Rationality properties of complex representations of reductive p-adic groups

TL;DR

This paper investigates the rationality properties of complex representations of reductive p-adic groups, establishing conditions under which these representations can be realized over algebraic closures of Q and analyzing Galois actions.

Contribution

It characterizes when elliptic and square-integrable representations can be realized over Qbar and examines the Galois stability of these representation sets.

Findings

Elliptic G-representations can be realized over Qbar iff their central character takes values in Qbar.

The sets of essentially square-integrable and elliptic representations are stable under Gal(C/Q).

The paper provides criteria linking representation rationality to central character values.

Abstract

For a reductive group G over a non-archimedean local field, we compare smooth representations over C with smooth representations over Qbar (an algebraic closure of Q). We show that an elliptic G-representation (in the sense of Arthur) can be realized over Qbar if and only if its central character takes values in Qbar. That applies in particular to all essentially square-integrable G-representations. We also study the action of the automorphism group of C/Q on complex G-representations. We prove that the sets of essentially square-integrable representations and of elliptic representations are stable under Gal(C/Q).