Local character expansion for mod-$\ell$ representations

TL;DR

This paper introduces a local character expansion for mod-$ ext{ extlbrackdbl}\ell$ representations of p-adic groups, using Moy-Prasad types, providing a new perspective without directly employing the character of the representation.

Contribution

It defines a local character expansion for mod-$ ext{ extlbrackdbl}\ell$ representations based on Moy-Prasad types, offering a novel approach distinct from previous methods.

Findings

Defines local character expansion coefficients for mod-$ ext{ extlbrackdbl}\ell$ representations.

Uses multiplicities of degenerate Moy-Prasad types instead of characters.

Establishes existence of local character expansion for these representations.

Abstract

Let $G$ be a $p$-adic reductive group with $p$ ``very large.'' For any irreducible admissible representation $\pi$ of $G$ over an algebraically closed field $C$ of characteristic $\not=p$, we define a ``local character expansion'' of $\pi$ with coefficients $c_{\mathcal{O}}(\pi)\in\mathbb{Q}$, that does not use the character of $\pi$ directly but instead use the multiplicities of degenerate Moy-Prasad types. Note that the existence of local character expansion for mod-$\ell$ representations is shown by another paper of the author using a different and quicker method.