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

TL;DR

The paper develops a theory of lifted mod-$\ell$ characters for $p$-adic reductive groups, defining $\Theta^{\dag}_{\pi}$ on $G_{\ell'}$ and proving a Harish-Chandra--Howe local character expansion with coefficients in $\mathbb{Q}$, later extended to positive characteristic via Waldspurger--DeBacker homogeneity. By base-changing to $K(C)$ and using degenerate characters, the author connects the mod-$\ell$ expansion to the usual complex-analytic framework and derives polynomial growth of parahoric-fixed spaces, tied to nilpotent orbits through the coefficients $c_{\mathcal{O}}(\pi,\psi)$. The work further transfers Var14/MW87 techniques to the mod-$\ell$ setting, establishing analogues for degenerate Whittaker models: nonvanishing models correspond to leading nilpotent orbits and, when the orbit is exactly $\operatorname{Ad}(G)Y$, the coefficient equals the model dimension. These results unify and extend classical character theory to mod-$\ell$ representations, with implications for endoscopy and harmonic analysis on $G_{\ell'}$.

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.

Theorems & Definitions (17)

• Definition 1
• Remark 2
• Theorem 3
• proof
• Lemma 4
• Corollary 5
• Corollary 6
• proof
• Remark 7
• Remark 8