Star operations for affine Hecke algebras

TL;DR

This paper classifies and analyzes star operations on affine Hecke algebras, revealing two main types and exploring their properties and implications for unitary modules.

Contribution

It identifies the only two star operations on graded affine Hecke algebras and connects one to p-adic groups and the other to real groups, with applications to module unitarity.

Findings

Two main star operations classified: $ ext{*}$ and $ullet$.

The $ullet$ operation relates to p-adic groups via Bernstein projectives.

Results on signatures and unitarity of $ullet$-invariant modules.

Abstract

In this paper, we consider the star operations for (graded) affine Hecke algebras which preserve certain natural filtrations. We show that, up to inner conjugation, there are only two such star operations for the graded Hecke algebra: the first, denoted $\star$, corresponds to the usual star operation from reductive $p$-adic groups, and the second, denoted $\bullet$ can be regarded as the analogue of the compact star operation of a real group considered by \cite{ALTV}. We explain how the star operation $\bullet$ appears naturally in the Iwahori-spherical setting of $p$-adic groups via the endomorphism algebras of Bernstein projectives. We also prove certain results about the signature of $\bullet$-invariant forms and, in particular, about $\bullet$-unitary simple modules.