Local characters of mod-$\ell$ representations of a $p$-adic reductive group

TL;DR

This paper introduces a new notion of lifted characters for mod-$ell$ representations of p-adic groups, extending classical local character expansion results and validating existing models in this context.

Contribution

It defines lifted characters for mod-$ell$ representations and generalizes classical local character expansion results to these characters.

Findings

Lifted characters are well-defined on certain compact elements.

Classical local character expansion results extend to lifted characters.

Degenerate Whittaker models' results are validated for these characters.

Abstract

We define the ``lifted character'' of mod-$\ell$ representations of $p$-adic reductive groups where $\ell\not=p$, on compact elements with pro-orders not divisible by $\ell$. We generalize the local character expansion results of Howe, Harish-Chandra and DeBacker to such lifted characters. We show that the result of Moeglin-Waldspurger and Varma on degenerate Whittaker models is valid for the character expansion.