A twisted Hecke algebra, then and now, and a Klein bottle of tempered representations

TL;DR

This paper explicitly describes a twisted Hecke algebra associated with a depth-zero cuspidal pair in a special linear group over a non-archimedean field, revealing a Klein bottle structure in the spectrum and comparing it to classical cases.

Contribution

It provides an explicit algebraic and geometric description of the Bernstein block and spectrum for a specific twisted Levi subgroup, including the Klein bottle model of the unitary principal series.

Findings

The twisted Hecke algebra is explicitly presented with generators and relations.

All simple modules of the algebra are 2-dimensional.

The primitive spectrum forms a complex algebraic variety with a Klein bottle as the maximal compact form.

Abstract

Let $F$ be a non-archimedean local field such that $4|q-1$, with $q$ the order of the residue field of $F$, and let $(M^0,\sigma^0)$ be the depth-zero cuspidal pair for the twisted Levi subgroup $G^0$ of $\mathrm{SL}_8$ arising from quadratic and quartic field extensions, as defined in the recent article by Adler-Fintzen-Ohara [AFO]. Then the corresponding Bernstein block is described by a twisted Hecke algebra $\mathcal{H}^0$. We describe $\mathcal{H}^0$ explicitly as a noncommutative $\mathbb{C}$-algebra with generators and relations. We describe explicitly the simple modules of $\mathcal{H}^0$. All the simple modules are $2$-dimensional. The primitive spectrum of $\mathcal{H}^0$ is then an explicit complex algebraic variety $\mathfrak{X}$. The maximal compact real form of $\mathfrak{X}$ is homeomorphic to a Klein bottle. This Klein bottle is a model of the unitary principal series of $G^0$ attached to the cuspidal pair $(M^0, \sigma^0)$. We make a full comparison with the classical situation in which $G = \mathrm{SL}_8$ and $(M,\sigma)$ is a cuspidal pair for $G$. The supercuspidal representation $\sigma$ is constructed from the same quadratic and quartic extensions of $F$. Let $\mathfrak{s}$ be the point in the Bernstein spectrum $\mathfrak{B} G$ determined by $(M,\sigma)$ and let $\mathfrak{s}^0$ be the point in the Bernstein spectrum $\mathfrak{B} G^0$ determined by $(M^0, \sigma^0)$. We compare the two points $\mathfrak{s}$ and $\mathfrak{s}^0$ and show explicitly that the corresponding Bernstein varieties are isomorphic. In that case, the Klein bottle re-appears, this time floating in the tempered dual of $\mathrm{SL}_8$.