Dirac cohomology, the projective supermodules of the symmetric group and the Vogan morphism

In this paper we will 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 2015 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 derived the branching graph using Dirac theory and combinatorics relating to the cohomology of Borel varieties $\mathcal{B}_e$ of $\mathfrak{g}$ and were able to use Dirac cohomology to construct an explicit model for the projective representations. We also described Vogan's morphism for Hecke algebras in type A using spectrum data of the Jucys-Murphy elements

Dirac cohomology and simple modules of the Dunkl-Opdam subalgebra via inherited Drinfeld properties

In this paper we define a new presentation for the Dunkl-Opdam subalgebra of the rational Cherednik algebra. This shows that the Dunkl-Opdam subalgebra is a Drinfeld algebra. We use this fact to define Dirac cohomology for the DO subalgebra. We also formalise generalised graded Hecke algebras and define a Langlands classification to generalised graded Hecke algebras

Two families of Dirac-like operators for Drinfeld's Hecke algebra

In this paper we define two generalisations of Dirac operators for Drinfeld's Hecke algebra. One generalisation, Parthasarathy operators inherit the notion of the Dirac inequality. The second generalisation, Vogan operators, inherit Dirac cohomology; if an operator has non-zero cohomology then it relates the infinitesimal character with a character of the a group. We prove properties about these operators and give a family of operators in each class

Dirac operators for the Dunkl Angular Momentum Algebra

We define a family of Dirac operators for the Dunkl angular momentum algebra depending on certain central elements of the group algebra of the Pin cover of the Weyl group inherent to the rational Cherednik algebra. We prove an analogue of Vogan's conjecture for this family of operators and use this to show that the Dirac cohomology, when non-zero, determines the central character of representations of the angular momentum algebra. Furthermore, interpreting this algebra in the framework of (deformed) Howe dualities, we show that the natural Dirac element we define yields, up to scalars, a square root of the angular part of the Calogero-Moser Hamiltonian

The centre of the Dunkl total angular momentum algebra

For a finite dimensional representation $V$ of a finite reflection group $W$, we consider the rational Cherednik algebra $\mathsf{H}_{t,c}(V,W)$ associated with $(V,W)$ at the parameters $t\neq 0$ and $c$. The Dunkl total angular momentum algebra $O_{t,c}(V,W)$ arises as the centraliser algebra of the Lie superalgebra $\mathfrak{osp}(1|2)$ containing a Dunkl deformation of the Dirac operator, inside the tensor product of $\mathsf{H}_{t,c}(V,W)$ and the Clifford algebra generated by $V$. We show that, for every value of the parameter $c$, the centre of $O_{t,c}(V,W)$ is isomorphic to a univariate polynomial ring. Notably, the generator of the centre changes depending on whether or not $(-1)_V$ is an element of the group $W$. Using this description of the centre, and using the projection of the pseudo scalar from the Clifford algebra into $O_{t,c}(V,W)$, we establish results analogous to ``Vogan's conjecture'' for a family of operators depending on suitable elements of the double cover $\tilde{W}$.Comment: 27 page

Clifford algebras, symmetric spaces and cohomology rings of Grassmannians

We study various kinds of Grassmannians or Lagrangian Grassmannians over $\mathbb{R}$, $\mathbb{C}$ or $\mathbb{H}$, all of which can be expressed as $\mathbb{G}/\mathbb{P}$ where $\mathbb{G}$ is a classical group and $\mathbb{P}$ is a parabolic subgroup of $\mathbb{G}$ with abelian unipotent radical. The same Grassmannians can also be realized as (classical) compact symmetric spaces $G/K$. We give explicit generators and relations for the de Rham cohomology rings of $\mathbb{G}/\mathbb{P}\cong G/K$. At the same time we describe certain filtered deformations of these rings, related to Clifford algebras and spin modules. While the cohomology rings are of our primary interest, the filtered setting of $K$-invariants in the Clifford algebra actually provides a more conceptual framework for the results we obtain

Variants of Schur-Weyl 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 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 &otimes; V &otimes;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.</p