#### 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

Done