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.