Signatures of representations of Hecke algebras and rational Cherednik algebras

Abstract

Determining whether an irreducible representation of a group (or ⁎-algebra) admits a non-degenerate invariant, positive-definite Hermitian form is an important problem in representation theory. In this paper, we study the related notion of signatures. Let q∈C with |q|=1, κ=−1/c with c∈Q<0 with certain additional (mild) restrictions, and n∈Z≥2. We study representations Sλ(q) of Hn(q), the Hecke algebra of type An−1, and representations Mκ(λ) of Hκ, the rational Cherednik algebra of type An−1, which have unique (up to scaling) invariant Hermitian forms (here λ is a partition of n). The signature is the number of elements with positive norm minus the number of elements with negative norm, and we analogously define the signature character in the case that there is a natural grading on the module. We provide formulas for (1) signatures of modules over Hn(q) and (2) signature characters of modules over Hκ. We study the limit c→−∞, in which case the signature character has a simpler form in terms of inversions and descents of permutations in Sn. We provide examples corresponding to some special shapes, and small values of n. Finally, when q=e2πic, we show that the asymptotic signature character of the Hκ-module Mκ(τ) is the signature of the Hn(q)-module Sτ(q).