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SUMMARY:David Vogan (MIT)
DTSTART:20220203T153000Z
DTEND:20220203T170000Z
DTSTAMP:20260422T145929Z
UID:atlas/5
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/atlas/5/">Un
 derstanding $K$: how atlas understands Cartan's theory of maximal compact 
 subgroups</a>\nby David Vogan (MIT) as part of Real reductive groups/atlas
 \n\n\nAbstract\nTo 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.\n\nTo study a reductive algebraic group $G$ over ${\\mathb
 b R}$\, Cartan introduced the Cartan involution $\\theta$\, an algebraic a
 utomorphism 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 fiel
 d 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 ide
 a is at the heart of almost all that ${\\tt atlas}$ does.\n
LOCATION:https://researchseminars.org/talk/atlas/5/
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