View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Algebra 252 (2002) 205–227 www.academicpress.com Filtrations on the Mackey decomposition for cyclotomic Hecke algebras M. Vazirani 1 Department of Mathematics, Berkeley, CA 94720-3840, USA Received 10 April 2000 Communicated by Georgia Benkart Abstract λ → λ λ → In [3], Grojnowski defines functors ei :RepHn Rep Hn−1 and fi :RepHn λ λ Rep Hn+1 shows that in the Grothendieck group K(RepHn ) [ei,fi ]M = hi(M)M, where hi(M) is an integer depending only on λ and the central character of M.Inthis λ paper, we show that the above relation makes sense when M is an irreducible Hn -module before passing to the Grothendieck group. That is, in the case hi(M) 0 eifi fiei M ⊕ hi(M)M and in the case hi(M) 0 eifi M ⊕ hi(M) M fiei M, k where for k 0, kM denotes the direct sum j=1 M. 2002 Elsevier Science (USA). All rights reserved. E-mail address: [email protected]. 1 The author was supported in part by an NSF Mathematical Sciences Postdoctoral Research Fellowship. 0021-8693/02/$ – see front matter 2002 Elsevier Science (USA). All rights reserved. PII:S0021-8693(02)00075-3 206 M. Vazirani / Journal of Algebra 252 (2002) 205–227 1. Introduction λ λ Let Hn (or Hn (q)) denote the cyclotomic Hecke algebra, where λ is an r-tuple ∈ × λ of nonzero elements from a field R,andq R as well. Hn is a deformation of n the group algebra of the wreath product (Z/rZ) Sn (and has dimension r n!). A precise definition is given in Section 2 below. Certain λ can be identified with level r dominant integral weights of the affine Lie algebra sl when q = 1 (or of gl∞ when the parameter q is not a root of unity). In this case, results of Ariki [1] and Grojnowski [3] show that the direct sum of Grothendieck groups λ K Rep Hn n has the structure of an irreducible sl-module (gl∞-module) with highest weight λ. In particular, they each define operators ei , fi ,andhi and prove they satisfy the Chevalley relations. Grojnowski’s operators come from functors λ → λ λ → λ ei :RepHn Rep Hn−1 and fi :RepHn Rep Hn+1, and many of his results λ come from a careful study of the structure of Hn -modules before passing to the λ Grothendieck group K(Rep Hn ). However, one must pass to the Grothendieck group to show the ei and fi satisfy the Serre relations, and to define the hi . Here, we show that an analogue of [ei,fi ]=hi λ holds in Rep Hn , before passing to the Grothendieck group. Given an irreducible module M, we prove that one of the modules eifi M and fi eiM is a direct summand of the other and that the quotient is a direct sum of copies of M. The multiplicity of M in this quotient is |ϕi(M) − εi(M)|. The quantities ϕi(M) and εi(M) are defined in terms of the functors fi and ei , and have many equivalent characterizations as outlined in Theorems 2.3 and 2.4 from [3] cited below. The map that gives the decomposition e f M f e M ⊕ ϕ (M) − ε (M)M i i i i i i or eifi M ⊕ ϕi(M) − εi(M) M fi eiM is a refinement of the cyclotomic Mackey decomposition, λ λ H λ H λ ⊕ Hn Hn n+1 n+1 rM Ind λ Res λ M Res λ Ind λ M. Hn−1 Hn−1 Hn Hn The main new ingredient is the study of canonical filtrations on the modules eiM, fiM, eifi M,andfi eiM that are induced by the action of certain central elements. This enables us to refine the proofs of [3, Sections 9 and 12] and obtain Theorem 7.1. In particular, we describe how copies of the irreducible modules ˜ e˜iM, fi M,andM sit in these filtrations and show the center acts on them as M. Vazirani / Journal of Algebra 252 (2002) 205–227 207 nilpotent Jordan blocks of sizes εi(M) and ϕi(M) and in terms of the tensor product of two such blocks. We warn the reader of a minor shift in notation: that we will consider functors × i ea and fa where a ∈ R , generalizing the case a = q which are denoted ei and fi above (and in [3]). 2. Definitions The definitions and propositions below are repeated from [3], [4] or [5], and we refer the reader there for proofs and for placing them in a larger context. Throughout the paper we fix an algebraically closed field R, and an invertible element q ∈ R×. We further assume that q = 1. For the case q = 1 we refer the reader to the appendix of [3]. If A is an R-algebra, we write Rep A for the category of left A-modules which are finite-dimensional as R-modules. We also write Z(A) forthecenterofthe algebra A. Recall that the socle of a module M, denoted soc(M), is the largest semisimple submodule of M, and that the cosocle of M, denoted cosoc(M), is its largest semisimple quotient. If M is an A-module and M is an A-module, M M denotes the A ⊗ A -module which is isomorphic to M ⊗R M as an R-module and has A ⊗ A action given by (a ⊗ a) · (m ⊗ m) = am⊗ am. For algebras A ⊂ B, we let Res denote restriction, and Ind denote induction. We take Ind to be the left adjoint of restriction, i.e. B = ⊗ IndA M B A M, whereas the right adjoint is given by ˘ B = IndA M HomA(B, M). fin λ 2.1. Hn , Hn, and Hn fin The finite Hecke algebra, Hn is the R-algebra with generators T1,...,Tn−1 and relations called braid relations: TiTi+1Ti = Ti+1TiTi+1, TiTj = Tj Ti , |i − j| > 1; quadratic relations: (Ti + 1)(Ti − q)= 0. 208 M. Vazirani / Journal of Algebra 252 (2002) 205–227 aff The affine Hecke algebra Hn (or Hn )istheR-algebra, which as an R-module is isomorphic to fin ⊗ ±1 ±1 Hn R R X1 ,...,Xn . fin [ ±1 ±1] The algebra structure is given by requiring that Hn and R X1 ,...,Xn are subalgebras, and that TiXi Ti = qXi+1,XiTj = Tj Xi if i = j,j + 1. Proposition 2.1 (Bernstein). The center of Hn, Z(Hn), is isomorphic to symmetric Laurent polynomials ±1 ±1 Sn Z(Hn) R X1 ,...,Xn . From this one can deduce that irreducible Hn-modules are finite-dimensional. It also follows that if M is an irreducible Hn-module, its central character can × n be identified with an orbit s = Sn · (s1,...,sn) in (R ) . [ ±1 ±1] ∈ In this case, there is a simultaneous R X1 ,...,Xn eigenvector v M and σ ∈ Sn such that Xi v = sσ(i)v for all i.(σ and v are not unique.) Observe that any indecomposable module has a well-defined central character. λ The cyclotomic Hecke algebra or Ariki–Koike algebra [2] Hn is the quotient r − = Hn/Iλ,whereIλ is the ideal generated by j=1(X1 λj ),andλ (λ1,...,λr ) is an r-tuple in (R×)r . λ In other words, Hn -modules are those Hn-modules on which X1 has eigenvalues in {λj } with Jordan block size bounded by the multiplicities in λ. i The case that all λj ∈{q | i ∈ Z} is of special interest. In particular, one can then associate λ with a weight of sl if q is a primitive th root of unity, or a ⊕ weight of gl∞ otherwise. (And so the result nK(Repq Hn) L(λ) from [1,3] mentioned in the introduction holds). The surjective algebra homomorphism evλ : Hn → Hn/Iλ ∗ λ → induces an exact functor evλ :RepHn Rep Hn, with left adjoint prλ, and right λ adjoint pr˘ λ. The functor prλ picks off the largest quotient of N that is an Hn - module, and can be defined via = prλ(N) N/IλN. λ The functor pr˘ λ picks off the largest submodule of N that is an Hn -module, i.e. = Iλ that is killed by Iλ, and so can be defined via pr˘ λ(N) N . λ Given an Hn -module M, we will often find it more useful to view it as an Hn- module. That is, we will confuse it with ev∗ M, but be sloppy about writing the ∗ λ evλ. ⊗ ⊗···⊗ → Observe that there is an embedding Hn1 Hn2 Hnk ' Hn if the nj are positive integers summing to n. When we induce and restrict modules between M. Vazirani / Journal of Algebra 252 (2002) 205–227 209 these algebras, we will often abbreviate as Indn and Resn or simply as n1,...,nk n1,...,nk Ind and Res. We will be more careful with subscripts and superscripts on the rare occasion λ λ Hn+1 when inducing in Rep Hn . In general, we avoid Ind λ , as the power of the Hn definitions of ea and fa below come from working within the affine Hecke algebra. 2.2. The functor ea Let N be any module in Rep Hn.Define∆aN to be the generalized a- H eigenspace of X in Res n N.Then∆ N is an H − ⊗ H -module because n Hn−1⊗H1 a n 1 1 Xn − a ⊆ Z(Hn− −⊗H ).Furthermore 1 1 ResHn N = ∆ N.