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Title: Variants of Schur-Weyl duality and Dirac cohomology
Author: Calvert, Kieran
ISNI:       0000 0004 7971 7185
Awarding Body: University of Oxford
Current Institution: University of Oxford
Date of Award: 2019
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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 Sn using Dirac cohomology and the branching graph for the irreducible genuine projective representations of Sn. In [15] Ciubotaru and He, using the extended Dirac index, showed that the characters of the projective representations of Sn 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 Be 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 Jucys-Murphy elements. Chapter 2: Dirac cohomology of the Dunkl-Opdam subalgebra We define a new presentation for the Dunkl-Opdam subalgebra of the rational Cherednik algebra. This presentation uncovers the Dunkl-Opdam subalgebra as a Drinfeld algebra. The Dunkl-Opdam subalgebra is the first natural occurrence of a non-faithful 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 Dunkl-Opdam 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 EndK(X ⊗ V ⊗k). We study functors Fμ,k from the category of admissible O(p; q) or Sp2n(R) modules to representations of the type B/C affine Brauer algebra. Furthermore, these functors take non-spherical principal series modules to principal series modules for the graded Hecke algebra of type Dk, Cn-k or Bn-k.
Supervisor: Ciubotaru, Dan Sponsor: Engineering and Physical Sciences Research Council
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: Lie Groups ; Lie algebras ; Representations of groups