The Steinberg Variety and Representations of Reductive Groups

THE STEINBERG V ARIETY AND REPRESENT A TIONS OF R EDUCTIVE GR OUPS J. MA TTHEW DOUGLASS AND GERHARD R ¨ OHRLE De dic ate d to Gu s L ehr er on the o c c asion of his 60th birthday Abstract. W e g ive an overview of s o me of the main results in geometric r e presentation theory that hav e been pr ov ed b y means of the Stein ber g v ariety . Stein b e r g’s insight w as to use such a v ariety of triples in order to pr ov e a conjectured for mu la by Grothendieck. The Stein b erg v ariet y w as later used to give an a lternative approach to Springer’s r e presentations and play ed a central r ole in the pro of of the Deligne-Lang lands co njecture for Hecke algebra s by Ka zhdan and Lusztig. 1. Intro duction Supp ose G is a connected, reductiv e algebraic group deﬁned o v er an algebraically c losed ﬁeld k , B is the v ariety o f Borel subgroups of G , and u is a unip oten t elemen t in G . Let B u denote t he closed sub v ariet y of B consisting of those Borel subgroups that con tain u , let r denote the rank of G , a nd let C denote the conjugacy class of u . In 1976, motiv ated b y the problem o f pro ving the equalit y conjectured b y Grothendiec k ( ∗ ) dim Z G ( u ) = r + 2 dim B u , in order to get the multiplic it y 2 in ( ∗ ) in the picture, Stein b erg [Ste76] in tro duced a v ariety of triples S = { ( v , B , B ′ ) ∈ C × B × B | v ∈ B ∩ B ′ } . By analyzin g the geometry of the v ariety S , he was able to prov e ( ∗ ) in most cases. In addition, b y exploiting the fa ct that t he G -orbits on B × B ar e canonically indexed b y elemen ts of the W ey l group of G , he sho wed that S could b e used to establish relationships b et w een W eyl group elemen ts a nd unip otent elemen ts in G . No w let g denote the Lie alg ebra of G , and let N denote the v ariet y of nilp oten t elemen ts in g . The Steinb er g variety of G is Z = { ( x, B , B ′ ) ∈ N × B × B | x ∈ Lie ( B ) ∩ Lie( B ′ ) } . If the c haracteristic of k is zero or go o d for G , then there is a G -equiv aria nt isomorphis m b et w een N a nd U , the v ariet y of unip oten t elemen ts in G , and so Z ∼ = { ( u, B , B ′ ) ∈ U ×B ×B | u ∈ B ∩ B ′ } . In the thirt y years since Stein b erg ﬁrst exploited the v ariet y S , the Stein b erg v ariety has pla y ed a k ey role in adv ancing our unde rstanding of ob jects that at ﬁrst seem to be quite unrelated: Date : May 28, 2018. 2000 Mathematics Subje ct Classiﬁc ation. P rimary 22E46, 19L4 7 , 20 G05; Secondary 14F99, 20G99 . The authors would like to thank their charming wives for their un wa vering supp ort during the prepar ation of this pap er. 1 2 J.M. D OUGLASS A ND G. R ¨ OHRLE • Represen tations of the W eyl group W of G • The geometry o f nilp oten t o rbits in g a nd their cov ers • Diﬀeren tia l op erators on B • Primitiv e ideals in the univ ersal env eloping algebra of g • Represen tations of p -adic groups and the lo cal Lang lands program In this pap er we hop e to g ive readers who are familiar with some asp ects of the represen- tation the ory of semisimple algebraic groups, or Lie groups, but who are not sp ecialists in this particular ﬂa vor of geometric represen tation theory , a n ov erview o f the main results that ha v e b een pro ved using the Stein b erg v ariet y . In the pro cess w e hop e to mak e these results more a ccessible to non-exp erts a nd at the same time emphasize the unifying role pla yed by the Stein b erg v ariet y . W e assume that the reader is quite familiar with the basics of the study of algebraic groups, especially reductiv e algebraic groups and their Lie algebras, as con ta ined in the b o oks by Springer [Spr98 ] and Carter [Car85] for example. W e will more or less follow the historical dev elopment, b eginning with concrete, geometric constructions and then progressing to increasingly more adv anced and abstract not ions. In § 2 we analyze the geometry of Z , including applications to orbital v arieties, character- istic v arieties and primitiv e ideals, and g eneralizations. In § 3 w e study the Borel-Mo or e homology o f Z and the relation with represe n tations of W eyl groups. So on after Stein b erg intro duced his v ariet y S , Kazhdan and Lusztig [KL80] deﬁned an action of W × W on the top Borel-Mo ore homology group of Z . F ollo wing a suggestion of Springer, t hey sho wed that the represen ta tion of W × W on the top homology group, H 4 n ( Z ), is the t w o-sided regular represen tat ion of W . Somewhat later, Ginzburg [Gin86] and indep enden tly Ka shiw ara and T anisak i [KT84], deﬁned a mu ltiplication on the total Borel-Mo ore homolo gy of Z . With this m ultiplication, H 4 n ( Z ) is a subalgebra isomor- phic to the group algebra of W . The a utho rs [DR 08a] [D R 08b] hav e used Ginzburg’s construction t o describ e the top Bo rel- Mo ore homology groups of the g eneralized Steinberg v arieties X P , Q 0 , 0 and X P , Q reg , reg (see § 2.4) in terms of W , a s w ell as to give an explicit, elemen tary , computation o f the to tal Borel-Mo ore homology of Z as a gra ded a lgebra: it is isomorphic to the smash pro duct of the coinv arian t algebra of W and the gro up algebra of W . Orbital v a r ieties arise naturally in t he geometry of the Steinberg v ariet y . Using the con vo- lution pro duct formalism, Hinic h and Joseph [HJ05] hav e recen tly prov ed an old conjecture of Joseph a b out inclusions of closures o f orbital v arieties. In § 4 w e study the equiv arian t K -theory of Z and what is undoubtedly the most importa nt result to date in v o lving the Stein b erg v ariet y: the Kazhdan-Lusztig isomorphism [KL87] b et w een K G × C ∗ ( Z ) and the extended, aﬃne Hec k e algebra H . Using this isomorphism, Kazhdan and L usztig w ere able to classify the irreducible represen tations of H and hence to classify the represen ta tions con taining a v ector ﬁxed b y an Iw ahori subgroup of the p -adic group with the same type as the Langlands dual L G of G . In this wa y , the Steinberg v ar iety pla ys a k ey role in the lo cal Langlands prog r am and also leads to a b etter understanding of the extended aﬃne Hec ke algebra. V ery recen t w ork in v o lving the Stein b erg v ariet y cen ters around a t tempts to categor ify the isomorphism b etw een the sp ecialization of K G × C ∗ ( Z ) at p and the Hec k e algebra of Iw ahori bi-in v ariant functions on L G ( Q p ). Because o f time and space constrain ts, w e leav e a discussion o f this researc h to a future article. STEINBERG V ARIETY AN D R EPRESENT A TIONS 3 2. Geometr y F or the res t of this paper, in order to simplify the exposition, we assume that G is con- nected, the deriv ed g roup of G is simply connec ted, and that k = C . Most of the r esults b elo w hold, with ob vious mo diﬁcations, fo r an arbitrary reductiv e algebraic gro up when the c hara cteristic of k is zero or ve ry go o d for G ( for the deﬁnition of “ve ry go o d ch aracteristic” see [Car85, § 1.1 4 ]). Fix a Borel subgroup B in G and a maximal torus T in B . Deﬁne U to b e the unip otent radical of B and deﬁne W = N G ( T ) /T to b e the W eyl gro up o f ( G, T ). Set n = dim B and r = dim T . W e will use the con v en tion that a low ercase f raktur letter denotes the Lie algebra of the algebraic gro up denoted b y the cor r espo nding upp ercase roman letter. F or x in N , deﬁne B x = { g B g − 1 | g − 1 x ∈ b } , the S pringer ﬁbr e a t x . 2.1. Irreducible comp onen t s of Z , W eyl group elemen ts, and nilp otent orbits. W e b egin analyzing the g eometry of Z using ideas that go bac k to Steinberg [Ste76] and Spaltenstein [Spa82]. The group G acts o n B by conjugatio n and on N by t he adjoint action. This latter action is denoted b y ( g , x ) 7→ g · x = g x . Th us, G acts “diagonally” on Z . Let π : Z → B × B b e the pro jection on the second and third factors. By the Bruhat Lemma, the elemen ts of W parametrize the G -orbits on B × B . An elemen t w in W corre- sp onds t o the G -orbit containing ( B , wB w − 1 ) in B × B . D eﬁne Z w = π − 1  G ( B , w B w − 1 )  , U w = U ∩ w U w − 1 , and B w = B ∩ w B w − 1 . The v arieties Z w pla y a k ey role in the rest o f this pap er. F or w in W , the restriction of π to Z w is a G -equiv ariant mo r phism from Z w on to a transitiv e G -space. The ﬁbre ov er the po in t ( B , w B w − 1 ) is isomorphic to u w and so it follo ws f rom [Slo80, I I 3.7 ] that Z w is isomorphic to the a sso ciated ﬁbre bun dle G × B w u w . Th us, Z w is ir r educible and dim Z w = dim G − dim B w + dim u w = 2 n . F urthermore, eac h Z w is lo cally closed in Z and so it follow s that { Z w | w ∈ W } is the set o f irreducible comp onen t s of Z . No w let µ z : Z → N denote the pro jection on the ﬁrst component. F or a G -orbit, O , in N , set Z O = µ − 1 z ( O ) and ﬁx x in O . Then the restriction of µ z to Z O is a G -equiv a rian t morphism from Z O on to a transitiv e G -space. The ﬁbre o v er x is isomorphic to B x × B x and so it follows from [Slo80, I I 3.7] that Z O ∼ = G × Z G ( x ) ( B x × B x ). Spaltenstein [Spa82, § I I.1] has sho wn that the v ar iety B x is equidimensional a nd Stein b erg and Spaltenstein ha v e sho wn that dim Z G ( x ) = r + 2 dim B x . This implies the follo wing results due to Stein b erg [Ste76, Prop osition 3.1]: (1) dim Z O = dim G − dim Z G ( x ) + 2 dim B x = dim G − r = 2 n . (2) Ev ery irreducible comp o nent of Z O has the fo rm G ( { x } × C 1 × C 2 ) = G ( { x } × ( Z G ( x )( C 1 × C 2 ))) where C 1 and C 2 are irreducible comp onents of B x . (3) A pair, ( C ′ 1 , C ′ 2 ), of irreducible comp onen ts of B x determines the same irreducible comp onen t o f Z O as ( C 1 , C 2 ) if and only if there is a z in Z G ( x ) with ( C ′ 1 , C ′ 2 ) = ( z C 1 z − 1 , z C 2 z − 1 ). 4 J.M. D OUGLASS A ND G. R ¨ OHRLE F rom (2) w e see that Z O is equidimensional with dim Z O = 2 n = dim Z and from (3) w e see that there is a bij ection b etw een irreducible comp onen ts of Z O and Z G ( x )-orbits on the set of irreducible comp onen ts of B x × B x . The closures of the irreducible c omp onen ts of Z O are closed, irreducible, 2 n -dimensional sub v arieties of Z and so eac h irreducible comp o nen t of Z O is of the form Z O ∩ Z w for some unique w in W . Deﬁne W O to be the subs et of W that parametrizes the irreducible comp onen t s of Z O . Then w is in W O if and only if Z O ∩ Z w is an irreducible comp onen t of Z O . Clearly , W is the disjoin t union of the W O ’s as O v aries o ver the nilp oten t orbits in N . The subsets W O are called two-side d Steinb er g c el ls . Tw o-sided Stein b erg cells hav e sev eral prop erties in common with t w o-sided Kazhdan-Lusztig c ells in W . Some of the prop erties of tw o-sided Stein b erg cells will b e describ ed in the next subsection. Kazhdan-Lusztig cells w ere in tro duced in [K L79, § 1]. W e will brieﬂy review this theory in § 4.4. In general there are more tw o-sided Stein b erg cells than tw o-sided Ka zhdan- L usztig cells. This ma y b e seen as follo ws. Clearly , t w o - sided Stein b erg ce lls are in bijection w ith the set of G -orbits in N . Tw o-sided K azhdan-Lusztig cells may b e related to nilp oten t orbits through the Springer corresp ondence using Lusztig’s analysis o f Kazhdan-Lusztig cells in W eyl groups. W e will review the Springer corresp ondence in § 3.4 b elo w, where w e will see that there is an injection from the set of nilp oten t orbits to the set o f irreducible represen tations of W giv en b y asso ciating with O the represen tatio n of W on H 2 d x ( B x ) C ( x ) , where x is in O a nd C ( x ) is the comp onen t group of x . Tw o-sided Kazhdan- Lusztig cells determine a ﬁltration of the group algebra Q [ W ] b y t wo-sided ideals (see § 4.4) and in the asso ciated graded W × W -mo dule, eac h su mmand contains a distinguished represen tation that is called sp e cial (see [Lus79] and [Lus84, Chapter 5]). The case-b y-case computation of the Springer correspondence sho ws that ev ery sp ecial represen tatio n of W is equiv alen t to the represen tation of W on H 2 d x ( B x ) C ( x ) for some x . The resulting nilp oten t orbits ar e called sp e cial nilp otent orbits. If G has t yp e A l , then ev ery irreducible represen ta t ion of W and eve ry nilp otent o rbit is sp ecial but otherwise there are non-sp ecial irreducible represen tation of W and nilp oten t orbits. Although in general there are f ew er t w o- sided Kazhdan-L usztig cells in W t ha n tw o - sided Stein b erg cells, Lusz tig [Lus89b, § 4] ha s constructed a bijection b et wee n the set of t wo-sided Ka zhdan- L usztig cells in the extended, aﬃne, W eyl gro up, W e , and the set of G - orbits in N . Th us, t here is a bijection b et w een t w o- sided Stein b erg cells in W and tw o- sided Kazhdan-Lusztig cells in W e . W e will describ e this bijection in § 4.4 in connection with the computation of the equiv ariant K -theory of the Stein b erg v ariet y . Supp ose O is a nilp oten t or bit and x is in O . W e can explicitly describ e the bijection in (c) ab o ve b etw een W O and the Z G ( x )-orbits o n the set of pairs of irreducible comp onen ts of B x as follow s. If w is in W O and ( C 1 , C 2 ) is a pair of irreducible comp onents o f B x , then w corresp onds to the Z G ( x )-orbit of ( C 1 , C 2 ) if and o nly if G ( B , w B w − 1 ) ∩ ( C 1 × C 2 ) is dense in C 1 × C 2 . Using the isomorphism Z w ∼ = G × B w u w w e see that Z O ∩ Z w ∼ = G × B w ( O ∩ u w ). Therefore, w is in W O if and only if O ∩ u w is dense in u w . This sho ws in particular that W O is closed under taking inv erses. W e conclude this subsection with some examples of t w o-sided Steinberg cells. STEINBERG V ARIETY AN D R EPRESENT A TIONS 5 When x = 0 w e ha v e Z { 0 } = Z w 0 = { 0 } × B × B whe re w 0 is t he longest elemen t in W . Therefore, W { 0 } = { w 0 } . A t the other extreme, let N reg denote the regular nilp oten t orbit. Then it follows from the fact that ev ery regular nilp o t en t elemen t is contained in a unique Borel subalgebra that W N reg con ta ins just the iden tity elemen t in W . F or G of type A l , it follo ws fro m a result of Spaltenstein [Spa76] that tw o elemen ts of W lie in the same t w o-sided Stein b erg cell if and only if they yie ld the same Y oung diagram under the Robinson-Sc hensted corresp ondence. A more reﬁned result due to Stein b erg will b e discussed at the end of the next subsection. 2.2. Orbital v ariet ies. Supp ose that O is a nilp oten t orbit. An orbital variety for O is an irreducible component of O ∩ u . An orbital variety is a subv ariet y of N that is orbital for some nilp oten t orbit. The reader sh ould b e a w are that some times an orbital v ariety is deﬁned as the closure of an irreducible comp onent of O ∩ u . W e will see in this subsection that orbita l v arieties can b e used to decomp ose t w o-sided Stein b erg cells into left and righ t Stein b erg cells and to reﬁne the relationship b etw een nilp oten t orbits and elemen ts of W . When G is of t yp e A l and W is the symmetric group S l +1 , the decomp osition of a t w o -sided Stein b erg ce ll into left and righ t Stein b erg ce lls can b e view ed as a geometric realization of the Robinson-Sc hensted corresp ondence. W e will see in the next subsection that orbital v arieties arise in the theory of asso ciated v arieties of ﬁnitely generated g -mo dules. Fix a nilp oten t or bit O and an elemen t x in O ∩ u . Deﬁne p : G → O b y p ( g ) = g − 1 x a nd q : G → B b y q ( g ) = g B g − 1 . Then p − 1 ( O ∩ u ) = q − 1 ( B x ). Spaltenstein [Spa82, § I I.2] has sho wn that (1) if C is an irreducible comp onen t of B x , then pq − 1 ( C ) is an o rbital v ariety for O , (2) ev ery orbital v ariety for O has the form pq − 1 ( C ) for some irreducible comp onent C of B x , and (3) pq − 1 ( C ) = pq − 1 ( C ′ ) for comp onen ts C and C ′ of B x if and only if C and C ′ are in the same Z G ( x )-orbit. It follows immediately that O ∩ u is equidimensional and all orbital v arieties for O ha ve the same dimension: n − dim B x = 1 2 dim O . W e decomp ose t wo-sided Stein b erg cells in to left and righ t Stein b erg cells following a construction of Joseph [Jos84, § 9]. Supp ose V 1 and V 2 are or bita l v arieties f or O . Choo se irr educible comp onen t s C 1 and C 2 of B x so t ha t pq − 1 ( C 1 ) = V 1 and pq − 1 ( C 2 ) = V 2 . W e ha ve seen that there is a w in W O so that Z O ∩ Z w = G ( { x } × Z G ( x )( C 1 × C 2 )). Clearly , µ − 1 z ( x ) ∩ Z w ⊆ µ − 1 z ( x ) ∩ Z w . Since b oth sides are closed, b o th sides a re Z G ( x )-stable, and the right hand side is the Z G ( x )-saturation of { x } × C 1 × C 2 , it follows that µ − 1 z ( x ) ∩ Z w = µ − 1 z ( x ) ∩ Z w . Let p 2 denote the pro jection of Z O to B by p 2 ( x, B ′ , B ′′ ) = B ′ . Then pq − 1 p 2 ( µ − 1 z ( x ) ∩ Z w ) = B ( O ∩ u w ). Also, pq − 1 p 2  µ − 1 z ( x ) ∩ Z w  = pq − 1 p 2 ( { x } × Z G ( x )( C 1 × C 2 )) = pq − 1 ( Z G ( x ) C 1 ) = V 1 . Since O ∩ u w is dense in u w w e hav e B u w ∩ O = B ( O ∩ u w ) ⊆ V 1 . Ho w ev er, since µ − 1 z ( x ) ∩ Z w is a dense, Z G ( x )-stable subse t of µ − 1 z ( x ) ∩ Z w , it follows that dim B ( O ∩ u w ) = dim pq − 1 p 2  µ − 1 z ( x ) ∩ Z w  6 J.M. D OUGLASS A ND G. R ¨ OHRLE = dim p 2  µ − 1 z ( x ) ∩ Z w  + dim B − dim Z G ( x ) = dim B x + dim B − r − 2 dim B x = n − dim B x and so B u w ∩ O = V 1 . A similar a rgumen t sho ws that B u w − 1 ∩ O = V 2 . This prov es the follow ing theorem. Theorem 2.1. I f O is a n i l p otent orbit an d V 1 and V 2 ar e orbital varieties for O , then ther e is a w in W O so that V 1 = B u w ∩ O and V 2 = B u w − 1 ∩ O . Con vers ely , if w is in W O , then u w is irreducible and the arguments ab ov e sho w that u w ∩ O is dense in u w and then that B u w ∩ O is an o rbital v ariet y . This prov es the next pro p osition. Prop osition 2.2. Orbital varieties ar e the subsets of u of the form B u w ∩ O , wher e u w ∩ O is dense in u w . F or w in W , deﬁne V l ( w ) = B u w − 1 ∩ O when w is in W O . F or w 1 and w 2 in W , deﬁne w 1 ∼ l w 2 if V l ( w 1 ) = V l ( w 2 ). Then ∼ l is an equiv alence relation a nd the equiv alence classes are called left Steinb er g c el ls. Similarly , deﬁne V r ( w ) = B u w ∩ O when w is in W O and w 1 ∼ r w 2 if V r ( w 1 ) = V r ( w 2 ). The equiv alence classes fo r ∼ r are called right S teinb er g c e l ls. Clearly , eac h t w o -sided Steinberg cell is a disjoint union o f left Stein b erg cells and is also the disjoin t union of right Steinberg cells. Precis ely , if w is in W O , then W O = a y ∈ V r ( w ) V l ( y ) = a y ∈ V l ( w ) V r ( y ) . It follow s from Theorem 2.1 that the rule w 7→ ( V r ( w ) , V l ( w )) deﬁnes a surjection fro m W to the set of pairs of orbital v a rieties for the same nilp oten t orbit. W e will see in § 3.4 that the n um b er of o rbital v arieties for a nilp otent orbit O is the dimension of the Springer represen tation of W corresp onding to the trivial represen t a tion of the comp onen t group of a ny elemen t in O . Denote this represen tation of W by ρ O . The n the nu m b er of pairs ( V 1 , V 2 ), whe re V 1 and V 2 are orbital v arieties f o r the same nilp oten t orbit, is P O (dim ρ O ) 2 . In general this sum is strictly smaller tha n | W | . Equiv alen t ly , in general, there are more irreducible represen ta t io ns of W than G -orbits in N . Ho w ev er, if G has t yp e A , f o r example if G = SL n ( C ) or GL n ( C ), then eve ry irreducib le represen tation o f W is of the f o rm ρ O for a unique nilp oten t orbit O . In this case w 7→ ( V r ( w ) , V l ( w )) deﬁnes a bijection f rom W to the set of pairs o f orbital v arieties for the same nilp oten t orbit. Stein b erg has shown that t his bijection is essen tially giv en b y the Robinson-Sc hensted corresp o ndence. In more detail, using the notation in the pro of of Theorem 2.1, supp ose that O is a nilp oten t orbit, V 1 and V 2 are orbita l v arieties for O , and C 1 and C 2 are the corresp onding irreducible compo nen ts in B x . In [Ste88] Stein b erg deﬁnes a function from B to the s et of standard Y oung tableaux and sho ws that G ( B , w B w − 1 ) ∩ ( C 1 × C 2 ) is dens e in C 1 × C 2 if and only if the pair o f standard Y o ung tableaux asso ciated to a generic pair ( B ′ , B ′′ ) in C 1 × C 2 is the same as the pair of standard Y oung tableaux asso ciated to w by the Robinson-Sc hensted corresp ondence. F or more details, see a lso [Dou96]. An op en pr o blem, eve n in t yp e A , is determining the orbit closures of orbital v arieties. Some rudimen tary information may b e o btained b y considering the top Borel-Mo ore homol- ogy group o f Z (see § 3 b elow and [HJ05, § 4, § 5]). STEINBERG V ARIETY AN D R EPRESENT A TIONS 7 2.3. Asso ciated v ariet ies and c haracteristic v arieties. The Stein b erg v a r iet y and or- bital v arieties also arise naturally in the Beilins on-Bernstein theory of algebraic ( D , K ) - mo dules [BB81]. This w as ﬁrst observ ed b y Borho and Brylinski [BB85] and Ginzburg [Gin86]. In this subsection w e b egin with a review of the Beilinson-Bernstein Lo calization Theorem and its connection with the computation of c ha r acteristic v arieties a nd asso ciated v arieties. Then we describ e an equiv ariant version of this theory . It is in the equ iv arian t theory that the Stein b erg v ariety naturally o ccurs. F or a v ar iety X (ov er C ), let O X denote the structure sheaf of X , C [ X ] = Γ( X , O X ) the algebra of global, regular functions o n X , and D X the sheaf of algebraic diﬀerential op erators on X . On an op en sub v ariety , V , of X , Γ( V , D X ) is the subalgebra of Hom C ( C [ V ] , C [ V ]) generated by m ultiplication b y eleme n ts of C [ V ] and C -linear deriv ations of C [ V ]. Deﬁne D X = Γ( X , D X ), the algebra of global, alg ebraic, diﬀeren tial op erators on X . A quasi-c oher ent D X -mo dule is a left D X -mo dule that is quasi-coheren t when considered as an O X -mo dule. Generalizing a familiar result for aﬃne v arieties, Beilinson-Bernstein [BB8 1, § 2] ha v e pro v ed that f or X = B , the glo ba l section functor, Γ( B , · ), deﬁnes an eq uiv alence of categories b etw een the category of quasi-coheren t D B -mo dules and the category o f D B - mo dules. In turn, the algebra D B is isomorphic to U ( g ) /I 0 , where U ( g ) is the univ ersal en veloping algebra of g and I 0 denotes the tw o-sided ideal in U ( g ) generated by the k ernel of the trivial c haracter of the cen ter of U ( g ) (see [BB8 2, § 3]). Thus , the category of D B -mo dules is equiv alen t to the category of U ( g ) /I 0 -mo dules, that is, the category of U ( g )-mo dules with trivial cen tral character. Comp osing these t w o equiv alences we see that the category of quasi-coheren t D B -mo dules is equiv alen t to the category of U ( g )- mo dules with trivial cen tral c haracter. In this e quiv- alence, coheren t D B -mo dules (that is, D B -mo dules that are coheren t when considered as O B -mo dules) corresp ond to ﬁnitely generated U ( g )-mo dules with trivial cen tral c haracter. The equiv alence of categories b etw een coheren t D B -mo dules and ﬁnitely generated U ( g )- mo dules with trivial cen tral c haracter has a geometric shado w that can b e described using the “momen t map” of the G - action on t he cotangen t bundle o f B . Let B ′ b e a Borel subgroup of G . Then using the Killing form on g , the cotangen t space to B at B ′ ma y b e iden tiﬁed with b ′ ∩ N , the nilradical of b ′ . D eﬁne e N = { ( x, B ′ ) ∈ N × B | x ∈ b ′ } and let µ : e N → N b e the pro jection on the ﬁrst factor. Then e N ∼ = T ∗ B , the cotangen t bundle of B . It is easy to see that Z ∼ = e N × N e N ∼ = T ∗ B × N T ∗ B . Using t he orders of diﬀerential op erators, w e obtain a ﬁltr a tion of D X . With respect to this ﬁltration, the asso ciated graded sheaf gr D B is isomorphic to the direct image p ∗ O T ∗ B , where p : T ∗ B → B is t he pro jection. Let M b e a coheren t D B -mo dule. Then M has a “go o d” ﬁltration suc h that g r M is a coheren t gr D B -mo dule. Since gr D B ∼ = p ∗ O T ∗ B , we see that gr M has the structure of a coheren t O T ∗ B -mo dule. The ch ar acteristic variety of M is the supp ort in T ∗ B of the O T ∗ B - mo dule gr M . Using the isomorphism T ∗ B ∼ = e N , we iden tify the c haracteristic v ariety o f M with a closed sub v ariet y of e N and denote this latter v ariet y by V e N ( M ). It is kno wn that V e N ( M ) is indep enden t of the choice of go o d ﬁltration. 8 J.M. D OUGLASS A ND G. R ¨ OHRLE No w consider the en v eloping algebra U ( g ) with t he standard ﬁltration. By the PBW Theorem, gr U ( g ) ∼ = Sym( g ), t he symmetric algebra of g . Using the Killing form, w e iden tify g with its linear dual, g ∗ , and gr U ( g ) with C [ g ]. Let M be a ﬁnitely generated U ( g )-mo dule. Then M has a “go o d” ﬁltration suc h that the asso ciated graded mo dule, gr M , a mo dule for gr U ( g ) ∼ = C [ g ], is ﬁnitely generated. The asso ciate d variety of M , denoted b y V g ( M ), is the supp ort o f the C [ g ]- mo dule gr M – a closed sub v ariet y of g . It is kno wn that V g ( M ) is indep enden t of the c ho ice of g o o d ﬁltration. Borho and Br ylinski [BB85, § 1.9] ha ve pro v ed the following theorem. Theorem 2.3. Supp ose that M is a c oher en t D B -mo dule and let M denote the sp ac e of glob al se ctions of M . Then V g ( M ) ⊆ N and µ ( V e N ( M )) = V g ( M ) . There are equiv ar ian t v ersions of the ab ov e constructions whic h incorpo rate a s ubgroup of G that acts on B with ﬁnitely many orbits. It is in this equiv ariant con text that the Stein b erg v ariety and orbital v arieties make their app earance. Supp ose t hat K is a closed, connected, algebraic subgroup of G that acts on B with ﬁnitely man y or bits. The tw o sp ecial cases w e are in terested in are the “hig hest we igh t” case, when K = B is a Borel subgroup of G , and the “Harish-Chandra” case, when K = G d is the diagonal subgroup of G × G . In the gene ral setting, we supp ose that W is a ﬁnite set that indexes the K -o rbits on B b y w ↔ X w . Of course, in t he examples w e ar e intereste d in, we kno w that the W eyl group W indexes t he set of o r bits of K on B . F or w in W , let T ∗ w B denote the conormal bundle to the K -orbit X w in T ∗ B . Then letting k ⊥ denote the subspace of g or t hogonal to k with resp ect to the Killing form and using our iden tiﬁcation of T ∗ B with pair s, w e may iden tify T ∗ w B = { ( x, B ′ ) ∈ N × B | B ′ ∈ X w , x ∈ b ′ ∩ k ⊥ } . Deﬁne Y k ⊥ = µ − 1 ( k ⊥ ∩ N ). Then Y k ⊥ is closed, Y k ⊥ = ` w ∈ W T ∗ w B = ∪ w ∈ W T ∗ w B , and µ restricts to a surjection Y k ⊥ µ − → k ⊥ (see [BB85, § 2.4 ]) . Summarizing, w e hav e a comm utativ e diagram (2.4) Y k ⊥ / / µ   e N µ   k ⊥ ∩ N / / N where the horizontal arro ws are inclusions. Moreo v er, for w in W , dim T ∗ w B = dim B and T ∗ w B is lo cally closed in Y k ⊥ . Thus , the set of irr educible comp onents of Y k ⊥ is { T ∗ w B | w ∈ W } . A quasi-c oh er ent ( D B , K ) -mo dule is a K -equiv arian t, quasi-coheren t D B -mo dule (for the precise deﬁnition see [BB85, § 2]). If M is a coheren t ( D B , K )-mo dule, then V e N ( M ) ⊆ Y k ⊥ . Similarly , a ( g , K ) -mo dule is a g -mo dule with a compatible alg ebraic action of K (fo r the precise deﬁnition see [BB85, § 2]). If M is a ﬁnitely g enerated ( g , K )-mo dule, then V g ( M ) is con ta ined in k ⊥ . As in the no n- equiv ariant setting, Beilinson-Bernstein [BB81, § 2] hav e prov ed that the global section functor, Γ( B , · ), deﬁnes an equiv alence of catego ries b et w een the category of quasi-coheren t ( D B , K )-mo dules and the category of ( g , K )-mo dules with trivial cen tral char- acter. Under this equiv alence, coheren t ( D B , K )-mo dules corresp ond to ﬁnitely generated ( g , K )-mo dules with trivial cen tral c hara cter. The addition o f a K - action results in a ﬁner v ersion of Theorem 2.3 (see [BB85, § 4]). STEINBERG V ARIETY AN D R EPRESENT A TIONS 9 Theorem 2.5. Supp ose that M is a c oher ent ( D B , K ) -mo dule and let M denote the sp ac e of glob al se ctions of M . (a) The vari e ty V e N ( M ) is a union of irr e ducible c o mp onents of Y k ⊥ and so ther e is a subset Σ( M ) of W such that V e N ( M ) = S w ∈ Σ( M ) T ∗ w B . (b) The variety V g ( M ) is c o ntaine d in k ⊥ ∩ N and V g ( M ) = µ ( V e N ( M )) = [ w ∈ Σ( M ) µ  T ∗ w B  . No w it is time to unrav el the notation in the highest weigh t and Harish-Chandra cases. First consider the highest we igh t case when K = B . W e ha v e k ⊥ = b ⊥ = u . Hence, Y u ⊥ = µ − 1 ( u ) ∼ = { ( x, B ′ ) ∈ N × B | x ∈ u ∩ b ′ } . W e denote Y u ⊥ simply by Y and call it the c onormal variety. F or w in W , X w is t he set o f B -conjug a tes of w B w − 1 and T ∗ w B ∼ = { ( x, B ′ ) ∈ N × B | B ′ ∈ X w , x ∈ u ∩ b ′ } . The pro jection of T ∗ w B t o B is a B -equiv ariant surjection onto X w and so T ∗ w B ∼ = B × B w u w . The dia gram (2.4) b ecomes Y / / µ   e N µ   u / / N . Moreo ver, for w in W , µ ( T ∗ w B ) = B u w . Since µ is prop er, it f o llo ws that µ  T ∗ w B  = B u w is the closure of an orbital v ariety . Argumen ts in the spirit of those given in § 2.1 (see [HJ05, § 3]) sho w that if w e set Y w = T ∗ w B and Y O = µ − 1 ( O ∩ u ), then dim Y O = n , Y O is equidim ensional, and the se t of irreducible comp onen t s of Y O is { Y O ∩ Y w | w ∈ W O } . Next consider the Harish-Chandra case. In this setting, the am bient gro up is G × G and K = G d is the diag onal subgroup. Clearly , k ⊥ = g ⊥ d = { ( x, − x ) | x ∈ g } is isomorphic to g and so Y g ⊥ d = ( µ × µ ) − 1 ( g ⊥ d ) = { ( x, − x, B ′ , B ′′ ) ∈ g × g × B × B | x ∈ b ′ ∩ b ′′ ∩ N } . Th us, in this case, Y g ⊥ d is clearly isomorphic to the Steinberg v ariet y and w e may iden tify the restriction of µ × µ to Y g ⊥ d with µ z : Z → N . The diagram (2.4) b ecomes Z / / µ z   e N × e N µ × µ   N / / N × N where the b ottom horizon tal map is giv en by x 7→ ( x, − x ). Moreo v er, for w in W , T ∗ w ( B × B ) = { ( x, − x, B ′ , B ′′ ) | ( B ′ , B ′′ ) ∈ G ( B , w B w − 1 ) , x ∈ b ′ ∩ b ′′ ∩ N } ∼ = Z w . Let p 3 : Z → B b e the pro jection on the third factor . Then p 3 is G -equiv ariant, G acts transitiv ely on B , and the ﬁbre o v er B is isomorphic to Y . This giv es ye t another description of the Steinberg v ariet y: Z ∼ = G × B Y . No w consider the follo wing three cat ego ries: • coheren t ( D B×B , G d )-mo dules, Mo d ( D B×B , G d ) coh ; 10 J.M. D OUGLASS A ND G. R ¨ OHRLE • ﬁnitely g enerated ( g × g , G d )-mo dules with t rivial cen tral c haracter, Mod ( g × g , G d ) fg 0 , 0 ; and • ﬁnitely generated ( g , B )-mo dules with trivial cen tral character, Mo d ( g , B ) fg 0 . W e ha ve seen that the global se ction functor deﬁnes an equiv alence of categories be- t ween Mo d ( D B×B , G d ) coh and Mo d ( g × g , G d ) fg 0 , 0 . Bernstein and Gelfa nd [BG 80], a s well as Joseph [Jos79], ha v e constructed an equiv a lence of categories b etw een Mo d ( g × g , G d ) fg 0 , 0 and Mo d ( g , B ) fg 0 . Comp osing these tw o equiv alences of categories w e see tha t the category of coheren t ( D B×B , G d )-mo dules is equiv alen t to the category of ﬁnitely generated ( g , B )- mo dules with trivial cen tral c haracter, Mo d ( g , B ) fg 0 . Bo th equiv alences behav e w ell w ith r esp ect to c har- acteristic v arieties and associat ed v arieties and hence so do es their compo sition. This is the con ten t of the next theorem. The theorem extends Theorem 2.5 and summarizes the relationships betw een the v a rious construc tions in this subs ection. See [BB85, § 4] for the pro of. Theorem 2.6. Supp ose M is a c oher ent ( D B×B , G d ) -mo dule, M is the sp ac e of glob al se ction s of M , and L is the ﬁnitely ge n er ate d ( g , B ) -mo d ule with trivia l c entr al char acter c orr e sp ond- ing to M . L et Σ = Σ( M ) b e as in The or em 2.5. Then whe n µ × µ : Y g ⊥ d → g ⊥ d is ide n tiﬁe d with µ z : Z → N we ha ve: (a) The char acteristic variety of M i s V T ∗ ( B×B ) ( M ) = ∪ y ∈ Σ Z y , a union of irr e ducible c omp onents of the Steinb er g variety. (b) The asso ciate d variety of M is V g ( M ) = µ z ( V g ( M )) = ∪ y ∈ Σ G u y = G · V u ( L ) , so t he asso ciate d variety of M is the ima ge under µ z of the char acteristic variety of M and is also the G -satur a tion of the asso ciate d variety of L . (c) The asso ciate d variety of L is V u ( L ) = ∪ y ∈ Σ B u y , a union o f closur es of orbital va ri- eties. The c haracteristic v ariety o f a coherent ( D B×B , G d )-mo dule is the union of the c haracteris- tic v arieties of its compo sition factors. Similarly the associated v ariet y of a ﬁnitely g enerated ( g × g , G d )-mo dule or a ﬁnitely generated ( g , B )- mo dule dep ends only on its compo sition factors. Th us, computing c haracteristic and associated v arieties reduces to the case of simple mo dules. The simple ob jects in eac h of these categories a r e indexed b y W , s ee [BB81, § 3] and [BB85, § 2.7, 4.3, 4.8]. If w is in W O and M w , M w , and L w are corresp onding simple mo dules, t hen it is shown in [BB85, § 4.9] t ha t µ z ( V g ( M w )) = V ( M w ) = G · V ( L w ) = O . In general, explicitly computing the subset Σ = Σ( M w ) so that V Z ( M w ) = ∪ y ∈ Σ Z y and V u ( L w ) = ∪ y ∈ Σ B u y for w in W is a very diﬃcult and op en problem. See [BB85, § 4.3] and [HJ05, § 6] fo r examples a nd mo r e information. 2.4. Generalized St einberg v arieties. When analyzing the restriction of a Springer rep- resen tation to para b olic subgroups of W , Springer introduced a generalization of e N dep end- ing on a parab olic subgroup P and a nilp oten t orbit in a Levi subgroup of P . Springer’s ideas extend natura lly t o what w e call “generalized Stein b erg v arieties.” The results in this subsection may b e found in [D R04]. Supp ose P is a conjugacy class of pa r a b olic subgro ups of G . The unip oten t radical of a subgroup, P , in P will b e denoted by U P . A G -equiv ariant function, c , from P to the p ow er set of N with the prop erties STEINBERG V ARIETY AN D R EPRESENT A TIONS 11 (1) u P ⊆ c ( P ) ⊆ N ∩ p and (2) the image of c ( P ) in p / u P is the closure of a single nilp oten t adjo in t P /U P -orbit is called a L e vi class function on P . D eﬁne e N P c = { ( x, P ) ∈ N × P | x ∈ c ( P ) } . Let µ P c : e N P c → N denote the pro jection on the ﬁrst f a ctor. Notice that µ P c is a prop er morphism. If Q is another conjugacy class of parab olic subgroups of G a nd d is a Levi class function on Q , then the gener ali z e d Steinb er g variety determined by P , Q , c , and d is X P , Q c,d = { ( x, P , Q ) ∈ N × P × Q | x ∈ c ( P ) ∩ d ( Q ) } ∼ = e N P c × N e N Q d . Since G acts on N , P , and Q , t here is a diago nal a ction of G on X P , Q c,d for all P , Q , c , and d . The v arieties arising from this construction for some particular choice s of P , Q , c , and d are w orth noting. (1) When P = Q = B , then c ( B ′ ) = d ( B ′ ) = { u B ′ } for ev ery B ′ in B , and so X B , B 0 , 0 = Z is the Stein b erg v ariety of G . (2) In the sp ecial case w hen c ( P ) and d ( Q ) are as small as p ossible and cor r esp o nd to the zero o rbits in p / u P and q / u Q resp ectiv ely: c ( P ) = u P and d ( Q ) = u Q , we denote X P , Q c,d b y X P , Q 0 , 0 . W e ha ve X P , Q 0 , 0 ∼ = T ∗ P × N T ∗ Q . (3) When P = Q = { G } , O 1 and O 2 are tw o nilp otent orbits in g , c ( G ) = O 1 and d ( G ) = O 2 , then X { G } , { G } c,d ∼ = O 1 ∩ O 2 . A sp ecial case that will arise frequen tly in the sequel is when c ( P ) and d ( Q ) are a s large as p o ssible and corresp ond to the regular, nilp oten t o rbits in p / u P and q / u Q resp ectiv ely: c ( P ) = N ∩ p and d ( Q ) = N ∩ q . W e denote this generalized Stein b erg v ariet y simply b y X P , Q . Abusing notation sligh tly , w e let µ : X P , Q c,d → N de note the pro jection on the ﬁrst co o r - dinate and π : X P , Q c,d → P × Q the pro jection on the second and third co o r dina t es. W e can then inv estigate the v arieties X P , Q c,d using preimag es of G -orbits in N and P × Q under µ and π as w e did in § 2.1 for the Stein b erg v ariety . Sp ecial cas es when at least one of c ( P ) or d ( Q ) is smooth turn out to b e the most tractable. W e will des crib e these cases in more detail below and refer the reader to [D R04] for more general res ults for arbitrary P , Q , c , and d . Fix P in P and Q in Q with B ⊆ P ∩ Q . Let W P and W Q denote the W eyl groups of ( P , T ) and ( Q, T ) resp ectiv ely . W e consider W P and W Q as subgroups of W . F or B ′ in B , deﬁne π P ( B ′ ) to b e the unique subgroup in P con ta ining B ′ . Then π P : B → P is a prop er morphism with ﬁbres isomorphic to P /B . D eﬁne η : Z → X P , Q b y η ( x, B ′ , B ′′ ) = ( x, π P ( B ′ ) , π Q ( B ′′ )) . Then η depends on P and Q and is a prop er, G -equiv ariant, surjectiv e morphism. Next, set Z P , Q = η − 1  X P , Q 0 , 0  and denote the restriction of η to Z P , Q b y η 1 . Then η 1 is also a prop er, surjectiv e, G -equiv a rian t morphism. Moreo v er, the ﬁbres of η 1 are all isomorphic to the smo o t h, complete v ariety P /B × Q/B . More generally , deﬁne Z P , Q c,d = η − 1  X P , Q c,d  . 12 J.M. D OUGLASS A ND G. R ¨ OHRLE The v ario us v arieties and morphisms w e hav e deﬁned ﬁt t o gether in a comm utative diagram where the horizon ta l arro ws are closed em b eddings, the v ertical arrows are pro p er maps, and the squares ar e cartesian: Z P , Q / / η 1   Z P , Q c,d / / η   Z η   X P , Q 0 , 0 / / X P , Q c,d / / X P , Q . F or w in W , deﬁne Z P , Q w to b e the in tersection Z P , Q ∩ Z w . Since (0 , B , w B w − 1 ) is in Z P , Q w and η 1 is G -equiv ariant, it is straightforw ard to c hec k that Z P , Q w ∼ = G × B w ( u P ∩ w u Q ). Th us Z P , Q w is smo oth and irreducible. The follow ing statemen ts are prov ed in [D R04]. (1) F o r w in W , dim η ( Z w ) ≤ 2 n with equality if and only if w has minimal length in W P w W Q . The v ariety X P , Q is equidimensional with dimension equal to 2 n and the set of irreducible comp onen ts of X P , Q is { η ( Z w ) | w has minimal length in W P w W Q } . (2) F o r w in W , Z P , Q w = Z w if a nd only if w has maximal length in W P w W Q . The v ariety Z P , Q is equidimensional with dimension equal to 2 n and t he set of irreducible comp onen t s of Z P , Q is { Z w | w has maximal length in W P w W Q } . (3) The v ariety X P , Q 0 , 0 is equidimensional with dimens ion equal to dim u P + dim u Q and the set of ir reducible comp onents of X P , Q 0 , 0 is { η 1 ( Z w ) | w has maximal length in W P w W Q } . (4) F o r a Levi class function d on Q , deﬁne ρ d to b e the n um b er of irreducible comp o nen ts of d ( Q ) ∩ ( u ∩ l Q ), where L Q is the Levi factor of Q that con ta ins T . Then ρ d is the n umber o f orbital v arieties for the op en dense L Q -orbit in d ( Q ) / u Q in the v ar iety of nilpo ten t elemen ts in q / u Q ∼ = l Q . The v arieties X B , Q 0 ,d are equidimens ional with dimension 1 2 (dim u + dim d ( Q ) + dim u Q ) and | W : W Q | ρ d irreducible comp onents. Notice that the ﬁrst statemen t relates minimal double coset represen tatives to regular orbits in Levi subalgebras and the third statemen t r elat es maximal double coset represen ta t ives to the zero o r bits in Levi subalgebras. The quan tit y ρ d in the fourt h statemen t is the degree of an irreducible represen tation of W Q (see § 3.5) and so | W : W Q | ρ d is t he degree of an induced represen tation o f W . The f act that X B , Q 0 ,d has | W : W Q | ρ d irreducible comp onen ts is num erical evidence for Conjecture 3.1 9 b elo w. 3. Homology In this section w e ta ke up the rat io nal Borel-Mo o re homology of the Stein b erg v ariet y and generalized Stein b erg v arieties. As mentioned in the In tro duction, soon after Stein b erg’s original pa p er, Kazhdan and Lusztig [KL80] deﬁned an action of W × W on the to p Borel- Mo ore homology gro up of Z . T hey constructed this action b y deﬁning an action of the simple reﬂections in W × W on H i ( Z ) and showing that the deﬁning relations of W × W are STEINBERG V ARIETY AN D R EPRESENT A TIONS 13 satisﬁed. They then pro ve d tha t the represen tation o f W × W on H 4 n ( Z ) is equiv alen t to the t wo-sided regula r represen tation of W , and fo llo wing a suggestion o f Springer, they gav e a decomp osition of H 4 n ( Z ) in terms of Springer represen tatio ns of W . Springer represen ta t io ns of W will b e describ ed in § 3.4– § 3.6. In the mid 1990s Ginzburg [CG97, Chapter 3] p opularized a quite general con v olutio n pro duct construction that deﬁnes a Q -algebra structure on H ∗ ( Z ), the total Borel-Moo re homology o f Z , and a r ing structure K G ( Z ) (see the next section for K G ( Z )). With this m ultiplication, H 4 n ( Z ) is a subalgebra isomorphic t o the group algebra of W . In this section, following [CG97, Chapter 3], [D R08b], and [HJ05] w e will ﬁrst describe the algebra structure o f H ∗ ( Z ), the decomp osition o f H 4 n ( Z ) in terms of Springer represen- tations, a nd the H 4 n ( Z )-mo dule structure on H 2 n ( Y ) using elemen tary top ological construc- tions. Then we will use a more sophis ticated she af-theoretic approac h to giv e an alt ernat e description of H ∗ ( Z ), a diﬀeren t vers ion of the decomp osition of H 4 n ( Z ) in terms of Springer represen tations, and t o describe the Borel-Mo ore homolog y of some generalized Stein b erg v arieties. 3.1. Borel-Mo ore homology and con volution. W e begin with a brief review of Borel- Mo ore homology , including the con volution and specialization constructions. The deﬁnitions and constructions in this subse ction mak e se nse in a very g eneral setting, ho w ev er for sim- plicit y w e will consider only complex algebraic v arieties. More details and pro ofs may b e found in [CG97, Chapter 2]. Supp ose that X is a d - dimensional, quasi-pro jectiv e, complex algebraic v ariety (not nec- essarily irreducible). T op olog ical notions will refer to the Euclidean topo logy on X unless otherwise speciﬁed. Tw o exceptions to this con v ention are that w e con tin ue to denote the dimension of X as a complex v ariet y b y dim X and that “irreducible” means irreducible with resp ect to the Zariski top ology . In particular, the top ological dimension of X is 2 dim X . Let X ∪ {∞} b e the one-p o in t compactiﬁcation o f X . Then the i th Borel-Mo ore ho molo gy space of X , denoted by H i ( X ), is deﬁned b y H i ( X ) = H sing i ( X ∪ {∞} , {∞} ) , the relative, singular homology with rational co eﬃcien ts of the pair ( X ∪ {∞} , {∞} ). Deﬁne a graded Q -v ector space, H ∗ ( X ) = X i ≥ 0 H i ( X ) – the B or el-Mo or e homolo gy of X . Borel-Mo ore ho mology is a biv arian t theory in the sense o f F ulto n and MacPherson [FM81]: Supp ose t ha t φ : X → Y is a morphism of v arieties. • If φ is prop er, then there is an induced direct imag e map in Borel-Mo ore homology , φ ∗ : H i ( X ) → H i ( Y ). • If φ is smo oth with f -dimensional ﬁbres, then there is a pullbac k map in Bo r el- Mo ore homology , φ ∗ : H i ( Y ) → H i +2 f ( X ). Moreo ver, if X is smo oth and A and B are closed sub v arieties of X , then there is an in tersection pairing ∩ : H i ( A ) × H j ( B ) → H i + j − 2 d ( A ∩ B ). Although not reﬂected in the notation, this pairing dep ends o n the triple ( X , A, B ). In particular, the interse ction pairing dep ends on the smo oth ambien t v ariety X . In dimensions greater than or equal 2 dim X , the Borel-Mo ore homology spaces of X are easily described. If i > 2 d , then H i ( X ) = 0, while the space H 2 d ( X ) has a natural 14 J.M. D OUGLASS A ND G. R ¨ OHRLE basis indexed b y the d - dimensional irreducible comp onen ts of X . If C is a d - dimensional irreducible comp o nen t o f X , then the homology class in H 2 d ( X ) determined b y C is denoted b y [ C ]. F or example, for the Stein b erg v ariety , H i ( Z ) = 0 for i > 4 n and the set { [ Z w ] | w ∈ W } is a basis of H 4 n ( Z ). Similarly , for the conormal v ariety , H i ( Y ) = 0 fo r i > 2 n and the set { [ Y w ] | w ∈ W } is a basis of H 2 n ( Y ). Supp ose that for i = 1 , 2 , 3, M i is a smo o t h, connected, d i -dimensional v ariet y . F or 1 ≤ i < j ≤ 3, let p i,j : M 1 × M 2 × M 3 → M i × M j denote the pro jection. Notice that eac h p i,j is smo oth and so the pullback maps p ∗ i,j in Borel-Mo ore homolo gy are deﬁned. No w supp ose Z 1 , 2 is a closed subset of M 1 × M 2 and Z 2 , 3 is a closed sub v ariety of M 2 × M 3 . Deﬁne Z 1 , 3 = Z 1 , 2 ◦ Z 2 , 3 to b e t he comp osition of the relations Z 1 , 2 and Z 2 , 3 . Then Z 1 , 3 = { ( m 1 , m 3 ) ∈ M 1 × M 3 | ∃ m 2 ∈ M 2 with ( m 1 , m 2 ) ∈ Z 1 , 2 and ( m 2 , m 3 ) ∈ Z 2 , 3 } . In order to deﬁne the conv olutio n pro duct, w e a ssume in additio n that the restriction p 1 , 3 : p − 1 1 , 2 ( Z 1 , 2 ) ∩ p − 1 2 , 3 ( Z 2 , 3 ) → Z 1 , 3 is a prop er morphism. Th us, there is a direct image map ( p 1 , 3 ) ∗ : H i  p − 1 1 , 2 ( Z 1 , 2 ) ∩ p − 1 2 , 3 ( Z 2 , 3 )  → H i ( Z 1 , 3 ) in Borel-Mo ore homolo g y . The c o n volution pr o d uct , H i ( Z 1 , 2 ) × H j ( Z 2 , 3 ) ∗ − → H i + j − 2 d 2 ( Z 1 , 3 ) is then deﬁned by c ∗ d = ( p 1 , 3 ) ∗  p ∗ 1 , 2 ( c ) ∩ p ∗ 2 , 3 ( d )  where ∩ is the inters ection pairing determined b y the subsets Z 1 , 2 × M 3 and M 1 × Z 2 , 3 of M 1 × M 2 × M 3 . It is a straigh tforward ex ercise to sho w tha t the conv o lutio n pro duct is asso ciativ e. The con volution construction is particularly w ell adapted to ﬁbred pro ducts. Fix a “base” v ariety , N , whic h is no t necess arily smo oth, and supp ose that for i = 1 , 2 , 3, f i : M i → N is a prop er morphism. Then ta king Z 1 , 2 = M 1 × N M 2 , Z 2 , 3 = M 2 × N M 3 , and Z 1 , 3 = M 1 × N M 3 , w e ha v e a conv o lution pro duct H i ( M 1 × N M 2 ) × H j ( M 2 × N M 3 ) ∗ − → H i + j − 2 d 2 ( M 1 × N M 3 ). As a sp ecial case, when M 1 = M 2 = M 3 = M and f 1 = f 2 = f 3 = f , then taking Z i,j = M × N M for 1 ≤ i < j ≤ 3, the con v olutio n pro duct deﬁnes a m ultiplication on H ∗ ( M × N M ) so that H ∗ ( M × N M ) is a Q -algebra with iden tit y . The iden tity in H ∗ ( M × N M ) is [ M ∆ ] where M ∆ is the diag onal in M × M . If d = dim M , then H i ( M × N M ) ∗ H j ( M × N M ) ⊆ H i + j − 2 d ( M × N M ) and so H 2 d ( M × N M ) is a subalgebra and ⊕ i< 2 d H i ( M × N M ) is a nilpotent, t wo-sided ideal. Another sp ecial case is when M a nd M ′ are smo o t h and f : M → N and f ′ : M ′ → N are prop er maps. The n taking Z 1 , 2 = M × N M and Z 2 , 3 = M × N M ′ , the conv olut io n pro duct deﬁnes a left H ∗ ( M × N M )- mo dule structure on H ∗ ( M × N M ′ ). A further special case of this construction is when M ′ = A is a smo oth, closed subset of N and f ′ : A → N is the inclusion. Then M × N A ∼ = f − 1 A and the con v olution pro duct deﬁnes a left H ∗ ( M × N M )- mo dule structure on H ∗ ( f − 1 ( A )). This construction will b e exploited extensiv ely in § 3.5. As an example, recall that Z ∼ = e N × N e N where µ : e N → N is a proper map. Applying the constructions in the last t wo para g raphs to Z and to M ′ , where M ′ = Y = µ − 1 ( u ) and M ′ = B x = µ − 1 ( x ) for x in N , w e obtain the following pr o p osition. STEINBERG V ARIETY AN D R EPRESENT A TIONS 15 Prop osition 3.1. The c onv olution pr o duct deﬁnes a Q -algebr a structur e on H ∗ ( Z ) so that H 4 n ( Z ) is a | W | -dim ensional s ub al g ebr a and L i< 4 n H i ( Z ) is a two-side d , ni lp otent ide a l . Mor e o ver, the c on volution pr o duct deﬁ n es left H ∗ ( Z ) -mo dule structur es on H ∗ ( Y ) and on H ∗ ( B x ) for x in N . In the next tw o subsections we will mak e use of the following sp ecialization c onstruction in Borel-Mo ore homolo gy due to F ulton and MacPherson [FM81, § 3.4]. Supp ose t ha t our base v ariet y N is smo oth and s - dimensional. Fix a distinguished p oin t n 0 in N and set N ∗ = N \ { n 0 } . Let M b e a v ariet y , not necessarily smo oth, and supp ose tha t φ : M → N is a surjectiv e morphism. Set M 0 = φ − 1 ( n 0 ) and M ∗ = φ − 1 ( N ∗ ). Assume tha t the restriction φ | M ∗ : M ∗ → N ∗ is a lo cally trivial ﬁbration. Then there is a “sp ecialization” map in Borel-Mo o re homology , lim : H i ( M ∗ ) → H i − 2 s ( M 0 ) (se e [CG97, § 2.6]) . It is sho wn in [CG97, § 2.7] that when all the v ario us constructions are deﬁned, sp ecialization comm utes with con v o lutio n: lim( c ∗ d ) = lim c ∗ lim d . 3.2. The specialization construct ion and H 4 n ( Z ) . Chriss and Ginzburg [CG97, § 3.4] use the sp ecialization construction to sho w that H 4 n ( Z ) is isomorphic to the group algebra Q [ W ]. W e presen t their construction in this subsection. In the next subsection w e sho w that the specialization construction can also b e used to sho w that H ∗ ( Z ) is isomor phic to the smash pro duct of the group algebra of W and the coin v ariant algebra of W . W e w ould like to apply the sp ecialization construction when the v ariety M 0 is equal Z . In order to do this, w e need v arieties that are larger than N , e N , and Z . Deﬁne e g = { ( x, B ′ ) ∈ g × B | x ∈ b ′ } and b Z = { ( x, B ′ , B ′′ ) ∈ g × B × B | x ∈ b ′ ∩ b ′′ } . Abusing not a tion ag ain, let µ : e g → g and µ z : b Z → g denote the pro jections o n the ﬁrst factors and let π : b Z → B × B denote the pro jection on the second and t hird factors. F or w in W deﬁne b Z w = π − 1 ( G ( B , w B w − 1 )). Then b Z w ∼ = G × B w b w . Therefore, dim b Z w = dim g and the closures of the b Z w ’s for w in W are t he irreducible comp onen ts of b Z . As with Z , we hav e an alternate description of b Z as ( e g × e g ) × g × g g . Ho w eve r, in con trast to the situation in § 2.3, where Z ∼ = { ( x, − x, B ′ , B ′′ ) ∈ N × N × B × B | x ∈ b ′ ∩ b ′′ ∩ N } , in this section w e use that b Z ∼ = { ( x, B ′ , x, B ′′ ) ∈ g × B × g × B | x ∈ b ′ ∩ b ′′ } . In particular, we will frequen tly iden tify b Z with the sub v ariety o f e g × e g consisting o f a ll pairs (( x, B ′ ) , ( x, B ′′ )) with x in b ′ ∩ b ′′ . Similarly , w e will f r equen tly identify Z with the sub v a riet y of e N × e N consisting of all pa irs (( x, B ′ ) , ( x, B ′′ )) with x in N ∩ b ′ ∩ b ′′ . F or ( x, g B g − 1 ) in e g , deﬁne ν ( x, g B g − 1 ) to b e the pro jection of g − 1 · x in t . Then ν : e g → t is a surjectiv e morphism . F or w in W , let Γ w − 1 = { ( h, w − 1 · h ) | h ∈ t } ⊆ t × t denote the graph of the action of w − 1 on t and deﬁne Λ w = b Z ∩ ( ν × ν ) − 1 (Γ w − 1 ) = { ( x, B ′ , B ′′ ) ∈ b Z | ν ( x, B ′′ ) = w − 1 ν ( x, B ′ ) } . 16 J.M. D OUGLASS A ND G. R ¨ OHRLE The spaces w e hav e deﬁned so f ar ﬁt in to a comm utativ e diagram with cartesian squares where δ : g → g × g is the diag onal map: (3.2) Λ w / /   b Z µ z / /   g δ   ( ν × ν ) − 1 (Γ w − 1 ) / /   e g × e g µ × µ / / ν × ν   g × g Γ w − 1 / / t × t . Let ν w : Λ w → Γ w − 1 denote t he composition of the le ftmost v ertical maps in (3.2), so ν w is the restriction of ν × ν to Λ w . W e will consider subsets of b Z of the for m ν − 1 w ( S ′ ) for S ′ ⊆ Γ w − 1 . Th us, for h in t w e deﬁne Λ h w = ν − 1 w ( h, w − 1 h ). Notice in pa r t icular that Λ 0 w = Z . More g enerally , for a subset S of t we deﬁne Λ S w = ` h ∈ S Λ h w . Then Λ S w = ν − 1 w ( S ′ ), where S ′ is the gra ph of w − 1 restricted to S . Let t reg denote the set o f r egula r elemen ts in t . F or w in W , deﬁne e w : G/T × t reg → G/T × t reg b y e w ( g T , h ) = ( g w T , w − 1 h ). The rule ( g T , h ) 7→ ( g · h, g B ) deﬁnes an isomorphism of v arieties f : G/T × t reg ∼ = − → e g rs , whe re e g rs = µ − 1 ( G · t reg ). W e denote the a utomorphism f ◦ e w ◦ f − 1 of e g rs also by e w . W e now ha ve all the notation in place fo r the specialization construction. Fix an elemen t w in W and a one-dimensional subspace, ℓ , of t so that ℓ ∩ t reg = ℓ \ { 0 } . The line ℓ is our base space and the distinguished p oin t in ℓ is 0 . As ab ov e, w e set ℓ ∗ = ℓ \ { 0 } . W e denote the restriction of ν w to Λ ℓ w again b y ν w . Then ν w : Λ ℓ w → ℓ is a surjectiv e morphism with ν − 1 w (0) = Z and ν − 1 w ( ℓ ∗ ) = Λ ℓ ∗ w . W e will see below that the restriction Λ ℓ ∗ w → ℓ ∗ is a lo cally trivial ﬁbration and so a sp ecialization map (3.3) lim : H i +2 (Λ ℓ ∗ w ) → H i ( Z ) is deﬁned. It is not hard to chec k that the v ariety Λ ℓ ∗ w is t he graph of e w | e g ℓ ∗ : e g ℓ ∗ → e g w − 1 ( ℓ ∗ ) , where for an arbitrary subset S of t , e g S is deﬁned to be ν − 1 ( S ) = { ( x, B ′ ) ∈ e g | ν ( x, B ′ ) ∈ S } . It follo ws t ha t for h in ℓ ∗ w e ha ve ν − 1 w ( h ) = Λ h w ∼ = G/T and that Λ ℓ ∗ w → ℓ ∗ is a locally trivial ﬁbration. Moreo v er, Λ ℓ ∗ w ∼ = e g ℓ ∗ and hence is an irreducible, (2 n + 1)-dimensional v ariety . Therefore, H 4 n +2 (Λ ℓ ∗ w ) is one-dimensional with basis { [Λ ℓ ∗ w ] } . T aking i = 4 n in (3.3), w e deﬁne λ w = lim([Λ ℓ ∗ w ]) in H 4 n ( Z ). Because Λ ℓ ∗ w is a graph, it follo ws easily from the deﬁnitions that for y in W , there is a con volution pro duct H ∗ (Λ ℓ ∗ w ) × H ∗ (Λ w − 1 ℓ ∗ y ) ∗ − → H ∗ (Λ ℓ ∗ w y ) and that [Λ ℓ ∗ w ] ∗ [Λ w − 1 ℓ ∗ y ] = [Λ ℓ ∗ w y ]. Becaus e sp ecialization comm utes with con v olutio n, we hav e λ w ∗ λ y = λ w y for all w and y in W . Chriss and Ginzburg [CG97, § 3.4] ha ve pro v ed the following: (1) The elemen t λ w in H 4 n ( Z ) do es not dep end on the c hoice of ℓ . STEINBERG V ARIETY AN D R EPRESENT A TIONS 17 (2) The expansion of λ w as a linear combination of the basis elemen ts [ Z y ] of H 4 n ( Z ) has the form λ w = [ Z w ] + P y w , then ther e is a subset F s,w of { x ∈ W | x < w , sx < x } so that [ Z s ] ∗ [ Z w ] = [ Z sw ] + P x ∈ F s,w n x [ Z x ] with n x > 0 . Using this result, Hinic h a nd Joseph [HJ05, Theorem 5.5] pro ve a result analogous to Prop osition 3.7 for right Stein b erg cells. Recall that f o r w in W w e ha v e deﬁned V r ( w ) = B u w ∩ O when w is in W O . F or an orbital v a r iet y V , deﬁne W V = { y ∈ W | V r ( y ) ⊆ V } . Theorem 3.10. F or w in W , t he smal lest subset, S , of W with the pr op erty that [ Z w ] ∗ λ y is in the sp a n of { [ Z x ] | x ∈ S } for al l y in W is V r ( w ) . In p articular, if V is any orbital variety, then the sp an of { [ Z x ] | x ∈ W V } is a right ide al in H 4 n ( Z ) . 3.6. Sheaf-theoretic decomp osition of H 4 n ( Z ) and H i ( B x ) . F or a v ariety X , the Q - v ector space H i ( X ) has more a sophisticated alternate description in terms of sheaf coho- mology (see [CG 9 7, § 8.3]). The properties of shea ves and pervers e she a v es w e use in this section may b e found in [KS90, Chapter 2,3], [D im04] and [Bo r84]. Let D ( X ) denote the full subcategory of the deriv ed category of shea v es of Q -v ector spaces on X consisting of complex es with b ounded, constructible cohomology . If f : X → Y is a morphism, then there are functors Rf ∗ : D ( X ) → D ( Y ) , Rf ! : D ( X ) → D ( Y ) , f ∗ : D ( Y ) → D ( X ) , and f ! : D ( Y ) → D ( X ) . The pair of functors ( f ∗ , R f ∗ ) is an adjoin t pair, as is ( Rf ! , f ! ). If f is proper, then Rf ! = Rf ∗ and if f is smo oth, then f ! = f ∗ [2 dim X ]. W e consider the constan t sheaf, Q X , as a complex in D ( X ) concen trated in degree zero. The dualizing sheaf, D X , of X is deﬁned by D X = a ! X Q { pt } , where a X : X → { pt } . If X is a rational homology manifo ld, in par t icular, if X is smo oth, then D X ∼ = Q X [2 dim X ] in D ( X ). It follows from t he deﬁnitions and b ecause f ∗ and f ! are functors that if f : X → Y , then (3.11) Q X ∼ = f ∗ Q Y and D X ∼ = f ! D Y in D ( X ). The cohomology and Borel-Mo ore homology of X ha ve v ery conv enient descriptions in sheaf-theoretic terms: (3.12) H i ( X ) ∼ = Ext i D ( X ) ( Q X , Q X ) and H i ( X ) ∼ = Ext − i D ( X ) ( Q X , D X ) where for F a nd G in D ( X ), Ext i D ( X ) ( F , G ) = Hom D ( X ) ( F , G [ i ]). No w supp o se that f i : M i → N is a pro p er morphism for i = 1 , 2 , 3 and that d 2 = dim M 2 . In con tra st to our assumptions in the conv olut io n se tup from § 3.1 where t he M i w ere assu med 22 J.M. D OUGLASS A ND G. R ¨ OHRLE to b e smo oth, in t he fo llo wing computation we assume only that M 2 is a ratio nal homology manifold. Consider the cartesian dia g ram M 1 × N M 2 f 1 , 2 / / δ 1   N δ   M 1 × M 2 f 1 × f 2 / / N × N where f 1 , 2 is the induced map. Using the argumen t in [CG97, § 8.6], w e hav e isomorphisms H i ( M 1 × N M 2 ) ∼ = Ext − i D ( M 1 × N M 2 ) ( Q M 1 × N M 2 , D M 1 × N M 2 ) (3.12) ∼ = Ext − i D ( M 1 × N M 2 ) ( f ∗ 1 , 2 Q N , δ ! 1 D M 1 × M 2 ) (3.11) ∼ = Ext − i D ( N ) ( Q N , R ( f 1 , 2 ) ∗ δ ! 1 D M 1 × M 2 ) (adjunction) ∼ = Ext − i D ( N ) ( Q N , δ ! R ( f 1 × f 2 ) ∗ D M 1 × M 2 ) (base c hange) ∼ = Ext − i D ( N ) ( Q N , δ ! ( R ( f 1 ) ∗ D M 1 ⊠ R ( f 2 ) ∗ D M 2 )) (K ¨ unneth) ∼ = Ext − i D ( N ) ( Q N , H om ( R ( f 1 ) ∗ Q M 1 , R ( f 2 ) ∗ D M 2 )) ([Bor84, 10.2 5 ]) ∼ = Ext − i D ( N ) ( Q N , H om ( R ( f 1 ) ∗ Q M 1 , R ( f 2 ) ∗ Q M 2 [2 d 2 ])) ( D M 2 ∼ = Q M 2 [2 d 2 ]) ∼ = Ext 2 d 2 − i D ( N ) ( Q N , H om ( R ( f 1 ) ∗ Q M 1 , R ( f 2 ) ∗ Q M 2 )) ∼ = Ext 2 d 2 − i D ( N ) ( R ( f 1 ) ∗ Q M 1 , R ( f 2 ) ∗ Q M 2 ) . Let ǫ 1 , 2 denote the comp o sition o f the ab o ve isomorphisms, so (3.13) ǫ 1 , 2 : H i ( M 1 × N M 2 ) ∼ = − → Ext 2 d 2 − i D ( N ) ( R ( f 1 ) ∗ Q M 1 , R ( f 2 ) ∗ Q M 2 ) . Chriss and Ginzburg [CG97, § 8.6] hav e shown tha t the isomorphisms ǫ 1 , 2 in tertwine the con volution pro duct on the left with the Y oneda pro duct (comp osition of mor phisms) on the righ t: giv en c in H i ( M 1 × N M 2 ) and d in H j ( M 2 × N M 3 ), we ha ve ǫ 1 , 3 ( c ∗ d ) = ǫ 2 , 3 ( d ) ◦ ǫ 1 , 2 ( c ). W e ma y apply the computation in equation (3.13) to H ∗ ( Z ). W e hav e seen that Z ∼ = e N × N e N and so H i ( Z ) ∼ = Ext 4 n − i D ( N ) ( Rµ ∗ Q e N , R µ ∗ Q e N ) . In particular, taking i = 4 n , w e conclude that are a lg ebra isomorphisms Q [ W ] ∼ = H 4 n ( Z ) ∼ = End D ( N ) ( Rµ ∗ Q e N ) op . The category D ( N ) is a tr iangulated category . It contains a full, ab elian sub category , denoted by M ( N ), consisting of “perve rse s hea v es on N ” (with resp ect to the middle p er- v ersity ). It follo ws from the Decomp osition Theorem of Beilinson, Bernstein, and Deligne [BBD82, § 5] that the complex Rµ ∗ Q e N is a semisimple ob ject in M ( N ). The simple ob jects in M ( N ) hav e a geometric description. Suppo se X is a smo oth, lo cally closed sub v ariety of N with co dimension d , i : X → N is the inclusion, and L is an irreducible lo cal syste m on X . Let IC( X , L ) denote the in tersection complex of Goresky and MacPherson [GM83, § 3]. The n i ∗ IC( X , L )[ − 2 d ] is a simple ob ject in M ( N ) and ev ery simple ob ject arises in this w a y . In addition to the original sources, [BBD82] and [GM83], w e refer the reader to [Sho88 , § 3] and [CG97, § 8.4] f or short in t r o ductions to the theory of STEINBERG V ARIETY AN D R EPRESENT A TIONS 23 in tersection complexes and p erv erse shea ves a nd to [Bor84] and [Dim04] for mor e thorough exp o sitions. Returning to Rµ ∗ Q e N , Borho and MacPherson [BM81 ] hav e sho wn that its decomp osition in to simple p erv erse sheav es is given by (3.14) Rµ ∗ Q e N ∼ = M x,φ j x ∗ IC( Gx, L φ )[ − 2 d x ] n x,φ where x runs o v er the set o f orbit repres en ta t iv es S in N , and for each x , j x : Gx → N is the inclusion, φ is in [ C ( x ), L φ is t he lo cal system on Gx corresp o nding to φ , and n x,φ is a non-negativ e in teger. Deﬁne IC x,φ = j x ∗ IC( Gx, L φ ). Then IC x,φ [ − 2 d x ] is a simple ob ject in M ( N ). It f ollo ws from the computation of the gr o ups C ( x ) that End D ( N ) (IC x,φ ) ∼ = Q . Therefore, (3.15) H 4 n ( Z ) ∼ = End D ( N ) ( Rµ ∗ Q e N ) op ∼ = End D ( N ) ( ⊕ x,φ IC x,φ [ − 2 d x ] n x,φ ) op ∼ = M x,φ End D ( N ) (IC n x,φ x,φ ) op ∼ = M x,φ M n x,φ  End D ( N ) (IC x,φ )  op ∼ = M x,φ M n x,φ ( Q ) op . This is a decomp osition of H 4 n ( Z ) as a direct sum of matrix rings and hence is the W edder- burn decomp osition of H 4 n ( Z ). Supp ose now that O is a G - orbit in N and x is in O . It is straigh tforw ard to c heck that H O ∼ = M φ ∈ [ C ( x ) End D ( N ) ((IC x,φ ) n x,φ ) ∼ = M φ ∈ [ C ( x ) M n x,φ ( Q ) . As in Prop osition 3.7, this is the decomp osition of H O in to minimal tw o-sided ideals. F or a second application of (3.13), let i x : { x } → N denote the inclusion. Then B x ∼ = e N × N { x } and so H i ( B x ) ∼ = Ext − i D ( N ) ( Rµ ∗ Q e N , R ( i x ) ∗ Q { x } ) ∼ = M y ,ψ Ext − i D ( N ) (IC y ,ψ [ − 2 d y ] n y ,ψ , R ( i x ) ∗ Q { x } ) ∼ = M y ,ψ Ext 2 d y − i D ( N ) (IC n y ,ψ y ,ψ , R ( i x ) ∗ Q { x } ) ∼ = M y ,ψ  V y ,ψ ⊗ Ext 2 d y − i D ( N ) (IC y ,ψ , R ( i x ) ∗ Q { x } )  where V y ,ψ is an n y ,ψ -dimensional v ector space. Because Q [ W ] ∼ = H 4 n ( Z ) ∼ = End D ( N ) ( Rµ ∗ Q e N ) acts b y p erm uting the simple summands, it follo ws from (3.15) that eac h V y ,ψ aﬀords an irreducible represen tation of W and that Ext 2 d y − i D ( N ) (IC y ,ψ , R ( i x ) ∗ Q { x } ) records the m ultiplicit y of V y ,ψ in H i ( B x ). Using that i ∗ x is left a djoin t to R ( i x ) ∗ , denoting the stalk of IC y ,ψ at x by 24 J.M. D OUGLASS A ND G. R ¨ OHRLE (IC y ,ψ ) x , and setting m x,i y ,ψ = dim Ext 2 d y − i D ( N ) (IC y ,ψ , R ( i x ) ∗ Q { x } ), w e obtain the decomposition of H i ( B x ) into ir reducible represen tatio ns of W : H i ( B x ) ∼ = M y ,ψ  V y ,ψ ⊗ Ext 2 d y − i D ( { x } ) ((IC y ,ψ ) x , Q { x } )  ∼ = M y ,ψ V m x,i y ,ψ y ,ψ . In the next subsection w e apply (3.13) to compute the Borel-Mo ore ho mology of some generalized Steinberg v a rieties. 3.7. Borel-Mo ore homology of generalized Stein b erg v arieties. Recall from § 2.4 t he generalized Steinberg v a riet y X P , Q = { ( x, P ′ , Q ′ ) ∈ N × P × Q | x ∈ p ′ ∩ q ′ } ∼ = e N P × N e N Q where e N P = { ( x, P ′ ) ∈ N × P | x ∈ p ′ } , ξ P : e N P → N is pro jection on the ﬁrst factor, and e N Q and ξ Q are deﬁned similarly . R ecall a lso that η : Z → X P , Q is a prop er, G -equiv arian t surjection. The ma in result of [DR08 a, Theorem 4.4], which is pro v ed using the constructions in the last subsection, is the following theorem describing the Borel-Mo ore homology of X P , Q . Theorem 3.16. C o nsider H 4 n ( Z ) as a W × W -mo dule using the isomorphis m H 4 n ( Z ) ∼ = Q [ W ] . Then ther e is a n isom orphism α : H ∗ ( X P , Q ) ∼ = − → H ∗ ( Z ) W P × W Q so that the c o mp osition α ◦ η ∗ : H ∗ ( Z ) → H ∗ ( Z ) W P × W Q is the aver agin g map. As a sp ecial case o f the theorem, if w e let e P (resp. e Q ) denote the primitiv e ide mp oten t in Q [ W P ] (resp. Q [ W Q ]) corresp onding to the trivial represen tation, then (3.17) H 4 n ( X P , Q ) ∼ = e P Q [ W ] e Q . Next recall the generalized Stein b erg v ariety X P , Q 0 , 0 ∼ = T ∗ P × N T ∗ Q . Set m = dim P / B + dim Q/B . Let ǫ P (resp. ǫ Q ) denote the primitiv e idemp ot ent in Q [ W P ] (resp. Q [ W Q ]) corre- sp onding to the sign represen tation. Then dim X P , Q 0 , 0 = 4 n − 2 m and it is shown in [DR08a, § 5] that (3.18) H 4 n − 2 m ( X P , Q 0 , 0 ) ∼ = ǫ P Q [ W ] ǫ Q . No w suppose that c is a Levi class function on P . Let L b e a Levi subgroup of P and c ho ose x in c ( P ) ∩ l . T hen w e ma y consider the Springer represen ta tion o f W P on H 2 d L x ( B L x ) C L ( x ) where C L ( x ) is the comp onen t group of Z L ( x ), B L x is the v ariet y of Borel subalgebras of l that c on tain x , and d L x = dim B L x . This is an irreducible represen tation of W P . Let f P denote a primitive idemp oten t in Q [ W P ] so t ha t Q [ W P ] f P ∼ = H 2 d L x ( B L x ) C L ( x ) . Set δ P , Q c,d = 1 2 (dim c ( P ) + dim u P + dim d ( Q ) + dim u Q ). Then it is show n in [DR04, Coro llary 2.6] that dim X P , Q c,d ≤ δ P , Q c,d . Generalizing the computat io ns (3.1 7) and (3.18), w e conjecture that the following statemen t is true. Conjecture 3.19. With the n otation ab ov e, H δ P , Q c,d ( X P , Q c,d ) ∼ = f P Q [ W ] f Q . The Borel-Mo ore homology of X P , Q ma y also b e computed using the sheaf theoretic metho ds in the last subse ction. W e ha ve X P , Q ∼ = e N P × N e N Q and Borho and MacPherson [BM83, 2.11] ha ve sho wn tha t e N P and e N Q are ratio nal homology manifolds. Therefore, as in (3.13) : H i ( X P , Q ) ∼ = Ext 4 n − i D ( N ) ( Rξ P ∗ Q e N P , R ξ Q ∗ Q e N Q ) . STEINBERG V ARIETY AN D R EPRESENT A TIONS 25 Borho and MacPherson [BM83, 2.11] ha ve also sho wn that Rξ P ∗ Q e N P is a semisimple ob ject in M ( N ) and describ ed its decomposition into simple p erv erse shea ves : Rξ P ∗ Q e N P ∼ = M ( x,φ ) IC x,φ [ − 2 d x ] n P x,φ , where the sum is ov er pairs ( x, φ ) as in equation (3.1 4), and n P x,φ is t he m ultiplicity of the irreducible represen tation H 2 d x ( B x ) φ of W in the induced represen tation Ind W W P (1 W P ). Th us, H i ( X P , Q ) ∼ = M x,φ M y ,ψ Ext 4 n − i D ( N )  IC x,φ [ − 2 d x ] n P x,φ , IC y ,ψ [ − 2 d y ] n Q y ,ψ  and so (3.20) H 4 n ( X P , Q ) ∼ = M x,φ M y ,ψ Hom D ( N )  IC x,φ [ − 2 d x ] n P x,φ , IC y ,ψ [ − 2 d y ] n Q y ,ψ  ∼ = M x,φ M n Q x,φ ,n P x,φ ( Q ) . Using the fact tha t n P x,φ is the m ultiplicit y of the irreducible represen tation H 2 d x ( B x ) φ of W in the induced represen tatio n Ind W W P (1 W P ), w e see that (3.20 ) is consisten t with (3.1 7 ). 4. Equiv ariant K -theor y Certainly the most imp ortant result to date in volving the Stein b erg v ariety is its appli- cation b y Kazhdan and Lusztig to the Langlands program [KL87]. T hey sho w that the equiv ariant K -theory of Z is isomorphic to the t w o- sided regular repres en ta tion of the ex- tended, aﬃne Hec ke algebra H . They then use this represen tation of H to classify simple H -mo dules a nd hence to classify represen tations o f L G ( Q p ) con taining a ve ctor ﬁxed b y an Iwahori subgroup, where L G ( Q p ) is the group of Q p -p oints of the Langla nds dual of G . As with homology , Chriss and Ginzburg hav e applied the con volution pro duct formalism to the equiv arian t K -theory of Z and recast Kazhdan and Lusztig’s results as an algebra isomorphism. Recall w e are assuming that G is simply connected. In this section w e describ e the isomorphism H ∼ = K G ( Z ), where G = G × C ∗ , and we giv e some applications to the study of nilp otent orbits. W e emphasize in particular the relatio nship b et w een nilp otent orbits, Kazhdan-Lusztig theory for the extended, aﬃne W eyl group, and (generalized) Stein b erg v arieties. 4.1. The generic, extended, aﬃne Hec k e algeb ra. W e begin b y describing the Bern- stein-Zelevinski presen tatio n of the extended, aﬃne Hec ke algebra follo wing the construction in [Lus89a]. Let v b e an indeterminate and set A = Z [ v , v − 1 ]. The ring A is the base ring of scalars for most of the constructions in this section. Let X ( T ) denote the c ha r acter group of T . Since G is simply connected, X ( T ) is the w eigh t lattice of G . Deﬁne X + to be the set of dominan t w eights with resp ect to the base of t he ro ot sys tem of ( G, T ) determined by B . The extende d , aﬃn e Weyl gr oup is W e = X ( T ) ⋊ W . There is a “length function” ℓ on W e that extends the usual length f unction on W . The br aid gr oup of W e is the group B r , with generators { T x | x ∈ W e } and relations T x T x ′ = T xx ′ if ℓ ( x ) + ℓ ( x ′ ) = ℓ ( xx ′ ). The generic, extende d, a ﬃ n e He cke algebr a, H , is the quotien t of 26 J.M. D OUGLASS A ND G. R ¨ OHRLE the group a lg ebra A [ B r ] by the t w o -sided ideal g enerated by the elemen ts ( T s + 1)( T s − v 2 ), where s runs through the simple reﬂections in W . Let L G denote the Langlands dual of G , so L G is an adjoin t g r oup. Let L G p denote t he algebraic group o ver Q p with the same type as L G . Supp ose tha t I is an Iw ahori subgroup of L G p and let C [ I \ L G p /I ] denote the space o f all compactly suppo r t ed functions L G p → C that are constan t on ( I , I )-double cosets. Consider C as an A -mo dule via the sp ecialization A → C with v 7→ √ p . The follo wing theorem, due to Iwahori a nd Mats umoto [IM65, § 3], relates H to represen tations of L G p con ta ining an I -ﬁxed v ector. Theorem 4.1. The ( I , I ) -double c osets of L G p ar e p ar a metrize d by W e . Mor e ov e r, if I x is the double c oset indexe d by x in W e , then the map which sends T x to the char acteristic function of I x extends to an algebr a isom o rphism C ⊗ A H ∼ = C [ I \ L G p /I ] . The algebra H has a factorization (as a tensor pro duct) analogous to the factorization W e = X ( T ) ⋊ W . Give n λ in X ( T ) one can write λ = λ 1 − λ 2 where λ 1 and λ 2 are in X + . Deﬁne E λ in H to b e the image of v ℓ ( λ 1 − λ 2 ) T λ . F o r x in W e , denote the image of T x in H again b y T x . Let H W denote the Iwahori-Hec k e a lg ebra of W (an A - algebra) with standard basis { t w | w ∈ W } . Lusztig [Lus89a , § 2] has prov ed the following theorem. Theorem 4.2. With the no tation ab ove we have: (a) E λ do es not dep end on the choic e of λ 1 and λ 2 . (b) The mapping A [ X ( T )] ⊗ A H W → H d eﬁne d by λ ⊗ t w 7→ E λ T w is an isomorph ism o f A -mo dules. (c) F or λ and λ ′ in X we have E λ E λ ′ = E λ + λ ′ and so the subsp ac e of H sp anne d by { E λ | λ ∈ X } is a sub algebr a isomorp h ic to A [ X ( T )] . (d) The c enter of H is isomorphic to A [ X ( T )] W via the isomorphism in (c). (e) The subsp ac e of H sp anne d by { T w | w ∈ W } is a sub algebr a isomo rp h ic to H W . Using parts (b) and (d) of the theorem, w e iden tify A [ X ( T )] with the subalgebra of H spanned by { E λ | λ ∈ X } , and A [ X ( T )] W with the cen ter of H . 4.2. Equiv arian t K -theory and con volu tion. Tw o in tro ductory references for the no- tions from equiv ariant K - theory w e use a re [BBM89, Chapter 2] and [CG9 7, Chapter 5]. F or a v ariet y X , le t C oh ( X ) den ote the category of coheren t O X -mo dules. Supp ose that H is a linear algebraic group acting on X . Let a : H × X → X b e the action morphism and p : H × X → X b e the pro jection. An H -e quivarian t c oher ent O X -mo dule is a pair ( M , i ), where M is a coheren t O X -mo dule and i : a ∗ M ∼ − → p ∗ M is an isomorphism satisfying sev eral conditions (se e [CG97, § 5.1] for the precise deﬁnition). With the obvious notion of morphism, H -equiv arian t O X -mo dules for m an ab elian category denoted b y C oh H ( X ). The Grothendiec k group of C oh H ( X ) is denoted b y K H ( X ) a nd is c alled the H -e quivariant K - gr oup of X . If X = { pt } is a p oint, then K H (pt) ∼ = R ( H ) is the represen tation ring of H . F or a ny X , K H ( X ) is naturally an R ( H )-mo dule. If H is the tr ivial group, then C oh H ( X ) = C oh ( X ) and K H ( X ) = K ( X ) is the Grothendiec k group o f the category of coheren t O X -mo dules. As with Borel-Mo o re homology , equiv ariant K - theory is a biv ariant theory in the sense of F ulton and MacPherson [FM81]: Supp ose that X and Y are H -v arieties and that f : X → Y STEINBERG V ARIETY AN D R EPRESENT A TIONS 27 is an H - equiv ariant morphism. If f is pro p er, there is a direct image map in equiv ar ian t K - theory , f ∗ : K H ( X ) → K H ( Y ), and if f is smo oth there is a pullbac k map f ∗ : K H ( Y ) → K H ( X ) in eq uiv arian t K - theory . Moreo v er, if X is sm o oth and A and B are closed, H -stable sub v arieties of X , there is an intersec tion pairing ∩ : K H ( A ) × K H ( B ) → K H ( A ∩ B ) (called a T or- pro duct in [L us98, § 6.4]). This pairing dep ends o n ( X , A, B ). Th us, w e may apply the con volution pro duct construction fro m § 3.1 in the equiv ariant K -theory setting. In more detail, supp ose that for i = 1 , 2 , 3, M i is a smo oth v a riet y with an algebraic actio n of H and f i : M i → N is a prop er, H -equiv arian t morphism. Supp ose that fo r 1 ≤ i < j ≤ 3, Z i,j is a closed, H -stable subv ariet y of M i × M j and that p 1 , 3 : p − 1 1 , 2 ( Z 1 , 2 ) ∩ p − 1 2 , 3 ( Z 2 , 3 ) → Z 1 , 3 is a prop er morphism. Then as in § 3.1, the formula c ∗ d = ( p 1 , 3 ) ∗  p ∗ 1 , 2 ( c ) ∩ p ∗ 2 , 3 ( d )  , where ∩ is the in tersection pairing determined by the sub sets Z 1 , 2 × M 3 and M 1 × Z 2 , 3 of M 1 × M 2 × M 3 , deﬁnes an associative con v olutio n pro duct, K H ( Z 1 , 2 ) ⊗ K H ( Z 2 , 3 ) ∗ − → K H ( Z 1 , 3 ). In particular, t he conv olution pro duct deﬁnes a ring structure on K G ( Z ). It is shown in [CG97, Theorem 7.2.2] that with this ring structure, K G ( Z ) is isomor phic to the group ring Z [ W e ]. In the next subsection w e describ e a more general r esult with Z [ W e ] replaced b y H and G replaced by G × C ∗ , wh ere C ∗ denote t he m ultiplicative group of non-zero complex n umbers. The v ariable, v , in the deﬁnition of H is giv en a geometric meaning using the isomorphism X ( C ∗ ) ∼ = Z . Let 1 C ∗ denote the t r ivial represen tatio n of C ∗ . Then the rule v 7→ 1 C ∗ extends to a ring isomorphism Z [ v , v − 1 ] ∼ = R ( C ∗ ). F or the r est of this pap er we will use this isomorphism to identify A = Z [ v , v − 1 ] and R ( C ∗ ). 4.3. The Kazhdan-Lu sztig isomorphism. T o streamline the notation, set G = G × C ∗ . Then R ( G ) ∼ = R ( G ) ⊗ Z R ( C ∗ ) ∼ = R ( G ) ⊗ Z A = R ( G )[ v , v − 1 ]. Similarly , for a close d subgroup, H , of G , w e denote the subgroup H × C ∗ of G b y H . In particular, T = T × C ∗ and B = B × C ∗ . In the remainder of this pap er w e will neve r need to consider the closure of a subgroup of G and so this notation should not lead to any confusion. Deﬁne a C ∗ -action on g by ( ξ , x ) 7→ ξ 2 x . W e consider B as a C ∗ -set with the trivial action. Then the action of G on e N and Z extends to an action of G on e N and Z , and µ z and µ are G -equiv aria nt. Recall from § 4.1 that w e are viewing the gr o up algebra A [ X ( T )] as a subspace of H , and that the center of H is Z ( H ) = A [ X ( T )] W . Using the iden tiﬁcation A = R ( C ∗ ), w e ma y b egin to in terpret subspaces of H in K -theoretic terms: K G ( { pt } ) ∼ = R ( G ) ∼ = R ( G ) ⊗ R ( C ∗ ) ∼ = R ( G )[ v , v − 1 ] ∼ = A [ X ( T )] W = Z ( H ) . Recall that the “diagonal” sub v ariet y , Z 1 , of the Stein b erg v ariety is deﬁned by Z 1 = { ( x, B ′ , B ′ ) ∈ N × B × B | x ∈ b ′ } . F or suitable choices of f i : M i → N and Z i,j , and using the em b edding A ⊆ R ( G ), the con v olution pro duct induces v a rious A -linear maps: (1) K G ( Z ) × K G ( Z ) ∗ − → K G ( Z ); with this m ultiplication, K G ( Z ) is an A - algebra. (2) K G ( Z 1 ) × K G ( Z 1 ) ∗ − → K G ( Z 1 ); with this m ultiplication, K G ( Z 1 ) is a commutativ e A -algebra. (3) K G ( Z ) × K G ( e N × B ) ∗ − → K G ( e N × B ); this deﬁnes a left K G ( Z )-mo dule structure o n K G ( e N × B ). 28 J.M. D OUGLASS A ND G. R ¨ OHRLE The group K G ( Z 1 ) has a well-kno wn description. F irst, the rule ( x, B ′ ) 7→ ( x, B ′ , B ′ ) de- ﬁnes a G -equiv aria nt isomorphism b et w een e N and Z 1 and hence an isomorphism K G ( Z 1 ) ∼ = K G ( e N ). Second, t he pro j ection e N → B is a v ector bundle a nd so, using t he Thom isomor- phism in equiv ariant K -theory [CG97, § 5.4], we ha v e K G ( e N ) ∼ = K G ( B ). Third, B is isomor- phic to G × B { pt } b y a G -equiv aria nt isomorphism and so K G ( B ) ∼ = K B ( { pt } ) ∼ = R ( B ) by a ve rsion of F ro b enius recipro cit y in equiv a rian t K -theory [CG97, § 5.2.1 6]. Finally , since U is the unip o ten t radical of B , w e hav e R ( B ) ∼ = R ( B /U ) ∼ = R ( T ) ∼ = R ( T )[ v , v − 1 ] ∼ = A [ X ( T )] . Comp osing these isomorphisms , w e get an isomorphism K G ( Z 1 ) ∼ = − → A [ X ( T )], whic h is in fact an isomorphism of A -algebras. The inv erse isomorphism A [ X ( T )] ∼ = − → K G ( Z 1 ) may b e computed explicitly . Supp ose tha t λ is in X ( T ). Then λ lifts to a represen tation of B . Denote the r epresen tation space by C λ . Then the pro jection morphism G × B C λ → B is a G -equiv aria nt line bundle on B . The sheaf of sections of this line bundle is a G -equiv aria nt, coheren t sheaf of O B -mo dules that w e will denote b y L λ . Pulling L λ bac k ﬁrst thr o ugh the ve ctor bundle pro jection e N → B and then through the isomorphism Z 1 ∼ = e N , w e g et a G -equiv aria nt, coheren t she af of O Z 1 -mo dules w e denote by L λ . Let i 1 : Z 1 → Z be the inclusion. Deﬁne e λ = ( i 1 ) ∗ ([ L λ ]) in K G ( Z ). Then λ 7→ e λ deﬁnes an A -linear map from A [ X ( T )] to K G ( Z ). A concen tration theorem due to Thomason and the Cellular Fibration Lemma of Chriss and Ginzburg can b e used to prov e the following prop osition (see [CG97, 6.2 .7 ] and [Lus98, 7.15]). Prop osition 4.3. The close d emb e ddings i 1 : Z 1 → Z and j : Z → e N × B induc e inje ctive maps in e quivariant K -the ory, K G ( Z 1 ) ( i 1 ) ∗ − − → K G ( Z ) j ∗ − → K G ( e N × B ) . The map ( i 1 ) ∗ is an A -algebr a monom o rphism and the map j ∗ is a K G ( Z ) -mo dule monom o r- phism. In p articular, K G ( e N × B ) is a faithful K G ( Z ) -mo dule. F rom the pro p osition and the isomorphism K G ( { pt } ) ∼ = Z ( H ), w e see that there is a comm utative diagra m of A -algebras and A - algebra homomorphisms: Z ( H )   / / ∼ =   A [ X ( T )]   / / ∼ =   H K G ( { pt } )   / / K G ( Z 1 )   / / K G ( Z ) . W e will complete this dia gram with an isomor phism of A -a lgebras K G ( Z ) ∼ = H following the argumen t in [Lus98, § 7]. Fix a simple reﬂection, s , in W . Then there is a simple ro ot, α , in X ( T ) and a corre- sp onding co c haracter, ˇ α : C ∗ → T , so that if h · , · i is the pa ir ing b et w een c haracters and co c haracters of T , then h α , ˇ α i = 2 and s ( λ ) = λ − h λ, ˇ α i α for λ in X ( T ). Choose a w eigh t λ ′ in X ( T ) with h λ ′ , ˇ α i = − 1 and set λ ′′ = − λ ′ − α . Then L λ ′ ⊠ L λ ′′ is in C oh G ( B × B ). Lusztig STEINBERG V ARIETY AN D R EPRESENT A TIONS 29 [Lus98, 7.19] has sho wn that the restriction of L λ ′ ⊠ L λ ′′ to the closed sub v ariety G ( B , sB s ) do es not dep end on the choice of λ ′ . Denote the restriction of L λ ′ ⊠ L λ ′′ to G ( B , sB s ) by L s . It is easy to chec k that Z 1 ∩ Z s = { ( x, g B g − 1 , g B g − 1 ) ∈ Z 1 | g − 1 x ∈ u s } . It fo llo ws that Z s is smo oth and that π : Z s → G ( B , sB s ) is a v ector bundle pro jection with ﬁbre u s . Th us, there is a pullbac k map in equiv ar ia n t K -t heory , π ∗ : K G  G ( B , sB s )  → K G  Z s  , and so w e may consider π ∗ ([ L s ]) in K G  Z s  . Let i s : Z s → Z denote the inclusion. The n i s is a closed em b edding a nd so there is a direct image map ( i s ) ∗ : K G  Z s  → K G ( Z ). Deﬁne l s = ( i s ) ∗ π ∗ ([ L s ]). Then l s is in K G ( Z ). Lusztig [Lus98, 7.24] has prov ed the follo wing lemma. Lemma 4.4. T h er e is a unique left H -mo dule structur e on K G ( e N × B ) with the pr op erty that for every k in K G ( e N × B ) , λ in X ( T ) , and simple r eﬂe ction s in W we have (a) − ( T s + 1) · k = l s ∗ k and (b) E λ · k = e λ ∗ k . No w the H -mo dule and K G ( Z )-mo dule structures on K G ( e N × B ) determ ine A -linear ring ho mo mor phisms φ 1 : H → End A  K G ( e N × B )  and φ 2 : K G ( Z ) → End A  K G ( e N × B )  resp ectiv ely . It follows fro m Lemma 4.4 t hat the imag e of φ 1 is con tained in the image of φ 2 and it follo ws f rom Prop osition 4 .3 that φ 2 is a n injec tion. Therefore, φ − 1 2 ◦ φ 1 determines an A -algebra homomorphism from H to K G ( Z ) that w e denote by φ . The following theorem is pro ved in [Lus98, § 8] using a construction that go es back to [KL87]. Theorem 4.5. The A -algeb r a hom o morphism φ : H → K G ( Z ) is an isomorphism and Z ( H )   / / ∼ =   A [ X ( T )]   / / ∼ =   H ∼ = φ   K G ( { pt } )   / / K G ( Z 1 )   / / K G ( Z ) is a c ommutative diagr am of A -algebr a s and A -alge br a homomorphisms. In [CG97, § 7.6] Chriss and G inzburg construct an isomorphism H ∼ = K G ( Z ) that satisﬁes the conclusions of Theorem 4.5 using a v ariant of the ideas ab ov e. Set e = P w ∈ W T w in H . It is easy to c hec k that there is an A -mo dule isomorphism K G ( e N ) ∼ = H e and hence an A - algebra isomorphism End A ( K G ( e N )) ∼ = End A ( H e ). The c on- v olutio n product construction can b e used to de ﬁne the structure of a left K G ( Z )-mo dule on K G ( e N ) [CG97, § 5.4] and hence an A -alg ebra homomorphism K G ( Z ) → End A ( K G ( e N )). Similarly , the left H -mo dule structure on H e deﬁnes an A - algebra homomorphism H → End A ( H e ). Chriss and Ginzburg sho w that the dia gram H / / End A ( H e ) ∼ =   K G ( Z ) / / End A ( K G ( e N )) 30 J.M. D OUGLASS A ND G. R ¨ OHRLE can be c ompleted to a comm utative square of A - algebras and that the resulting A - algebra homomorphism H → K G ( Z ) is an isomorphism . W e will see in § 4.5 ho w this construction leads to a conjectural description of the equiv ariant K -theory of the g eneralized Steinberg v arieties X P , Q . 4.4. Irreducible represen tations of H , tw o-sided cells, and nilpot en t or bit s. The isomorphism in Theorem 4.5 has b een used by Kazhdan and Lusztig [KL87, § 7] to give a geo- metric construction and parametrization of irreducible H - mo dules. Using this construction, Lusztig [Lus89b, § 4] has found a bijection b et w een the set of t wo-side d Kazhdan-Lusztig cells in W e and the set of G -orbits in N . In order to describ e this bijection, as w ell as a con- jectural description of t w o- sided ideals in K G ( Z ) analogous to the decomp osition of H 4 n ( Z ) giv en in Prop osition 3.7, w e need to review the Kazhdan-Lusztig theory of t w o -sided cells and Lusztig’s based ring J . The rules v 7→ v − 1 and T x 7→ T − 1 x − 1 , for x in W e , deﬁne a ring inv olutio n of H denoted b y h 7→ h . The a rgumen t giv en b y Kazhdan and Lusztig in the pro of of [K L 79, Theorem 1.1] applies to H and sho ws that there is a unique ba sis, { c ′ y | y ∈ W e } , of H with the following prop erties: (1) c ′ y = c ′ y for all y in W e ; and (2) if w e write c ′ y = v − ℓ ( y ) P x ∈ W e P x,y c ′ x , then P y ,y = 1, P x,y = 0 unless x ≤ y , and P x,y is a p olynomial in v 2 with degree (in v ) a t most ℓ ( y ) − ℓ ( x ) − 1 when x < y . The p o lynomials P x,y are called K a zhdan-Lusztig p olynomials . F or x and y in W e , deﬁne x ≤ LR y if there exists h 1 and h 2 in H so that whe n h 1 c ′ y h 2 is expresse d as a linear com bination of c ′ z , the co eﬃcien t of c ′ x is non- zero. It follo ws fro m the results in [KL79, § 1] that ≤ LR is a preorder on W e . The equiv alence classes determ ined by this preorder ar e two-side d Kazhda n -Lusztig c el ls. Supp ose that Ω is a t wo-sided cell in W e and y is in W e . Deﬁne y ≤ LR Ω if there is a y ′ in Ω with y ≤ LR y ′ . Then by construction, the span of { c ′ y | y ≤ LR Ω } is a t wo-sided ideal in H . W e denote this tw o -sided ideal b y H Ω . The tw o sided ideals H Ω deﬁne a ﬁltratio n of H . In [Lus87 , § 2], L usztig ha s deﬁned a ring J whic h a f ter extending scalars is isomorphic to H , but for whic h the tw o-sided cells index a decomp osition into o rthogonal t w o- sided ideals, r ather than a ﬁltration by tw o-sided ideals. F or x , y , and z in W e , deﬁne h x,y ,z in A b y c ′ x c ′ y = P z ∈ W e h x,y ,z c ′ z . Next, deﬁne a ( z ) to b e the least non-negativ e in teger i with the prop erty that v i h x,y ,z is in Z [ v ] for all x and y . It is show n in [Lus85, § 7] that a ( z ) ≤ n . Finally , deﬁne γ x,y ,z to b e the constan t term of v a ( z ) h x,y ,z . No w let J b e the free ab elian group with basis { j y | y ∈ W e } and deﬁne a binary o p eration on J b y j x ∗ j y = P z ∈ W e γ x,y ,z j z . F or a tw o- sided cell Ω in W e , deﬁne J Ω to b e the span of { j y | y ∈ Ω } . In [Lus87, § 2], Lusztig prov ed tha t there a r e only ﬁnitely many t wo-sided cells in W e and deriv ed t he follo wing prop erties of ( J, ∗ ): (1) ( J, ∗ ) is an a sso ciativ e ring with iden tity . (2) J Ω is a tw o -sided ideal in J and ( J Ω , ∗ ) is a ring with iden tity . (3) J ∼ = ⊕ Ω J Ω (sum o v er all tw o-sided cells Ω in W e ). (4) There is a homomorphism of A -algebras, H → J ⊗ A . STEINBERG V ARIETY AN D R EPRESENT A TIONS 31 Returning to geometry , recall that U denotes the set of unip oten t elemen ts in G and that B u = { B ′ ∈ B | u ∈ B ′ } for u in U . Supp ose u is in U , s in G is semisimple, and u and s comm ute. Let h s i denote the smalles t closed, diagonalizable subgroup of G con taining s and se t h s i = h s i × C ∗ . In [Lus89b , § 2], Lusztig deﬁnes an a ctio n of h s i on B u using a ho mo mor phism SL 2 ( C ) → G corresp onding to u . Deﬁne A C = A ⊗ C , H C = H ⊗ A A C , and K u,s =  K h s i ( B u ) ⊗ C  ⊗ R ( h s i ) ⊗ C A C . In [Lus89b, § 2], Lusztig deﬁnes comm uting actions of H C and C ( u s ) on K u,s . F or an ir - reducible represen tation ρ of C ( us ), let K u,s,ρ denote the ρ -isotopic comp o nent of K u,s , so K u,s,ρ is an H C -mo dule. The next result is pr ov ed in [Lus89b, Theorem 4.2]. Theorem 4.6. Supp ose u and s a r e as ab ove and that ρ is an irr e ducible r epr esentation of C ( u s ) such that K u,s,ρ 6 = 0 . Then, up to isomorphis m , ther e is a unique simple J -mo dule, E , with the pr op erty that when E ⊗ C [ v ,v − 1 ] C ( v ) is c onsider e d as an H C ⊗ C [ v ,v − 1 ] C ( v ) - m o dule, via the homomorphis m H → J ⊗ A , then E ⊗ C [ v ,v − 1 ] C ( v ) ∼ = K u,s,ρ ⊗ C [ v ,v − 1 ] C ( v ) . Giv en u , s , and ρ as in the t heorem, let E ( u, s, ρ ) denote the correspo nding simple J - mo dule. Since J ∼ = ⊕ Ω J Ω and E ( u, s, ρ ) is simple, there is a unique tw o-sided cell Ω( u, s, ρ ) with t he prop ert y that J Ω( u,s,ρ ) E ( u, s, ρ ) 6 = 0. The main result in [Lus89b, Theorem 4.8] is the next theorem. Theorem 4.7. With the notation as a b ove, the two- side d c el l Ω( u, s, ρ ) dep e nds only o n the G -c onjugacy class of u . Mor e over, the rule ( u, s, ρ ) 7→ Ω( u, s, ρ ) determines a wel l-d eﬁne d bije ction b etwe en the set of unip otent c on j ugac y classes in G and the s e t of two-side d c el ls in W e . This bije ction has the pr op erty that a ( z ) = dim B u for any z in Ω( u , s, ρ ) . Using a Springer isomorphism U ∼ = N w e obtain the follo wing corollary . Corollary 4.8. Ther e is a bije ction b etwe en the set of nilp otent G -orbits in N and the set of two-side d c el ls of W e with the pr op erty that if x is in N and Ω is the two-side d c el l c orr esp onding to the G -orbit G · x , then a ( z ) = dim B x for every z in Ω . W e can no w work out some examples. Let Ω 1 denote the t wo-sided Kazhdan- L usztig cell corresp onding to the regula r nilp oten t orbit. Then a ( z ) = 0 for z in Ω 1 and Ω 1 is the unique t wo-sided Kazhdan-Lusztig cell on whic h the a -function tak es the v alue 0. Let 1 denote the ide n tit y eleme n t in W e . Then it follo ws imme diately from the deﬁnitions that { 1 } is a t wo-sided cell and tha t a (1) = 0 . Therefore, Ω 1 = { 1 } . A t the other extreme, let Ω 0 denote the tw o-sided Kazhdan-Lusztig cell cor r esp o nding to the nilp otent orbit { 0 } . Then a ( z ) = n for z in Ω 0 and Ω 0 is the unique tw o -sided Kazhdan-Lusztig cell on whic h the a -function take s the v alue n . Shi [Shi87] has show n that Ω 0 = { y ∈ W e | a ( y ) = n } = { y 1 w 0 y 2 ∈ W e | ℓ ( y 1 w 0 y 2 ) = ℓ ( y 1 ) + ℓ ( w 0 ) + ℓ ( y 2 ) } . The relation ≤ LR determines a partial o rder on the set of t w o- sided Kazhdan- L usztig cells and one of the imp ortant prop erties o f Lusztig’s a function is that a ( y 1 ) ≤ a ( y 2 ) whenev er y 2 ≤ LR y 1 (see [Lus85, Theorem 5.4]) . Therefore, Ω 1 is t he unique maximal t w o -sided cell and Ω 0 is the unique minimal t wo-sided cell. It fo llo ws that H Ω 1 = H and tha t H Ω 0 is the span of { c ′ y | y ∈ Ω 0 } . 32 J.M. D OUGLASS A ND G. R ¨ OHRLE Summarizing, w e ha ve seen that H is ﬁltered by the t wo sided ideals H Ω , wh ere Ω runs o v er the set of tw o- sided Kazhdan-Lusztig cells in W e , and that there is a bijection b et w een the set of tw o-sided cells in W e and the set of nilp otent orbit s n N . No w supp ose tha t O is a nilp oten t orbit and recall the subv ariet y Z O of Z deﬁned in § 3.5. Let i O : Z O → Z denote the inclusion. There are direct image maps, ( i O ) ∗ in Borel-Moor e homology and in K -theory . It follo ws fr o m the con volution construction that the imag es of these maps are tw o- sided ideals in H ∗ ( Z ) and K G ( Z ) resp ective ly . In § 3.5 w e described the image of ( i O ) ∗ : H 4 n ( Z O ) → H 4 n ( Z ), a tw o -sided ideal in H 4 n ( Z ). The argumen t in [KL87, § 5] sho ws that ( i O ) ∗ ⊗ id : K G ( Z O ) ⊗ Q → K G ( Z ) ⊗ Q is injectiv e. In contrast, ( i O ) ∗ : H j ( Z O ) → H j ( Z ) is an injection when j = 4 n , but fails to b e an injection in general. F or example, taking O = O = { 0 } , w e hav e that Z { 0 } = { 0 } × B × B and dim H ∗ ( Z { 0 } ) = dim H ∗ ( Z ) = | W 2 | . Ho we v er, dim H 4 n ( Z { 0 } ) = 1 and H 4 n ( Z ) = | W | and so ( i { 0 } ) ∗ : H j ( Z { 0 } ) → H j ( Z ) cannot b e an injection for all j . Deﬁne I O to b e the image of ( i O ) ∗ : K G ( Z O ) → K G ( Z ), a tw o-sided ideal in K G ( Z ). There is a n in triguing conjectural description of the image of I O under the isomorphism K G ( Z ) ∼ = H due to Ginzburg [G in87] that ties together all the themes in this subsection. Conjecture 4.9. Supp ose that O is a G -o rbit in N and Ω is the two-side d c e l l in W e c orr esp onding to O as in Cor ol la ry 4 .8. T h en φ ( I O ) = H Ω , wher e φ : K G ( Z ) ∼ = − → H is the isomorphism in The or em 4.5. This conjecture has b een prov ed when G has t yp e A l b y T anisaki and Xi [TX06]. Xi has recen tly sho wn that the conjecture is true after extending scalars to Q ([Xi08]). As a ﬁrst example, consider the case o f the regular nilp oten t orbit and the corresp onding t wo-sided cell Ω 1 . Then O = N , I N = K G ( Z ) a nd H Ω 1 = H . Th us the conjecture is easily seen to b e true in this case. F or a more interes ting example, consider the case of the zero nilp oten t orbit. Then Z { 0 } = { 0 } × B × B . The corresp onding t w o-sided cell, Ω 0 , has b een describ ed ab o ve a nd w e hav e seen that H Ω 0 is the span of { c ′ y | y ∈ Ω 0 } . It is easy to c hec k that P w ,w 0 = 1 for ev ery w in W and th us c ′ w 0 = v − n P w ∈ W T w = v − n e , where e is a s in § 4.3. Let H c ′ w 0 H denote the t wo sided ideal generated b y c ′ w 0 . In [Xi94 ], Xi has prov ed the fo llo wing theorem. Theorem 4.10. With the no tation as ab ove we have φ  I { 0 }  = H c ′ w 0 H = H Ω 0 . 4.5. Equiv arian t K -theory of general ized Stein b erg v arieties. Supp ose P and Q are conjugacy classes of parab olic subgroups of G a nd recall the generalized Steinberg v arieties X P , Q and X P , Q 0 , 0 , and the maps η : Z → X P , Q and η 1 : Z P , Q = η − 1 ( X P , Q 0 , 0 ) → X P , Q 0 , 0 from § 2.4 . W e hav e a cartesian square of prop er mor phisms (4.11) Z P , Q k / / η 1   Z η   X P , Q 0 , 0 k 1 / / X P , Q where k a nd k 1 are the inclusions. STEINBERG V ARIETY AN D R EPRESENT A TIONS 33 The morphism η 1 is smo oth and so there is a pullback map in equiv aria n t K -theory , η ∗ 1 : K G ( X P , Q 0 , 0 ) → K G ( Z P , Q ). W e can describ e the R ( G )-mo dule structure of K G ( Z P , Q ) and K G ( X P , Q 0 , 0 ) using the argumen t in [Lus98, 7.15] together with a stronger c oncen tr a tion theorem due to Thomason [Tho92, § 2]. Theorem 4.12. The homomorphisms η ∗ 1 : K G ( X P , Q 0 , 0 ) → K G ( Z P , Q ) and k ∗ : K G ( Z P , Q ) → K G ( Z ) ar e inje ctive . Mor e ov er, K G ( X P , Q 0 , 0 ) is a fr e e R ( G ) -mo dule with r ank | W | 2 / | W P || W Q | and K G ( Z P , Q ) is a fr e e R ( G ) -mo dule with r ank | W | 2 . The Cellular Fibration Lemma o f Chriss and Ginzburg [CG 9 7, 6.2.7] can b e used to describe the R ( G )-mo dule structure of K G ( X P , Q ) when P = B or Q = B . Prop osition 4.13. The e quivariant K - gr oup K G ( X B , Q ) is a fr e e R ( G ) -mo dule with r ank | W | 2 / | W Q | . W e exp ect that K G ( X P , Q ) is a free R ( G )-mo dule with rank | W | 2 / | W P || W Q for a r bitr ary P and Q . W e mak e a more general conjecture ab o ut K G ( X P , Q ) after ﬁrst cons idering a n example in whic h ev erything has b een explicitly computed. Consider the v ery sp ecial case when P = Q = { G } . In this case the spaces in (4.11 ) are w ell-known: X { G } , { G } 0 , 0 ≡ { 0 } , Z { G } , { G } = Z w 0 = Z { 0 } ∼ = B × B , and X { G } , { G } ≡ N . Also, η : Z → X { G } , { G } ma y b e iden tiﬁed with µ z : Z → N and k : Z { G } , { G } → Z ma y b e iden tiﬁed with the closed em b edding B × B → Z b y ( B ′ , B ′′ ) 7→ (0 , B ′ , B ′′ ) and so (4.11) b ecomes Z { G } , { G } = Z { 0 } i { 0 } / /   Z µ z   X { G } , { G } 0 , 0 = { 0 } / / N ∼ = X { G } , { G } . The image of ( i { 0 } ) ∗ : K G ( Z { 0 } ) → K G ( Z ) is I { 0 } and we sa w in Theorem 4.10 that I { 0 } ∼ = H c ′ w 0 H = H Ω 0 . Ostrik [Ost00] has describ ed the map ( µ z ) ∗ : K G ( Z ) → K G ( X { G } , { G } ). Recall that W e = X ( T ) ⋊ W . Because the fundamen tal W eyl cham b er is a fundamen t a l domain f or t he a ction of W on X ( T ) ⊗ R , it follo ws that eac h ( W , W )- double coset in W e con ta ins a unique elemen t in X + . Also, eac h ( W , W )-double coset in W e con ta ins a unique elemen t with minimal length. F or λ in X + w e let m λ denote the elemen t with minimal length in the double coset W λW . Theorem 4.14. F or x in W e , ( µ z ) ∗ ( c ′ x ) = 0 unless x = m λ for some λ in X + . Mor e over, the map ( µ z ) ∗ : K G ( Z ) → K G ( X { G } , { G } ) is surje ctive and { ( µ z ) ∗ ( c ′ m λ ) | λ ∈ X + } is an A -b asis of K G ( X { G } , { G } ) . Notice that the theorem is the K -theoretic analog of Theorem 3.16 in the v ery sp ecial case w e are considering. 34 J.M. D OUGLASS A ND G. R ¨ OHRLE T o prov e this result, Ostrik uses the description of Z as a ﬁbred pro duct a nd the t w o corresp onding facto r izat io ns of µ z : (4.15) Z = e N × N e N / /   X B , { G } ∼ = e N   e N ∼ = X { G } , B / / X { G } , { G } ∼ = N . It follows from the construction of the isomorphism K G ( Z ) ∼ = H giv en by Chriss a nd Ginzburg [CG97, § 7.6] (see the end o f § 4.3 ) that aft er applying the functor K G to ( 4 .15) the resulting comm utat ive diagram of equiv a rian t K -groups ma y b e iden tiﬁed with the fo llo wing comm utative diagra m subspaces of H : (4.16) H / /   H c ′ w 0   c ′ w 0 H / / c ′ w 0 H c ′ w 0 where the maps are give n by the appropriate righ t or left m ultiplication b y c ′ w 0 . W e conclude with a conjecture describing K G ( X P , Q ) for arbitrary P and Q . Recall f r o m § 3.7 that X P , Q ∼ = e N P × N e N Q . The pro j ection µ : e N → N fa ctors as e N η P − → e N P ξ P − → N where η P ( x, g B g − 1 ) = ( x, g P g − 1 ) and ξ P ( x, g P g − 1 ) = x . Using this factor izat io n, w e ma y expand diagram (4.15) to a 3 × 3 diag ram with X P , Q in the cen ter: (4.17) Z / /   X B , Q / /   e N   X P , B / /   X P , Q / /   e N P   e N / / e N Q / / N . Let w P and w Q denote the long est elemen ts in W P and W Q resp ectiv ely . Comparing (4.1 5), (4.16), and (4.17), w e mak e the follow ing conjecture. This conjecture is a K -theoretic analog of (3.17) a nd Conjecture 3.19 . Conjecture 4.18. With the n otation ab ov e, K G ( X P , Q ) ∼ = c ′ w P H c ′ w Q . If the conjecture is true, then a fter applying the functor K G to (4.17) the resulting com- m utat iv e diag ram of equiv ar ian t K -gro ups may b e iden tiﬁed with the follo wing comm utative STEINBERG V ARIETY AN D R EPRESENT A TIONS 35 diagram of subspaces of H : (4.19) H / /   H c ′ w Q / /   H c ′ w 0   c ′ w P H / /   c ′ w P H c ′ w Q / /   c ′ w P H c ′ w 0   c ′ w 0 H / / c ′ w 0 H c ′ w Q / / c ′ w 0 H c ′ w 0 . Reference s [BB81] A. Beilinson and J. Bernstein, L o c alisation de g -mo dules , C. R. Acad. Sci. P aris S ´ er. I Math. 292 (1981), no. 1, 15 –18. [BB82] W. Borho and J.L. Brylinski, Diﬀer ential op er ators on homo gene ous sp ac es. 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Dep ar tment of Ma thema tics, University o f Nor th Texas, Denton TX, USA 762 03 E-mail addr ess : dougl ass@un t.edu URL : http: //hil bert.math.unt.edu F akul t ¨ at f ¨ ur Ma thema tik, Ruhr-Universit ¨ at Bochum, D-447 80 Bochu m, G ermany E-mail addr ess : gerha rd.roe hrle@ rub.de URL : http: //www .ruhr-uni-bochum.de/ffm/Lehrstuehle/Lehrstuhl-VI/rubroehrle.html

The Steinberg Variety and Representations of Reductive Groups

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