Local character expansion for mod-$\mathrm{\ell }$ representations

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 $\pi$ 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-$\mathrm{\ell }$ representations is shown by another paper of the author using a different and quicker method.