The goals of this project are to determine novel algebraic structures, classify their representations and study the connection with Dirac cohomology.
By studying the symmetries of deformations of the Dirac operator interesting algebraic structures arise, which do not appear in the undeformed setting. More specifically, we work inside a deformation of the Weyl algebra of linear differential operators with polynomial coefficients, being either a rational Cherednik algebra associated with a reflection group, or an abstract generalization thereof. Within the tensor product with a Clifford algebra, we determine the subalgebra of all elements (anti)commuting with a known subalgebra containing the Dirac operator. This concept of two commuting algebraic structures is reminiscent of reductive dual pairs in Howe duality, of which Schur-Weyl duality and the decomposition of polynomials in terms of spherical harmonics are special cases.
In this project, on the one hand, we will extend the framework of dual pairs and symmetries to deformations of the abstract Dirac elements appearing in Dirac cohomology. On the other hand, for the symmetry algebras obtained in this way, we will examine their action on the space of polynomials and exploit the dual pair structure to identify and classify their representations. By combining these results, we aim to develop a notion of Dirac cohomology for the abstract generalization of the rational Cherednik algebras.