#### Alladi Ramakrishnan Hall

#### WEYL ORBITS OF π-SYSTEMS IN KAC-MOODY ALGEBRAS

#### Krishanu Roy

##### IMSc

*Given a symmetrizable Kac-Moody algebra g, a π-system of g *

is a subset of

its real roots such that pairwise differences are not roots. When g is

finite dimensional,

Dynkin showed that linearly independent π–systems arise precisely as

simple systems of reg-

ular semisimple subalgebras of g. He also computed the number of Weyl

group orbits for

each π-system in g. We prove if any symmetrizable Kac-Moody algebra g

admits a linearly

independent π-system of affine type, then the number of Weyl orbits of

π-systems of this type

is necessarily infinite. We also prove if g is simply-laced and the

π-system is (simply-laced) of

overextended type, then the number of Weyl group orbits is finite, and

can in fact be obtained

as a sum of the number of orbits of certain finite type π-systems inside finite root systems. This is joint work with L.Carbone, K N Raghavan, B.Ransingh and S.Viswanath.

Done