Local character expansion for mod-$\ell$ representations
Cheng-Chiang Tsai
TL;DR
The paper develops a local character expansion for finite-length admissible $\ell$-modular representations of a $p$-adic reductive group $G$ in the regime where $p$ is very large, using multiplicities of degenerate Moy-Prasad types rather than the character itself.It builds a linear-algebra framework with spaces $\operatorname{DM}^{>r}$, $\operatorname{DI}^{>r}$, and a basis indexed by nilpotent orbits to express $\dim_C\operatorname{Hom}_{G_{x\ge s}}(\psi_\phi,\pi)$ as a rational combination of orbital measures $\mu_{\mathcal O}([\phi+\mathfrak g^*_{x>-s}])$, for $s>\operatorname{depth}(\pi)$.The main theorem asserts the existence and uniqueness of coefficients $c_{\mathcal O}(\pi,\psi)\in\mathbb{Q}$ satisfying $\dim_C\operatorname{Hom}_{G_{x\ge s}}(\psi_\phi,\pi)=\sum_{\mathcal O} \mu_{\mathcal O}([\phi+\mathfrak g^*_{x>-s}])\,c_{\mathcal O}(\pi,\psi)$, and recovers the classical Harish-Chandra–Howe expansion when $\operatorname{char}(F)=0$ and $C=\mathbb{C}$. The approach also applies to mod-$\ell$ representations via a companion result, offering a robust framework for understanding local character expansions without direct reference to the character, with potential computational and theoretical implications in $p$-adic representation theory.
Abstract
Let $G$ be a $p$-adic reductive group with $p$ ``very large.'' For any irreducible admissible representation $π$ of $G$ over an algebraically closed field $C$ of characteristic $\not=p$, we define a ``local character expansion'' of $π$ with coefficients $c_{\mathcal{O}}(π)\in\mathbb{Q}$, that does not use the character of $π$ 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.