On local integrability results for $p$-adic reductive groups

Abstract

We present a short proof, based on local character expansions, of the celebrated theorem of Harish-Chandra about local integrability of complex characters of $p$-adic reductive groups. The proof gives an algebraic incarnation of the local integrability that works for some coefficients different from $\mathbb{C}$, verifies local integrability in cases that appear not covered in the literature, and shows that a character is locally-$L^{\hat{I}}\pm$ for some specified $\hat{I}\pm >1$ as in [GGH23].

8 pages. Comments welcome!