THESIS
2015
Abstract
This thesis consists of two parts. In the first part, we recall the definition of Dirac cohomology for category O
p and present a new proof of the character formulae of irreducible highest weight modules in category O
p in terms of Dirac cohomology and characters of parabolic Verma modules. In particular,
we obtain the character formulae of irreducible highest weight representations
for symmetric pair (U(m, n), U(m) × U(n)) in terms of Dirac cohomology and Littlewood-Richardson coefficients. In the second part, we systematically investigate the Dirac cohomology for basic Lie superalgebras as well as the relationship
between the central character of a representation of a basic Lie superalgebra and the infinitesimal characters appearing in its Dirac cohomology. Furthermore, we
obtain t...[
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This thesis consists of two parts. In the first part, we recall the definition of Dirac cohomology for category O
p and present a new proof of the character formulae of irreducible highest weight modules in category O
p in terms of Dirac cohomology and characters of parabolic Verma modules. In particular,
we obtain the character formulae of irreducible highest weight representations
for symmetric pair (U(m, n), U(m) × U(n)) in terms of Dirac cohomology and Littlewood-Richardson coefficients. In the second part, we systematically investigate the Dirac cohomology for basic Lie superalgebras as well as the relationship
between the central character of a representation of a basic Lie superalgebra and the infinitesimal characters appearing in its Dirac cohomology. Furthermore, we
obtain the Dirac cohomology of typical representations for gl(m∣n) and the Dirac cohomology of Kac modules for gl(2∣1).
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