A twisted Hecke algebra, then and now, and a Klein bottle of tempered representations
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866908862497423360 |
|---|---|
| author | Aubert, Anne-Marie Plymen, Roger |
| author_facet | Aubert, Anne-Marie Plymen, Roger |
| contents | 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 $(M^0,σ^0)$ 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 $(M^0, σ^0)$.
We make a full comparison with the classical situation in which $G = \mathrm{SL}_8$ and $(M,σ)$ is a cuspidal pair for $G$. The supercuspidal representation $σ$ 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 $(M,σ)$ and let $\mathfrak{s}^0$ be the point in the Bernstein spectrum $\mathfrak{B} G^0$ determined by $(M^0, σ^0)$. 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 $\mathrm{SL}_8$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_03027 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A twisted Hecke algebra, then and now, and a Klein bottle of tempered representations Aubert, Anne-Marie Plymen, Roger Representation Theory Number Theory 22E50 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 $(M^0,σ^0)$ 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 $(M^0, σ^0)$. We make a full comparison with the classical situation in which $G = \mathrm{SL}_8$ and $(M,σ)$ is a cuspidal pair for $G$. The supercuspidal representation $σ$ 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 $(M,σ)$ and let $\mathfrak{s}^0$ be the point in the Bernstein spectrum $\mathfrak{B} G^0$ determined by $(M^0, σ^0)$. 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 $\mathrm{SL}_8$. |
| title | A twisted Hecke algebra, then and now, and a Klein bottle of tempered representations |
| topic | Representation Theory Number Theory 22E50 |
| url | https://arxiv.org/abs/2603.03027 |