Title:

Variants of SchurWeyl duality and Dirac cohomology

This thesis is divided into the following three parts. Chapter 1: Realising the projective representations of Sn We derive an explicit description of the genuine projective representations of the symmetric group S_{n} using Dirac cohomology and the branching graph for the irreducible genuine projective representations of S_{n}. In [15] Ciubotaru and He, using the extended Dirac index, showed that the characters of the projective representations of S_{n} are related to the characters of elliptic graded modules. We derive the branching graph using Dirac theory and combinatorics relating to the cohomology of Borel varieties B_{e} of g. We use Dirac cohomology to construct an explicit model for the projective representations. We also describe Vogan's morphism for Hecke algebras in type A using spectrum data of the JucysMurphy elements. Chapter 2: Dirac cohomology of the DunklOpdam subalgebra We define a new presentation for the DunklOpdam subalgebra of the rational Cherednik algebra. This presentation uncovers the DunklOpdam subalgebra as a Drinfeld algebra. The DunklOpdam subalgebra is the first natural occurrence of a nonfaithful Drinfeld algebra. We use this fact to define Dirac cohomology for the DO subalgebra. We formalise generalised graded Hecke algebras and extend a Langlands classification to generalised graded Hecke algebras. We then describe an equivalence between the irreducibles of the DunklOpdam subalgebra and direct sums of graded Hecke algebras of type A. This equivalence commutes with taking Dirac cohomology. Chapter 3: Functors relating nonspherical principal series We define an extension of the affine Brauer algebra called the type B/C affine Brauer algebra. This new algebra contains the hyperoctahedral group, and it naturally acts on End_{K}(X ⊗ V ^{⊗k}). We study functors F_{μ,k} from the category of admissible O(p; q) or Sp_{2n}(R) modules to representations of the type B/C affine Brauer algebra. Furthermore, these functors take nonspherical principal series modules to principal series modules for the graded Hecke algebra of type D_{k}, C_{nk} or B_{nk}.
