Meng-Kiat Chuah

Double Vogan diagrams and semisimple symmetric spaces

A Vogan diagram is a set of involution and painting on a Dynkin diagram. It selects a real form, or equivalently an involution, from a complex simple Lie algebra. We introduce the double Vogan diagram, which is two sets of Vogan diagrams superimposed on an affine Dynkin diagram. They correspond to pairs of commuting involutions on complex simple Lie algebras, and therefore provide an independent classification of the simple locally symmetric pairs.

Finite order automorphisms on real simple Lie algebras

We add extra data to the affine Dynkin diagrams to classify all the finite order automorphisms on real simple Lie algebras. As applications, we study the extensions of automorphisms on the maximal compact subalgebras and also study the fixed point sets of automorphisms.

Regular principal models of split semisimple Lie groups

Abstract Let G be a semisimple Lie group. Geometric quantization is a machinery which transforms a symplectic G-manifold X to a unitary G-representation . Let C be a Cartan subgroup of G, and L the stabilizer of an element in the Lie algebra of C. Let , where L ss is the commutator subgroup of L, and is the Lie algebra of the centralizer of L in C. When G is split, we perform geometric quantization to G × H-invariant symplectic forms on X. As a result, we construct a regular principal model in the sense that every regular principal series representation of G occurs once in . We also perform symplectic reduction to X and show that “quantization commutes with reduction”.