A REMARK ON SOME BASES IN THE HECKE ALGEBRA R.VIRK 1 Contents 1 0 1. Introduction 1 2 2. The Hecke algebra 2 n 3. The Ext form 4 a 4. Dualities between the bases 5 J References 6 8 ] O C h. 1. Introduction t a InthisbriefnoteweconsidersomebasesintheHeckealgebraandexhibitcertain m dualities between them. Our results and arguments are completely combinatorial. [ However,toexplainthe motivationandputthecontentsofthisnoteinperspective we need to discuss some results from geometric representation theory. 2 LetGbeareductivegroup,B ⊆GaBorelsubgroupandletBbetheflagvariety v 4 of G. Then the Iwahori-Hecke algebra of G realizes the Grothendieck group of B- 2 equivariantperversesheavesonB.Multiplication inthe Heckealgebracorresponds 9 to convolution of sheaves. 1 In his streamlined treatment of the Hecke algebra, Soergel [So1] considers sev- . 2 eral pairwise commuting automorphisms (see [So1, Thm. 2.7]. The geometric sig- 1 nificance of some of these is well known. For instance, the automorphism d (2.3.1) 0 corresponds to Verdier duality on sheaves. This note was motivated by trying to 1 : understand the geometric significance of the automorphism b (3.4.1). I will now v outline the contents of this note from this perspective. i X A cornerstone of Kazhdan-Lusztig theory is the identification of IC-complexes r (=simple objects) on B with the Kazhdan-Lusztig basis of the Hecke algebra. In a Thm. 3.7 we identify/construct the basis corresponding to projective objects. In Thm.4.2weexplainhowbconnects the projectivebasiswith the Kazhdan-Lusztig basis. Actually, Thm. 4.2 implies that b switches the Kazhdan-Lusztig basis with the basis corresponding to tilting objects (see [Virk, §9] for the precise statement). Now,Soergel’s‘tiltingduality’[So2]switchestiltingobjectswithprojectiveobjects and the Koszul duality of [BGS] switches projective objects with simple objects. It follows that b is the composition of these two dualities. Another way to see this is to use [So3, Thm. 4.4] to directly deduce that b switches the Kazhdan-Lusztig basis with the tilting basis. The aforementioned Koszul duality does not commute with convolution (it doesn’t preserve the monoidal unit). In particular, it does not descendtoaringautomorphismofthe Heckealgebra.However,Thm.4.2isacom- binatorial shadow of the statement that the tilting duality intertwines convolution and Koszul duality (cf. [BG, Conjecture 5.18, Thm. 5.24, Thm. 6.10]). 1 2 R.VIRK Acknowledgments I am grateful to W. Soergel for some very helpful correspon- dence and to Arun Ram for his comments on a preliminary version of this note. This work is supported by the NSF grant DMS-0652641. 2. The Hecke algebra 2.1. Let(W,S) be aCoxetersystem, ℓ: W →Z the correspondinglength func- ≥0 tion and let ≤ denote the Bruhat order on W. In particular, x < y means x ≤ y and x6=y. The identity element in W is denoted by e. 2.2. The Hecke algebra H is the free Z[v,v−1]-module Z[v,v−1]T with x∈W x Z[v,v−1]-algebra structure given by L T T =T if ℓ(xy)=ℓ(x)+ℓ(y), (2.2.1) x y xy (T +1)(T −v−2)=0 for all s∈S. (2.2.2) s s Set H =vℓ(x)T for all x∈W. Then (2.2.2) implies that each H is invertible. In x x x particular, if s∈S, then H2 =1−(v−v−1)H . Hence, s s H−1 =H +(v−v−1). (2.2.3) s s Note that T =1. e 2.3. Define a ring involution d: H→H by d(v)=v−1, d(H )=H−1 . (2.3.1) x x−1 We will often write H instead of d(H). We call C ∈H self-dual if C =C. For each s∈S set C =H +v. (2.3.2) s s Then, by (2.2.3), each C is self-dual. In the sequel we will need the following well s known formulae: 2.4. Lemma. Let s∈S and let x∈W be arbitrary. Then H +vH if sx>x; sx x C H = s x (Hsx+v−1Hx if sxy (cid:1) =H +vH + h (H +v−1H )+ h (H +vH ). x sx y,sx sy y y,sx sy y yy We need to look slightly more carefully at the construction of C in Lemma 2.5. x Note that for y x; C C = syx, then once again by Lemma 2.9 C =C C − µ(y,x)C , sx s x y yy syx; sx,y x,y hC H ,H i= s x y (δsx,y+v−1δx,y if sxy; x,sy x,y hH ,C H i= x s y (δx,sy+v−1δx,y if sy x and sy > y, then the assertion is evident; if sx > x and sy < y, then certainly x 6= y and so the assertion holds; if sxy,thenonceagainx6=y andthe assertionholds;finally,ifsxx y x z x,z Proof.PProceed by induction. Let w be the longest element in W. We start our 0 induction with P = H . Now let x ∈ W be given and suppose we know the w0 w0 existence of P for all y > x. If x 6= w find s ∈ S such that sx > x. Then by the y 0 induction hypothesis b(C )P ∈H + h H s sx x y y y>x X for some h ∈Z[v]. For each y >x let p =hb(C )P ,C i. Set y y s sx y P =b(C )P − p P . x s sx y y y>x X Let’s show that P ∈ H + vZ[v]H . To do this we must demonstrate that x x y>x y (a) p ∈ Z[v] and (b) p (0) = h (0), for each y > x. Both of these follow from y y y P the fact that each h ∈ Z[v] and each C ∈ H + vZ[v]H . By construction y y y z z hP ,C i=δ for all z ∈W. (cid:3) x z x,z P 3.7. Theorem. Assume W is finite. Then for each x ∈ W there exists a unique P ∈H such that hP ,C i=δ for all y ∈W. x x y x,y Proof. Existence was established in the previous Lemma. Uniqueness follows from the evident fact that the form h·,·i is non-degenerate. (cid:3) 4. Dualities between the bases 4.1. Lemma. Assume W is finite. Let w be the longest element in W. Then for 0 all x∈W, P =CxH for some self-dual Cx ∈H. x w0 Proof. Proceed by induction. We start our induction with P = H . Now let w0 w0 x ∈ W be given and suppose we know that P = CyH , with Cy = Cy, for all y w0 y > x. If x 6= w find s ∈ S such that sx > x. Then, by the proof of Lemma 3.6 0 and the induction hypothesis, P =b(C )P − p P =b(C )CsxH − p CyH , x s sx y y s w0 y w0 y>x y>x X X where p = hb(C )P ,C i. To complete the proof it suffices to show that p ∈ y s sx y y Z[v+v−1] for each y. By Prop. 3.5 hb(C )P ,C i=hP ,b(C )C i. s sx y sx s y If sy < y, then b(C )C = (C −(v+v−1))C = 0 by Prop. 2.10. If sy > y, then s y s y once again using Prop. 2.10, b(C )C =(C −(v+v−1))C =C −(v+v−1)C + m C s y s y sy y z z z X for appropriate integers m ∈Z. The result follows. (cid:3) z 4.2. Theorem. Assume W is finite. Let w be the longest element in W. Then 0 b(C )H =P x w0 xw0 for all x∈W. Proof. By Lemma 4.1 P H−1 is self-dual. On the other hand, by Lemma 3.6, xw0 w0 P H−1 ∈H H−1+ vZ[v]H H−1. xw0 w0 xw0 w0 y w0 y>Xxw0 6 R.VIRK Using the fact that H H−1 =H H−1H−1 =H−1 and applying d we obtain z w0 z z w0z−1 w0z−1 P H−1 ∈H + v−1Z[v−1]H . xw0 w0 x yw0 y>Xxw0 By the uniqueness of C (Thm. 2.7) we must have that xw0 P H−1 =b(C ). (cid:3) xw0 w0 x References [BG] A. Beilinson, V. Ginzburg,Wall-crossing functors and D-modules, Represent. Theory 3 (1999), 1-31(electronic). [BGS] A.Beilinson,V.Ginzburg,W.Soergel,KoszuldualitypatternsinRepresentationThe- ory,JournalofA.M.S.9(1996), 473-527. [KL] D. Kazhdan, G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math.53(1979), 165-184. [So1] W. Soergel,Kazhdan-Lusztig polynomials and a combinatoric for tilting modules, Repre- sentationTheory1(1997), 37-68(electronic). [So2] W. Soergel, Character formulas for tilting modules over Kac-Moody algebras, Represen- tationTheory2(1998), 432-448(electronic). [So3] W.Soergel,Andersenfiltrationand hard Lefschetz,Geom.Funct.Anal.17(2008),no.6, 2066-2089. [Virk] R.Virk,A note on Hecke patterns incategory O,arXiv:1012.2416v1. DepartmentofMathematics,University of California,Davis, CA95616 E-mail address: virk@math.ucdavis.edu

A remark on some bases in the Hecke algebra

Similar A remark on some bases in the Hecke algebra