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# A factorization result for classical and similitude groups

Published:2017-11-01

• Alan Roche,
Dept. of Mathematics, University of Oklahoma, Norman, OK, USA
• C. Ryan Vinroot,
Dept. of Mathematics, College of William and Mary, Williamsburg, VA, USA
 Format: LaTeX MathJax PDF

## 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 Mœglin, Vignéras and Waldspurger on the existence of automorphisms of $p$-adic classical groups that take each irreducible smooth representation to its dual.
 Keywords: classical group, similitude group, involution, $p$-adic group, dual of representation
 MSC Classifications: 20G15 - Linear algebraic groups over arbitrary fields 22E50 - Representations of Lie and linear algebraic groups over local fields [See also 20G05]

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