aa r X i v : . [ m a t h . R T ] N ov THE CONVOLUTION ALGEBRA STRUCTURE ON K G ( B × B ) SIAN NIE
Abstract.
We show that the convolution algebra K G ( B × B )is isomorphic to the Based ring of the lowest two-sided cell of theextended affine Weyl group associated to G , where G is a connectedreductive algebraic group over the field C of complex numbers and B is the flag variety of G . Introduction
We are interested in understanding the equivariant group K G ( B × B ),where G is a connected reductive algebraic group over C and B is theflag variety of G .When G has simply connected derived subgroup, the K¨unneth for-mula K G ( B × B ) ≃ K G ( B ) ⊗ R G K G ( B ) is proved in Proposition 1.6of [KL] and plays an important role in Kazhdan-Lusztig’s proof ofDelinge-Langlands conjecture for affine Hecke algebra associated to G ,where R G denotes the representation ring of G . Furthermore, by The-orem 1.10 of [Xi], the convolution algebra structure on K G ( B × B ) isisomorphic to the based ring of the lowest two-sided cell of the extendedaffine Weyl group associated to G .In general, K G ( B × B ) is not isomorphic to K G ( B ) ⊗ R G K G ( B ). Toset a Deligne-Langlands-Lusztig classification for affine Hecke algebraassociated to G , it seems useful to understand the equivariant K -groups K G ( B × B ). The main result of this paper is Theorem 1.1, which saysthat the convolution algebra on K G ( B × B ) is isomorphic to the basedring of the lowest two-sided cell of the extended affine Weyl groupassociated to G . Since the based ring is known explicitly in [Xi], themain result gives an explicit description to the equivariant K -group K G ( B × B ). 1. Preliminary
Let G be a connected reductive algebraic group over C , B a Berolsubgroup of G and T a maximal torus of G , such that T ⊂ B . TheWeyl group W = N G ( T ) /T of G acts on the character group X =Hom( T, C ∗ ) of T . Using this action we define the extended affine Weylgroup W = X ⋊ W . By classification theorem for connected reductive algebraic groups,there exists a connected reductive algebraic group ˜ G with simply con-nected derived subgroup such that G is a quotient group of ˜ G moduloa finite subgroup of the center of ˜ G . Denote by π : ˜ G → G the quotienthomomorphism. Set ˜ B = π − ( B ), ˜ T = π − ( T ), ˜ X = Hom( ˜ T , C ∗ ) and˜ W = ˜ X ⋊ W . Note that X is naturally a subgroup of ˜ X of finite index,hence W is a naturally subgroup of ˜ W of finite index.Let R ⊂ X be the root of G and ˜ G . Let R − ⊂ R to be the set ofnegative roots determined by B . Set R + = R − R − . Let ∆ ⊂ R + beset of simple positive roots.Denote by λ α the dominant fundamental weight corresponding toa simple positive root α ∈ R + . For any w ∈ W , define x w = w − ( Q α ∈ ∆ ,w − ( α ) < λ α ) ∈ ˜ X . It is known that Z [ ˜ X ] is a free Z [ X ] W -module with a basis { x w | w ∈ W } .Let ℓ : ˜ W → N be the length function. Note that ℓ ( wλ ) = ℓ ( w )+ ℓ ( λ )for any w ∈ W and any dominant weight λ ∈ ˜ X . Also we have ℓ ( λ α s α ) = ℓ ( λ α ) − α ∈ ∆. Let Σ = { wx w | w ∈ W } . Then the lowest two-sided cell ˜ c of ˜ W consists of elements f − w χg with f, g ∈ Σ and χ ∈ ˜ X + . (See [Shi])Here w is the longest element of W and ˜ X + is the set of dominantweights of ˜ X . The lowest two-sided cell of W is c = ˜ c ∩ W . The ringstructure of J ˜ c of ˜ c is defined in § § Z -module, it is free with a basis t z , z ∈ ˜ c . The basedring J c of c is a subring of J ˜ c spanned by all t z , z ∈ c . For an algebraic group M over C and an variety Z over C whichadmits an algebraic action of M , denote by K M ( Z ) the Grothendieckgroup of M -equivariant coherent sheaves on Z . We refer to Chapter 5of [CG] for more about the equivariant K -group K M ( Z ).There is a natural map L : Z [ ˜ X ] → K ˜ G ( B ) which associates χ ∈ ˜ X to the unique equivariant line bundle [ L ( χ )] on B such that ˜ T acts onthe fibre L ( χ ) B over B ∈ B via χ . Here B = ˜ G/ ˜ B = G/B is theflag variety. It is well known that L is an isomorphism. By abuseof notation, we will use χ and [ L ( χ )] interchangeably in the followingcontext.The convolution on K G ( B × B ) is defined by F ∗ F ′ = Rp ∗ ( p ∗ F L O O B × B × B p ∗ F ′ ) , where F , F ′ ∈ K G ( B × B ) and p , p , p : B × B × B → B × B areobvious natural projections. Identifying K ˜ G ( B × B ) with K ˜ G ( B ) ⊗ R ˜ G K ˜ G ( B ) ≃ Z [ ˜ X ] ⊗ Z [ ˜ X ] W Z [ ˜ X ], the convolution becomes( χ ⊗ χ ) ∗ ( χ ′ ⊗ χ ′ ) = ( χ , χ ′ ) χ ⊗ χ ′ , where ( , ) : Z [ ˜ X ] ⊗ Z [ ˜ X ] W Z [ ˜ X ] → Z [ ˜ X ] W is given by( χ , χ ′ ) = δ − X w ∈ W ( − ℓ ( w ) w ( χ ρχ ′ ) . Here δ = Q α ∈ R + ( α − α − ) and ρ = Q α ∈ R + α .For f = wx w ∈ Σ, set x f = x w . Since ( , ) is a perfect pair-ing (See Proposition 1.6 in [KL]), we can find y f ∈ Z [ ˜ X ] such that( x f , y f ) = δ f,f ′ . The following result is due to N. Xi. (See Theorem1.10 in [Xi].)( ∗ ) The map σ : J ˜ c → K ˜ G ( B × B ) ≃ Z [ ˜ X ] ⊗ Z [ ˜ X ] W Z [ ˜ X ] given by t f − w χf ′ V ( χ ) y f ⊗ x f ′ for χ ∈ ˜ X + and f, f ′ ∈ Σ is an isomorphismof R ˜ G -algebras. Here V ( χ ) ∈ R ˜ G stands for the irreducible ˜ G -modulewith highest weight χ . Now we state the main result of this paper.
Theorem 1.1. (a) The natural map i : K G ( B × B ) → K ˜ G ( B × B ) isan injective of homomorphism of algebra.(b) As a Z -module, the image of i is spanned by { V ( χ ) y f ⊗ x f ′ ; χ ∈ ˜ X + , f, f ′ ∈ Σ , f − w χf ′ ∈ W } .(c) In particular, via the isomorphism σ in (*), J c is isomorphic tothe convolution algebra K G ( B × B ) as R G -algebras. Proof of Theorem 1.1
Set Ω = ˜
W /W = ˜
X/X = { λX ; λ ∈ ˜ X } , which is a finite abeliangroup. For a left coset λX , let Z [ λX ] be the Z -submodule of the groupalgebra Z [ ˜ X ] spanned by elements in λX . For any A ∈ Z [ λX ] and B ∈ Z [ µX ], we have AB ∈ Z [ λµX ]. Moreover if there is C ∈ Z [ ˜ X ]such that A = BC , then C ∈ Z [ λµ − X ]. Lemma 2.1.
For f ∈ Σ , we have y f ∈ Z [ x − f X ] .Proof. For f, f ′ ∈ Σ, set A f,f ′ = ( x f , x f ′ ) = δ − P w ∈ W ( − ℓ ( w ) w ( x f ρx f ′ )which lies in Z [ x f x f ′ X ]. Let ( A f,f ′ ) ( f,f ′ ) ∈ Σ × Σ be the inverse matrix of( A f,f ′ ) ( f,f ′ ) ∈ Σ × Σ . Then a direct computation shows A f,f ′ ∈ Z [ x − f x − f ′ X ].Since y f = P f ′ ∈ Σ A f,f ′ x f ′ , we have y f ∈ Z [ x − f X ]. (cid:3) For w ∈ W , let Y w ⊂ B × B be the G -orbit containing ( B, wB ).Then B × B = ` w ∈ W Y w and the projection to the first factor p : Y w → B is an affine bundle of rank ℓ ( w ). Numbering the elements of W as u , u , · · · , u r such that u i ≮ u j if j < i . Let F i = ` j ≤ i Y u j . Then F i is closed in B × B . We have the following commutative diagram SIAN NIE / / K G ( F i − ) (cid:15) (cid:15) / / K G ( F i ) (cid:15) (cid:15) / / K G ( Y u i ) (cid:15) (cid:15) / / / / K ˜ G ( F i − ) / / K ˜ G ( F i ) / / K ˜ G ( Y u i ) / / p : Y u i → B is an affinebundle, we have the following commutative diagram K G ( Y u i ) ∼ / / K G ( B ) / / ≀ (cid:15) (cid:15) K ˜ G ( B ) ∼ / / ≀ (cid:15) (cid:15) K ˜ G ( Y u i ) Z [ X ] / / Z [ ˜ X ]which shows that the natural morphism K G ( Y u i ) → K ˜ G ( Y u i ) is in-jective. Using induction on i , we see that the natural morphism i : K G ( B × B ) → K ˜ G ( B × B ) is injective. One shows directly that i isa homomorphism of convolution algebras. Part (a) of Theorem 1.1 isproved. For w ∈ W , let X w = ˜ Bw ˜ B/ ˜ B and X w = wB + ˜ B/ ˜ B , where ˜ B + ⊃ ˜ T is the opposite of ˜ B . Then X w is an ˜ T -invariant open neighborhoodof X w in B . Set X i = ` j ≤ i X u j . Note that B × B = ˜ B \ ( ˜ G × B ), wherethe action of ˜ B on ˜ G × B is given by b ( g, h ˜ B ) = ( gb − , bh ˜ B ). Hence K ˜ G ( B × B ) ≃ K ˜ B ( B ) ≃ K ˜ T ( B ). Similarly, K ˜ G ( F i ) ≃ K ˜ T ( X i ) and K ˜ G ( Y w ) ≃ K ˜ T ( X w ).Let j w : Y w → B × B be the natural G -equivariant inclusion. Since X w is a ˜ T -equivariant vector bundle over a single point. We have K ˜ G ( Y w ) ≃ K ˜ T ( X w ) ≃ Z [ ˜ X ]. Then a direct computation shows thatthe induced homomorphism Lj ∗ w of equivariant K -groups is given by Lj ∗ w : K ˜ G ( B × B ) → K ˜ G ( Y w ) ≃ Z [ ˜ X ] ,x ⊗ x x w ( x ) , where x ⊗ x ∈ Z [ ˜ X ] ⊗ Z [ ˜ X ] W Z [ ˜ X ] ≃ K ˜ G ( B × B ). Proposition 2.2.
Let l ∈ K ˜ G ( B × B ) , then l ∈ K G ( B × B ) if and onlyif Lj ∗ w ( l ) ∈ K G ( Y w ) for any w ∈ W .Proof. Denote by k w : X w ֒ → B , k i : X i ֒ → B and j i : F i ֒ → B × B thenatural immersions. Then we have K ˜ G ( B × B ) Lj ∗ i / / ≀ (cid:15) (cid:15) K ˜ G ( F i ) res / / ≀ (cid:15) (cid:15) K ˜ G ( Y u i ) ≀ (cid:15) (cid:15) K ˜ T ( B ) Lk ∗ i / / K ˜ T ( X i ) res / / K ˜ T ( X u i ) Hence our proposition is equivalent to the following statement
For any l ∈ K ˜ T ( B ) , l ∈ K T ( B ) if and only if Lk ∗ w ( l ) ∈ K T ( X w ) forany w ∈ W . The ”only if” part is obviously. We show the ”if” part. Note that thesupport supp( l ) of l belongs to some X i , we argue by induction on i . Ifsupp( l ) = ∅ , that is, l = 0, then the statement follows trivially. Supposethe proposition holds for any element in K ˜ T ( B ) whose support belongsto X i − . We show it also holds for z ∈ K ˜ T ( B ) with supp( l ) ⊂ X i .Since we have the following ˜ T -equivariant morphism X u i / / ≀ (cid:15) (cid:15) X u i ≀ (cid:15) (cid:15) L α ∈ R − ,u − i ( α ) > C α / / L β ∈ R + C u i ( β ) , where C α denotes the one dimensional vector space C on which ˜ T actsvia the character α . By Proposition 5.4.10 in [CG], Lk ∗ u i ( l ) = Y α ∈ R + ,u − i ( α ) > (1 − α − ) l | X ui ∈ K T ( X u i ) . Since Q α ∈ R + ,u − i ( α ) > (1 − α − ) ∈ K T ( X u i ), then l | X ui ∈ K T ( X u i ) by2.1. Since X u i is a ˜ T -stable open subset of X i , there exists l ′ ∈ K T ( X i )such that l ′ | X ui = l | X ui . Then supp( l − l ′ ) ⊂ X i − . Using inductionhypothesis, we have l − l ′ ∈ K T ( B ). Hence l = ( l − l ′ ) + l ′ ∈ K T ( B )and the proof is finished. (cid:3) Corollary 2.3.
Let z = f − w χf ′ with f, f ′ ∈ Σ and χ ∈ ˜ X + . Then z ∈ W if and only if σ ( z ) = V ( χ ) y f ⊗ x f ′ ∈ K G ( B × B ) .Proof. Obviously z ∈ W if and only if x − f χx f ∈ X . On the other hand, V ( χ ) ∈ Z [ χX ]. By Lemma 2.1, y f ∈ Z [ x − f X ]. Then by Proposition2.2, V ( χ ) y f ⊗ x f ′ ∈ K G ( B × B ) if and only if V ( χ ) y f x f ′ ∈ Z [ X ], whichis equivalent to x − f χx f ∈ X . (cid:3) Proof of part (b) and (c) of Theorem 1.1.
By Corollary 2.3, σ ( J ) ⊂ K G ( B × B ). It remains to show that K G ( B × B ) ⊂ σ ( J ). Let l ∈ K G ( B × B ). Due to ( ∗ ), we can assume that l = X f,f ′ ∈ Σ a f,f ′ V ( χ f,f ′ ) y f ⊗ x f ′ with a f,f ′ ∈ Z , χ f,f ′ ∈ ˜ X + . Since y f ⊗ x f = σ ( t f − w f ) ∈ K G ( B × B )for each f ∈ Σ, we have( y f ⊗ x f ) ∗ l ∗ ( y f ′ ⊗ x f ′ ) = a f,f ′ V ( χ f,f ′ ) y f ⊗ x f ′ ∈ K G ( B × B ) . SIAN NIE
Hence by Corollary 2.3, f − w χ f,f ′ f ′ ∈ J whenever a f,f ′ = 0. Hence l = P f,f ′ a f,f ′ σ ( t f − w χ f,f ′ f ′ ) ∈ σ ( J ). (cid:3) Some results on K G ( P × P ) Let I ⊂ ∆ be a subset and P the parabolic subgroup of type I containing B . Define P = G/P be the variety of all parabolic subgroupsof type I .Let D be the set of double cosets of W with respect to W I ⊂ W .Here W I is the parabolic subgroup generated by I . For each w ∈ W ,define Z ¯ w = { ( P, P ′ ) ∈ P × P | ( P, P ′ ) is conjugate to ( P, w P ) } , where ¯ w denotes the double coset W I wW I . Then we have P × P = a d ∈ D Z d . For any double coset d ∈ D , there is a unique element u d ∈ d suchthat u d is the smallest in d under the Bruhat order. Let d, d ′ ∈ D , wesay d ≥ d ′ if u d ≥ u d ′ under the Bruhat order. Lemma 3.1.
With notations as above, then we have d ≥ d ′ if and onlyif ¯ Z d ⊃ Z d ′ .Proof. Consider the natural projections Y u d → Z d and Y u d ′ → Z d ′ ,which are restrictions of the natural projection p : B × B → P × P to Y u d and Y u d ′ respectively. Since u d ≥ u d ′ , we have ¯ Y u d ⊃ Y u d ′ . Hence Z d ′ ⊂ p ( ¯ Y u d ′ ) ⊂ p ( ¯ Y u d ) = ¯ Z d . The “if part” follows from the fact thatthe morphism p above is projective. (cid:3) Proposition 3.2.
Let d ∈ D . We have the following short exact se-quence: → K G ( Z With notations in 3.2. The morphism R ( p | Y ud ) ∗ : K G ( Y u d ) −→ K G ( Z ¯ w ) defined above is surjective.Proof. Let’s compute R ( p | Y ud ) ∗ ([ θ χ ]). Note that p is smooth and projec-tive, and Z ¯ w is integral (as a scheme). Hence by Corollary 12.9 of [Har],we have R i p | Y ud ∗ ( θ χ ) is vector bundle over Z ¯ w and R i p ∗ ( θ χ ) | ( P, w P ) = H i ( p − ( P, w P ) , θ χ | p − ( P, w P ) ) for all i ≥ 0. Hence when χ is a dominantweight, Rp ∗ ( θ χ ) | ( P, w P ) = V χ , where V χ is the irreducible P ∩ w P -modulewith highest weight χ . Note that all [ V χ ] with χ dominant generates K P ∩ w P (pt) = K G ( Z ¯ w ). Hence Rp ∗ is surjective. (cid:3) Corollary 3.4. The natural morphism Rp ∗ : K G ( B × B ) → K G ( P × P ) is surjective.Proof. Let l ∈ K G ( P × P ). We show that l lies in the image of Rp ∗ byinduction on the dimension of the support of supp( l ). If supp( l ) = ∅ ,hence l = 0, it follows trivially. Now we assume that the statementholds for any l ′ such that supp( l ′ ) ⊂ Z References [CG] N. Chriss and V. Ginzburg, Representation Theory and Complex Geom-etry , Birkhauser, Boston, 1997.[Har] R. Hartshorne, Algebraic Geometry , Springer, New York, 1997.[KL] D. Kazhdan and G. Lusztig, Proof of The Delign-Langlands Conjecturefor Hecke Algebras , Ivent.Math., Vol.87, 1987, 153–215.[L1] G. Lusztig, Cells in Affine Weyl Groups, II , J.Alg., (1987), no. 2,536–548.[L2] G. Lusztig, Bases in Equivariant K-theory. II , Rerpresentation Theory,Vol. 3 (1999), 281–353.[Xi] N. Xi, The Based Ring of The Lowest Two-Sided Ring of an Affine WeylGroup , J.Alg., (1990), 356–368.[Shi] J. Shi,