Signatures of representations of Hecke algebras and rational Cherednik algebras

摘要

Determining whether an irreducible representation of a group (or $*$-algebra)admits a non-degenerate invariant, positive-definite Hermitian form is animportant problem in representation theory. In this paper, we study a relatednotion: that of signatures. We study representations $S^{lambda}(q)$ of$mathcal{H}_{n}(q)$, the Hecke algebra of type $A$ ($|q| = 1$), andrepresentations $M_{c}(lambda)$ of $mathbb{H}_{c}$, the rational Cherednikalgebra of type $A$ ($c in mathbb{R}$), which have unique (up to scaling)invariant Hermitian forms (here $lambda$ is a partition of $n$). The signatureis the number of elements with positive norm minus the number of elements withnegative norm, and we analogously define the signature character in the casethat 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 ofmodules over $mathbb{H}_{c}$. We study the limit $c ightarrow -infty$, inwhich case the signature character has a simpler form in terms of inversionsand descents of permutations in $S(n)$. We provide examples corresponding tosome special shapes, and small values of $n$. Finally, when $q = e^{2 pi ic}$, we show that the asymptotic signature character of the$mathbb{H}_{c}$-module $M_{c}(au)$ is the signature of the$mathcal{H}_{n}(q)$-module $S^{au}(q)$.