Smooth representations and Hecke algebras of $p$-adic $\mathrm{GL}_n(\mathcal{D})$

TL;DR

This paper investigates the representation theory of p-adic groups $ ext{GL}_n( ext{D})$, showing that the structure of cuspidal blocks in their categories of smooth representations is independent of the division algebra $ ext{D}$, especially for small n.

Contribution

It demonstrates that the cuspidal blocks in the Bernstein decomposition of $ ext{GL}_n( ext{D})$ representations do not depend on the division algebra $ ext{D}$, using Bushnell-Kutzko and Sécherre-Stevens theories.

Findings

Cuspidal blocks are independent of $ ext{D}$.

For $n=1,2$, the entire representation category is independent of $ ext{D}$.

The results unify the understanding of representation categories across different division algebras.

Abstract

The main question we are going to address in this paper is: How much does the representation theory of the $p$-adic group $\mathrm{GL}_n(\mathcal{D})$ depend on the $p$-adic division algebra $\mathcal{D}$? Let $\mathcal{D}$ be a central division algebra defined over some locally compact non-archimedean local field. Using Bushnell-Kutzko theory of types and S\'echerre-Stevens decomposition of spherical Hecke algebras associated to types, we obtain that the cuspidal blocks in the Bernstein decomposition of the category $\mathcal{R} \left( \mathrm{GL}_n(\mathcal{D}) \right)$ of smooth complex representations of $\mathrm{GL}_n(\mathcal{D})$ do not depend on the $p$-adic division algebra $\mathcal{D}$. In particular, when $n=1$ or $2$, the category $\mathcal{R} \left( \mathrm{GL}_n(\mathcal{D}) \right)$ does not depend on the $p$-adic division algebra $\mathcal{D}$.