Representations and Lie Theory Seminar

Schedule of talks:

Abstracts

January 25: Associated to a pair of algebras quantizing the same graded Poisson algebra there is a category of Harish-Chandra bimodules. These have been studied with some detail in the context of universal enveloping algebras, finite W-algebras and hypertoric enveloping algebras, among others. I will introduce this concept in the setting of rational Cherednik algebras, with an emphasis on the relationship between Harish-Chandra bimodules and category O, which can be more clearly seen in type A.

February 15: In the theory of representation of p-adic groups, the local converse problems ask if one can characterize a generic irreducible representation \pi of G(F) by the gamma factors of its various twists with GL_k(F), where G is a reductive group, F is a p-adic field, and k varies in a set depends on G. In this talk, we will sketch a proof of a local converse theorem for the unitary group U(2,2).

March 29: In representation stability we consider sequences $(V_n)$ of representations of groups $G_n$ (e.g. $G_n = Gl(n)$ or $S_n$) and look at possible stabilization phenomena. In recent years this area has become very active due to articles by Church-Ellenberg-Farb who realized (for $G_n = S_n$) that one should pack the information of this entire sequence into one object (an $FI$-module) and applied this succesfully to examples coming from topology. Closely related work by Sam-Snowden showed that an $FI$-module is nothing else but a module over a certain {\it twisted commutative algebra} and related the category of $FI$-modules to $Rep S_{\infty}$, the category of algebraic representations of the infinite symmetric group $S_{\infty}$.
A different approach is due to Deligne who defined interpolating categories $\underline{Rep} S_t$, $t \in \mathbb{C}$, interpolating the usual representation categories of the symmetric group. These categories also capture stabilizing phenomena. I will give a (superficial) overwiew of the subject and explain how Sam-Snowden's $Rep S_{\infty}$ is related to Deligne's $\underline{Rep} S_t$.

April 12: One of the basic results in the theory of simple Lie algebras is the classification of simple finite-dimensional modules in terms of their highest weights, and more generally of the collection of simple objects in category O. This classification can be reinterpreted in terms of the "B-algebra" of the universal enveloping algebra, an algebra which is isomorphic to the (equivariant) cohomology ring of the flag variety of the Langlands dual group.
We will discuss a generalization of this story -- through several examples -- studying the highest weight theory of algebras called truncated shifted Yangians. In this case, the B-algebra is conjecturally isomorphic to the equivariant cohomology ring of a Nakajima quiver variety. This allows for a combinatorial description of the set of highest weights. Via a conjecture of Hikita, this fits into an even more general framework related to symplectic duality.

April 19: Parabolic induction has played a crucial role in the classification of reductive groups over local fields. The techniques developed by Knapp-Stein, Harish-Chandra, and others, for reductive Lie groups was extended to the setting of p-adic groups by Silberger. The Langlands-Shahidi method shows that understanding this aspect of the harmonic analysis on these groups has deep arithmetic consequences, particularly in terms of understanding local L-functions. The theory of the Knapp-Stein R-group, gives a combinatorial algorithm for understanding the structure of induced representations, and these R-groups (and their construction on the dual side by Arthur et al) have played a key role in trace formula methods. We’ll give an overview of this program, including the known results for quasi-split groups. We’ll conclude by describing our joint work with Choiy on inner forms, as well as developing work with ban and Choiy on Spin groups.

April 26: A quasi-Coxeter category is a braided tensor category which carries an action of a generalised braid group B_W on the tensor powers of its objects. The data which defines the action of B_W is similar in flavour to the associativity constraints in a monoidal category, but is related to the coherence of a family of fiber functors on C. I will outline how to construct such a structure on integrable, category O representations of a symmetrisable Kac-Moody algebra g, in a way that incorporates the monodromy of the KZ and Casimir connections of g. The rigidity of this structure implies in particular that the monodromy of the latter connection is given by the quantum Weyl group operators of the quantum group U_h(g). This is joint work with Andrea Appel.

June 7: The classical Hodge identities are relations between certain natural differential operators acting on the sections of the exterior algebra bundle over a Kähler manifold. There are analogous relations for riemannian and hyper-Kähler manifolds. In this talk we will show how one can obtain integrated forms of these identities by a single uniform abstract procedure. The new identities are functorial in that they define a multiplicative functor from the category of manifolds with one of the three geometries to the category of representations of an algebraic structure depending only on the type of geometry.