A factorization result for classical and similitude groups

TL;DR

This paper proves a factorization property for elements of classical and similitude groups, showing they can be expressed as products of transformations with specific form-preserving properties, extending previous work and applications.

Contribution

It generalizes existing factorization results to a broader class of groups and applies this to automorphisms of p-adic classical groups, extending known theorems.

Findings

Every element can be written as a product of two transformations with specific properties.

Extended the result of Wonenburger and Vinroot to more groups.

Re-proved and extended a result on automorphisms of p-adic classical groups.

Abstract

For most classical and similitude groups, we show that each element can be written as a product of two transformations that a) preserve or almost preserve the underlying form and b) whose squares are certain scalar maps. This generalizes work of Wonenburger and Vinroot. As an application, we re-prove and slightly extend a well-known result of M{\oe}glin, Vign\'{e}ras and Waldspurger on the existence of automorphisms of $p$-adic classical groups that take each irreducible smooth representations to its dual.