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 a related notion: that of signatures. We study representations $S^{\lambda}(q)$ of $\mathcal{H}_{n}(q)$, the Hecke algebra of type $A$ ($|q| = 1$), and representations $M_{c}(\lambda)$ of $\mathbb{H}_{c}$, the rational Cherednik algebra of type $A$ ($c \in \mathbb{R}$), which have unique (up to scaling) invariant Hermitian forms (here $\lambda$ 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 $\mathcal{H}_{n}(q)$ and (2) signature characters of modules over $\mathbb{H}_{c}$. We study the limit $c \rightarrow -\infty$, in which case the signature character has a simpler form in terms of inversions and descents of permutations in $S(n)$. We provide examples corresponding to some special shapes, and small values of $n$. Finally, when $q = e^{2 \pi i c}$, we show that the asymptotic signature character of the $\mathbb{H}_{c}$-module $M_{c}(\tau)$ is the signature of the $\mathcal{H}_{n}(q)$-module $S^{\tau}(q)$.