Variants of Schur-Weyl duality and Dirac cohomology

Thesis

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 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 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 non-spherical principal series modules to principal series modules for the graded Hecke algebra of type D_k, C_n-k or B_n-k.

Authors: Calvert, K

• DOI: 10.5287/ora-mzxb5kp0w
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Subjects: Lie algebras; Representations of groups; Lie Groups