Talks
A Remark on the Langlands Correspondence for Tori
Abstract: For an algebraic torus defined over a local (or global) field F, Langlands established a natural homomorphism from the group of continuous cohomology classes of the Weil group, valued in the dual torus, onto the space of complex characters of the rational points of the torus (or automorphic characters in the global case). In this talk, based on a current paper co-authored with E. Opdam, I will explain topological aspects of Langlands' correspondence. We show that if we topologize the spaces of continuous homomorphisms and continuous cochains using the compact-open topology, Langlands's map becomes a homomorphism of abelian complex Lie groups. Moreover, we demonstrate that, in both the local and global settings, the subset of unramified characters is the identity component of the relevant space of characters. .
A Quantized Metaplectic Howe Duality in Rank One
Abstract: Dirac-type operators and Howe dualities provide a uniform approach to decomposition problems and symmetry algebras in the orthogonal and symplectic settings. Quantizing these structures is subtle, often convention-dependent, and in the orthogonal Dirac setting competing frameworks do not currently agree. In this talk I focus on the symplectic (metaplectic) Howe duality in the first nontrivial case, rank one, and present an explicit and computable quantization, based on joint work with M. Brito. The main outcome is a clean quantum duality in which both sides are Drinfeld–Jimbo sl(2)-type quantum groups (with different deformation parameters), realized via Hayashi’s deformed Weyl algebra. I will outline the resulting quantum analogues of the classical realization, Fischer-type structure, monogenics, and first-order symmetries, and conclude with the main obstacles to extending the construction beyond rank one.
Dirac operators for the Dunkl Angular Momentum Algebra
Abstract: 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. This is a joint work with K. Calvert (Lancaster University).
Symplectic Dirac Operators for Lie Algebras and graded Hecke algebras
Abstract: We define a pair of symplectic Dirac operators in an algebraic setting motivated by the analogy with the algebraic theory of orthogonal Dirac operators. We compute and study the commutator of these two elements in the context quadratic Lie algebras and graded affine Hecke algebras. This commutator can be seen as an analogue of Parthasarathy's formula for the square of orthogonal Dirac operators. This is a joint work with D. Ciubotaru and P. Meyer.
On the computational aspects of the spherical unramified automorphic spectrum
Abstract: In a joint work with E. Opdam and V. Heiermann, we studied the residual automorphic spectrum obtained from residues of the spherical Eisenstein series. In this talk, I will report on this project with emphasis on the computer-assisted part.