Filtrations on projective modules for Iwahori-Hecke algebras (Q2776263)

The purpose of this paper is to refine earlier results of the first named author [Represent. Theory 2, No. 7, 264-277 (1998; Zbl 0901.20004)] on decomposition numbers for Iwahori-Hecke algebras.NEWLINENEWLINENEWLINELet \(H\) be the generic Iwahori-Hecke algebra associated with a finite Weyl group \(W\), defined over \(A=\mathbb{Z}[v,v^{-1}]\) where \(v\) is an indeterminate. Let \(K\) be the field of fractions of \(A\) and let \({\mathcal O}\subseteq K\) be a discrete valuation ring containing \(A\) such that the image of \(v\) in \(k\), the residue field of \(A\), has finite order. The decomposition maps \(d^H_k\) of interest in this paper are maps between the Grothendieck groups of \(H_K\) and \(H_k\). The authors assume that the characteristic of \(k\) is either \(0\) or a good prime for \(W\). The decomposition numbers \(d_{V,M}\) are defined by the equation NEWLINE\[NEWLINEd^H_k([V])=\sum_{M\in\text{Irr}(H_k)}d_{V,M}[M]NEWLINE\]NEWLINE for \(V\in\text{Irr}(H_K)\). It is known [from loc. cit.] that \(d_{V,M}\neq 0\) implies that \(a_M\leq a_V\), where \(a\) is Lusztig's \(a\)-function. Furthermore, the decomposition matrix is lower triangular with ones on the diagonal, and the subset \(B\) of \(\text{Irr}(H_K)\) corresponding via the diagonal entries to \(\text{Irr}(H_k)\) can be described explicitly.NEWLINENEWLINENEWLINEThe refinements provided in this paper show that certain decomposition numbers \(d_{V,M}\) are zero: if \(d_{V,M}\neq 0\) and \(a_V=a_M\) then Corollary 4.3 shows that \(V\) lies in \(B\) and \(M\) corresponds to \(V\) in the sense of the previous paragraph. This shows that the subset \(B\) is uniquely defined and that the correspondence is unique. The filtrations of the title are defined in terms of the \(a\)-function and provide strong information in the case of projective modules (Theorems 3.4 and 4.2); the results on decomposition numbers follow from these.NEWLINENEWLINEFor the entire collection see [Zbl 0977.00027].