Geometric and algebraic parameterizations for dirac cohomology of simple modules in Ορ and their applications

Abstract

In this thesis, we show that the Dirac cohomology H_D(L(λ)) of a simple highest weight module L(λ) in O^p can be parameterized by a specific set of weights: a subset W_I(λ) of the orbit of the Weyl group W acting on λ+ρ. As an application, we show that any simple module in O^p is determined up to isomorphism by its Dirac cohomology. We describe four parameterizations of H_D(L(λ)) which are related to the Verma-BGG Theorem for regular λ. As an application, for Verma modules with regular infinitesimal character, we obtain an extended version of the Verma-BGG Theorem. We also investigate Dirac cohomology of Kostant modules. Using Dirac cohomology, we give new proofs of the simplicity criterion for parabolic Verma modules with I = ∅ or with regular infinitesimal character and describe a new simplicity criterion for parabolic Verma modules with regular infinitesimal character.

Date of Award	2019
Original language	English
Awarding Institution	The Hong Kong University of Science and Technology