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

TL;DR

This paper provides a new algebraic proof of Harish-Chandra's theorem on local integrability of complex characters in p-adic reductive groups, extending results to other coefficients and specific L^α conditions.

Contribution

It introduces an algebraic approach to local integrability, covering cases not previously addressed and demonstrating local L^α integrability for certain coefficients.

Findings

Proof based on local character expansions simplifies existing arguments.

Verifies local integrability for coefficients beyond complex numbers.

Shows characters are locally L^α for some α > 1, extending known results.

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-$L^{\alpha}$ for some specified $\alpha>1$ as in [GGH23].