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

We define the ``lifted character’’ of mod-$\mathrm{\ell }$ representations of $p$-adic reductive groups where $\mathrm{\ell }≠p$, on compact elements with pro-orders not divisible by $\mathrm{\ell }$. 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.