Dualizing involutions for classical and similitude groups over local non-archimedean fields

TL;DR

This paper provides an elementary proof demonstrating the existence of automorphisms in many p-adic classical and similitude groups that map each irreducible smooth representation to its dual, extending previous results.

Contribution

It offers a new, elementary proof of dualizing involutions for p-adic classical and similitude groups, building on prior work by M{\

Findings

Automorphisms exist that take irreducible smooth representations to their duals.

The proof applies to both classical and similitude groups.

Extends known results to a broader class of p-adic groups.

Abstract

Building on ideas of Tupan, we give an elementary proof of a result of M{\oe}glin, Vign\'{e}ras and Waldspurger on the existence of automorphisms of many $p$-adic classical groups that take each irreducible smooth representations to its dual. Our proof also applies to the corresponding similitude groups.