Alladi Ramakrishnan Hall

Compact forms of symmetric spaces

M S Raghunathan

IIT Bombay

We will give a proof of the theorem of A Borel asserting that
every symmetric space admits a compact quotient by a discrete group of
isometries; and Borel deduces it from the existence of a $\mathbb{Q}$-structure on
a semisimple Lie algebra admitting a Cartan involution defined over $\mathbb{Q}$.I
will give a proof of a somewhat stronger result: Let $G$ be a semisimple
Lie algebra and $F$ a finite (abelian) group of automorphiosms of $G$
consisting entirely of elements of order 2 and containing a Cartan
involution. Then $G$ admits a $\mathbb{Q}$ structure in for which every $f \in F$
is $\mathbb{Q}$-rational