THESIS
2019
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...[
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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.
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