Representations of $GL_n(D)$ near the identity

TL;DR

This paper characterizes the restriction of irreducible smooth representations of $GL_n(D)$ near the identity, expressing them as virtual sums of induced representations and determining fixed point dimensions via a polynomial for large congruence subgroups.

Contribution

It introduces a new description of representations of $GL_n(D)$ near the identity using virtual induced representations and establishes a polynomial formula for fixed point dimensions under congruence subgroups.

Findings

Restriction of representations matches a virtual sum of induced representations.

Fixed point dimensions are given by a degree polynomial independent of the subgroup choice.

The polynomial degree relates to the representation's depth and structure.

Abstract

For a central division algebra $D$ of dimension $d^2$ over a finite extension $F$ of $\mathbb Q_p$ or of $\mathbb F_p((t))$, a field $R$ of characteristic prime to $p$, and an irreducible smooth $R$-representation $\pi$ of $G=GL_n(D)$, we show that for small enough compact open pro-$p$ subgroup $K$ of $G$, the restriction of $\pi$ to $K$ is the same as that of a virtual representation $\sum c_\pi(\lambda) Ind_{P_\lambda}^G 1$, where the sum is over partitions $\lambda$ of $n$ and $P_\lambda$ a parabolic subgroup of $G$ associated to $\lambda$. When $K$ is a Moy-Prasad subgroup of $G$ we determine from the $c_\pi(\lambda)$ a polynomial $P_{\pi,K}$ of degree $d(\pi)$ independent of the choice of $K$, such that for large enough integers $j$ the dimension of the points of $\pi$ fixed under the congruence subgroup $K_j$ of $K$ is $P_{\pi,K}(q^j)$ where $q$ is the cardinality of the residue field of $D$.