researchseminars.org

David Vogan (MIT)
2022-02-03 15:30 - 17:00 UTC | https://researchseminars.org/talk/atlas/5/
Title: Understanding $K$: how atlas understands Cartan's theory of maximal compact subgroups (https://researchseminars.org/talk/atlas/5/)
by David Vogan (MIT) as part of Real reductive groups/atlas

Abstract
To study an algebraic variety $X$ over a finite field ${\mathbb F}_q$, we use the Frobenius map $F$, an algebraic map from $X$ to $X$ whose set of fixed points is precisely $X({\mathbb F}_q)$. The power of this tool stems from the fact that $F$ (unlike the Galois group action) is algebraic.

To study a reductive algebraic group $G$ over ${\mathbb R}$, Cartan introduced the Cartan involution $\theta$, an algebraic automorphism of $G$ having order 2. The group of fixed points of $\theta$ is an algebraic subgroup $K$ of $G$. It is not true (as in the finite field case) that $K$ is equal to $G({\mathbb R})$, but Cartan showed how to get a great deal of information about $G({\mathbb R})$ from $K$. This idea is at the heart of almost all that ${\tt atlas}$ does.
