Generalizations Of The Cartan And Iwasawa Decompositions For SL$_2(k)$

TL;DR

This paper generalizes the Cartan and Iwasawa decompositions for algebraic groups over arbitrary fields and involutions, extending their applicability beyond real reductive Lie groups.

Contribution

It introduces a broad framework for decompositions of algebraic groups over any field with arbitrary involutions, generalizing classical decompositions.

Findings

Extended decompositions to algebraic groups over arbitrary fields.

Applicable to groups with general involutions.

Provides a unified approach to classical and new decompositions.

Abstract

The Cartan and Iwasawa decompositions of real reductive Lie groups play a fundamental role in the representation theory of the groups and their corresponding symmetric spaces. These decompositions are defined by an involution with a compact fixed-point group, called a Cartan involution. For an arbitrary involution, one can consider similar decompositions. We offer a generalization of the Cartan and Iwasawa decompositions for algebraic groups defined over an arbitrary field $k$ and a general involution.