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...[ Read more ]

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).