Local character expansion for mod-$\ell$ representations

Abstract

Let $G$ be a $p$-adic reductive group with $p$ ``very large.'' For any irreducible admissible representation $\ddot{I}€$ of $G$ over an algebraically closed field $C$ of characteristic $\ne p$, we define a ``local character expansion'' of $\ddot{I}€$ with coefficients $c_{\mathcal{O}}(\ddot{I}€)\in \mathbb{Q}$, that does not use the character of $\ddot{I}€$ 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.

The result of this paper is mostly superseded by another paper of myself uploaded on the same day. The method presented here may still be of interest