly when G is replacedby G aff . However, in this case one can use the corresponding flag Uhlenbeck space U γ . Infact, as was mentioned in the Introduction, in [1] only the case of k of characteristic 0 isconsidered. In what follows we are going to assume that the results of loc. cit. are validalso in positive characteristic.The flag Uhlenbeck space U γ has properties similar to the space of quasi-maps QM γ considered above in the previous Section. Namely we have:a) U γ is an affine variety of dimension 2 | γ | , which contains M γ as a dense open subset.b) There is a factorization map π γ : U γ → Sym γ A ; it has a section i γ : Sym γ A → U γ .c) U γ is endowed with a G m -action which contracts U γ to the image of i γ .These properties are identical to the corresponding properties of QM γ from the previousSection. The next (stratification) property, however, is different (and it is in fact respon-sible for the additional term in Theorem 1.7). Namely, let δ denote the minimal positiveimaginary coroot of G ′ aff . Then we have ) There exists a stratification U γ = [ γ ′ ∈ Λ + ,n ∈ Z ,γ − γ ′ − nδ ∈ Λ + ( M γ − γ ′ − nδ × Sym γ ′ A ) × Sym n ( G m × A ) . (4.1)In particular, if we now set F γ = ( π γ ) − ( γ ·
0) we get F γ = ( [ γ ′ ∈ Λ + ,n ∈ Z ,γ − γ ′ − nδ ∈ Λ + F γ ′ ) × Sym n ( G m ) . (4.2)4.2. Description of the IC-sheaf.
In [1] we describe the IC-sheaf of U γ . To formulatethe answer, we need to introduce some notation. Let P ( n ) denote the set of partitions of n . In other words, any P ∈ P ( n ) is an unordered sequence n , · · · , n k ∈ Z > such that P n i = n . We set | P | = k . For a variety X and any P ∈ P ( n ) we denote by Sym P ( X ) thelocally closed subset of Sym n ( X ) consisting of all formal sums P n i x i where x i ∈ X and x i = x j for i = j . The dimension of Sym P ( X ) is | P | · dim X . Let alsoSym ∗ ( W [2](1)) P = k O i =1 Sym ∗ ( W [2](1)) n i . Theorem 4.3.
The restriction of IC U γ to M γ − γ ′ − nδ × Sym γ ′ ( A ) × Sym P ( G m × A ) isisomorphic to constant sheaf on that scheme tensored with Sym ∗ ( n + ) γ ′ ⊗ Sym ∗ ( W ) P [2 | γ − γ ′ − nδ | ]( | γ − γ ′ − nδ | ) . Corollary 4.4.
The restriction of IC U γ to F γ − γ ′ − nδ × Sym P ( G m ) is isomorphic to theconstant sheaf tensored with Sym ∗ ( n ∨ + ) γ ′ ⊗ Sym ∗ ( W ) P [2 | γ − γ ′ − nδ | ]( | γ − γ ′ − nδ | ) . Let now S γ denote the restriction of IC U γ to F γ . Then, as in (3.2) we get X γ ∈ Λ + χ k ( S γ ) e − γ = Y α ∈ R + − e − α ) m α . (4.3)On the other hand, arguing as in (3.3) we get that X γ ∈ Λ + χ k ( S γ ) e − γ = A ( q ) I g ( q ) Q α ∈ R + (1 − q − e − α ) m α . (4.4)where A ( q ) = ∞ X n =0 X P ∈P ( n ) Tr(Fr , H ∗ c (Sym P ( G m ) , Q l ) ⊗ Sym ∗ ( W [2](1)) P ) e − nδ . This implies that I g ( q ) = A ( q ) − Y α ∈ R + (cid:18) − q − e − α − e − α (cid:19) m α . t remains to compute A ( q ). However, it is clear that A ( q ) = ∞ X n =0 Sym n ( H ∗ c ( G m ) ⊗ W [2](1)) e − nδ = ∆ W ( e − δ )∆ W (1) ( e − δ ) . (4.5)This is true since H ic ( G m ) = 0 unless i = 1 ,
2, and we have H c ( G m ) = Q l , H c ( G m ) = Q l ( − , and thus if we ignore the cohomological Z -grading, but only remember the corresponding Z -grading, then we just haveSym ∗ ( H ∗ c ( G m ) ⊗ W [2](1)) = Sym ∗ ( W ) ⊗ Λ ∗ ( W (1)) , whose character is exactly the right hand side of (4.5). References [1] A. Braverman, M. Finkelberg and D. Gaitsgory,
Uhlenbeck spaces via affine Lie algebras , Theunity of mathematics, 17–135, Progr. Math., , Birkhuser Boston, Boston, MA, 2006.[2] A. Braverman, D. Kazhdan and M. Patnaik,
The Iwahori-Hecke algebra for an affine Kac-Moodygroup , in preparation.[3] I. Cherednik and X. Ma,
A new take on spherical, Whittaker and Bessel functions ,arXiv:0904.4324.[4] B Feigin, M. Finkelberg, A. Kuznetsov, I. Mirkovic,
Semi-infinite flags. II. Local and global inter-section cohomology of quasimaps’ spaces. Differential topology, infinite-dimensional Lie algebras,and applications , 113–148, Amer. Math. Soc. Transl. Ser. 2, 194, Amer. Math. Soc., Providence,RI, 1999.[5] S. Gaussent and G. Rousseau,
Kac-Moody groups, hovels and Littelmann paths , Annales del’Institut Fourier, no. 7 (2008), p. 2605-2657[6] R. P. Langlands, Euler products , A James K. Whittemore Lecture in Mathematics given atYale University, 1967. Yale Mathematical Monographs, . Yale University Press, New Haven,Conn.-London, 1971.[7] I. G. Macdonald, A formal identity for affine root systems. Lie groups and symmetric spaces ,195-211, Amer. Math. Soc. Transl. Ser. 2, , Amer. Math. Soc., Providence, RI, 2003.[8] J. Tits,
Groups and group functors attached to Kac-Moody data , Workshop Bonn 1984 (Bonn,1984), 193–223, Lecture Notes in Math., , Springer, Berlin, 1985
A.B.: Department of Mathematics, Brown University, 151 Thayer St., Providence RI 02912,USA; [email protected]
M.F.: IMU, IITP and State University Higher School of Economics, Department of Math-ematics, 20 Myasnitskaya st, Moscow 101000 Russia; [email protected]
D.K.: Einstein Institute of Mathematics, Edmond J. Safra Campus, Givat Ram The HebrewUniversity of Jerusalem Jerusalem, 91904, Israel; [email protected]@math.huji.ac.il