aa r X i v : . [ m a t h . R T ] J un OPEN PROBLEMS ON IWAHORI-HECKE ALGEBRAS
G. Lusztig
Below we state four open problems (see (a)-(d)) on Iwahori-Hecke algebras. Let I be a finite set and let ( m ij ) ( i,j ) ∈ I × I be a symmetric matrix whose diagonalentries are 1 and whose nondiagonal entries are integers ≥ ∞ . Let W be thegroup with generators { s i ; i ∈ I } and relations ( s i s j ) m ij = 1 for any i, j such that m ij < ∞ ; this is a Coxeter group. (Examples of Coxeter groups are the Weylgroups of simple Lie algebras; these are finite groups. Other examples are theaffine Weyl groups which are almost finite.) For w ∈ W let | w | be the smallestinteger n ≥ w is a product of n generators s i , i ∈ I . We assumethat we are given a weight function L : W −→ N that is, a function such that L ( w ) > w ∈ W − { } and L ( ww ′ ) = L ( w ) + L ( w ′ ) for any w, w ′ in W such that | ww ′ | = | w | + | w ′ | . (For example, w
7→ | w | is a weight function.) Let A = Z [ v, v − ] where v is an indeterminate. Let H be the free A -module with basis { T w ; w ∈ W } . There is a unique structure of associative A -algebra on H for which( T s i + v − L ( s i ) )( T s i − v L ( s i ) ) = 0 for i ∈ I and T w T w ′ = T ww ′ for any w, w ′ in W such that | ww ′ | = | w | + | w ′ | ; this is the Iwahori-Hecke algebra associated to W, L .For c ∈ C −{ } let H c = C ⊗ AH where C is viewed as an A -algebra via the ringhomomorphism A −→ C , v c . Now H c is also referred to as an Iwahori-Heckealgebra.(a) Show that the algebras associated in [S20] to a supercuspidal representationof a parabolic subgroup of a p -adic reductive group are (up to extension by a groupalgebra of a small finite group) of the form H q where q is a power of p , with H associated to an affine Weyl group W and with L in the collection Σ W of weightfunctions on W described in [L91, § , [L95] , [L02] . For example, if W is of affine type F , then Σ W consists of all L whose values on { s i ; i ∈ I } are (1 , , , ,
1) or (1 , , , ,
2) or (2 , , , ,
1) or (1 , , , , W is ofaffine type G , then Σ W consists of all L whose values on { s i ; i ∈ I } are (1 , , , ,
3) or (3 , ,
1) or (1 , , F q instead of p -adic groups is known to hold, without the words in parenthesis; Supported by NSF grant DMS-1855773. Typeset by
AMS -TEX G. LUSZTIG in that case, W is a Weyl group and Σ W consists of the weight functions on W described in [L78, p.35]. There is a unique group homomorphism¯: H −→ H such that v n T w T w − = v − n for n ∈ Z , w ∈ W ; it is a ring isomorphism. Let H ≤ = P w ∈ W Z [ v − ] T w ⊂ H .For any w ∈ W there is a unique element c w ∈ H ≤ such that ¯ c w = c w and c w − T w ∈ v − H ≤ (see [KL],[L03]). Then { c w ; w ∈ W } is an A -basis of H .For x, y, z in W we define f x,y,z ∈ A , h x,y,z ∈ A by T x T y = P z ∈ W f x,y,z T z , c x c y = P z ∈ W h x,y,z c z .(b) Show that there exists an integer N ≥ such that for any x, y, z in W wehave v − N f x,y,z ∈ Z [ v − ] . (See [L03, 13.4].) If W is finite this is obvious. If W is an affine Weyl group, thisis known.We will now assume that (b) holds. With N as in (b), we see that v − N h x,y,z ∈ Z [ v − ] for any x, y, z in W . It follows that for any z ∈ W there is a unique integer a ( z ) ≥ h x,y,z ∈ v a ( z ) Z [ v − ] for all x, y in W and h x,y,z / ∈ v a ( z ) − Z [ v − ]for some x, y in W . Hence for x, y, z in W there is a well defined integer γ x,y,z − such that h x,y,z = γ x,y,z − v a ( z ) mod v a ( z ) − Z [ v − ]. Let J be the free abeliangroup with basis { t w ; w ∈ W } . For x, y in W we set t x t y = P z ∈ W γ x,y,z − t z .(This is a finite sum.)(c) Show that this defines an (associative) ring structure on J (without ingeneral). Assume now that W is a Weyl group or an affine Weyl group and L = || . In thiscase, (c) is known to be true and the ring J does have a unit element.More generally, assume that W is an affine Weyl group and L ∈ Σ W (see(a)); in this case there is a (conjectural) geometric description [L16, 3.11] of theelements c w . From this one should be able to deduce (c) as well as the well-definedness of the C -algebra homomorphism H q −→ J in [L03, 18.9], where H q is as in (a) and J = C ⊗ J is independent of q . One should expect that theirreducible (finite dimensional) J -modules, when viewed as H q -modules, form abasis of the Grothendieck group of H q -modules. (This is indeed so if L = || .) Thisshould provide a construction of the “standard modules” of H q which, unlike theconstruction in [L95],[L02], does not involve the geometry of the dual group. Assume that W is finite and that L = || . Let C be a conjugacy class in W ; let C min be the set of all w ∈ C such that | w | is minimal. For w ∈ C let N w ∈ A bethe trace of the A -linear map H −→ H , h v | w | T w hT w − . We have N w ∈ Z [ v ].(Note that N w | v =1 is the order of the centralizer of w in W .) From [GP] onecan deduce that for w ∈ C min , N w depends only on C , not on w . We say that C is positive if N w ∈ N [ v ]. For example, if C is an elliptic regular conjugacyclasses (in the sense of [S74]) then C is positive (see [L18]). If W is of type A n ,the positive conjugacy classes are { } and the class of the Coxeter element. Inthe case where W is a Weyl group of exceptional type a complete list of positiveconjugacy classes in W is given in [L18]. PEN PROBLEMS ON IWAHORI-HECKE ALGEBRAS 3 (d)
Make a list of all positive conjugacy classes in W assuming that W is aWeyl group of type B n or D n . References [GP] M.Geck and G.Pfeiffer,
Characters of finite Coxeter groups and Iwahori-Hecke algebras ,Clarendon Press, Oxford, 2000.[KL] D.Kazhdan and G.Lusztig,
Representations of Coxeter groups and Hecke algebras , Inv.Math. (1979), 165-184.[L78] G.Lusztig, Representations of finite Chevalley groups , Regional Conf. Ser. in Math., vol. 39,Amer. Math. Soc..[L91] G.Lusztig,
Intersection cohomology methods in representation theory , Proc. Int. Congr.Math. Kyoto 1990, Springer Verlag, 1991.[L95] G.Lusztig,
Classification of unipotent representations of simple p -adic groups , Int. Math.Res. Notices (1995), 517-589.[L02] G.Lusztig, Classification of unipotent representations of simple p -adic groups,II , Repre-sent.Th. (2002), 243-289.[L03] G.Lusztig, Hecke algebras with unequal parameters , CRM Monograph Ser.18, Amer. Math.Soc., 2003, additional material in version 2 (2014), arxiv:math/0208154.[L16] G.Lusztig,
Nonsplit Hecke algebras and perverse sheaves , Selecta Math. (2016), 1953-1986.[L18] G.Lusztig, Positive conjugacy classes in Weyl groups , arxiv:1805.03772.[S20] M.Solleveld,
Endomorphism algebras and Hecke algebras for reductive p -adic groups ,arxiv:2005.07899.[S74] T.A.Springer, Regular elements of finite reflection groups , Invent. Math. (1974), 159-193.(1974), 159-193.