Jing Song Huang

Dirac cohomology, unitary representations and a proof of a conjecture of Vogan

U radu je dokazana slutnja D. Vogana, koja odredjuje infinitesimalni karakter ireducibilnog (g, K)-modula koji ima Diracovu kohomologiju razlicitu od nula. Infinitesimalni karakter je dan kao rho-translat najvece tezine K-tipa u Diracovoj kohomologiji. Diracova kohomologija se definira kao ker D / (im D \cap ker D), gdje D oznacava Diracov operator na zadanom (g, K)-modulu tenzoriranom sa spin modulom.

Dirac operators and Lie algebra cohomology

Dirac cohomology is a new tool to study unitary and admissible represen- tations of semisimple Lie groups. It was introduced by Vogan and further studied by Kostant and ourselves (V2), (HP1), (K4). The aim of this paper is to study the Dirac cohomology for the Kostant cubic Dirac operator and its relation to Lie algebra coho- mology. We show that the Dirac cohomology coincides with the corresponding nilpotent Lie algebra cohomology in many cases, while in general it has better algebraic behavior and it is more accessible for calculation.

Functions on the model orbit in E8

First published in Representation Theory in Vol.2,1998. Published by the American Mathematical Society.

Dirac cohomology, K-characters and branching laws

Inspired by work of Enright and Willenbring, we prove a generalized Littlewood’s restriction formula in terms of Dirac cohomology. Our approach is to use a character formula of irreducible unitary lowest weight modules instead of the Bernstein-Gelfand-Gelfand resolution, and the proof is much simpler. We also show that our branching formula is equivalent to the formula of Enright and Willenbring in terms of nilpotent Lie algebra cohomology. This follows from the close relationship between the Dirac cohomology and the corresponding nilpotent Lie algebra cohomology for unitary representations of semisimple Lie groups of Hermitian type, which was established by Huang, Pandzic, and Renard.

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.

Dirac cohomology of cohomologically induced modules for reductive Lie groups

We extend the setting and a proof of the Vogans conjecture on Dirac cohomology to a possibly disconnected real reductive Lie group

Weyl’s construction and tensor power decomposition for ₂

G

Dimensions of spaces of generalized spherical functions

in the Harish-Chandra class. We show that the Dirac cohomology of cohomologically induced module

Dirac cohomology, elliptic representations and endoscopy

{\mathcal L}_S(Z)

Dirac Cohomology and Generalization of Classical Branching Rules

is completely determined by the Dirac cohomology of the inducing module