Enhanced Langlands parameters for a p-adic group G are pairs formed by a Langlands parameter for G and an irreducible character of a certain component group attached to the parameter. We will first introduce a notion

of cuspidality for these pairs. The cuspidal pairs are expected to correspond to the supercuspidal irreducible representations of G via the local Langlands correspondence.

We will next describe a construction of a cuspidal support map for enhanced Langlands parameters, the key tool of which is an extension to disconnected complex Lie groups of the generalized Springer correspondence due to Lusztig.

Finally, we will use this map to decompose the set of enhanced Langlands parameters into Bernstein series.

This is joint work with Ahmed Moussaoui and Maarten Solleveld.

# Past Representation Theory Seminar

Let U be the quantized enveloping algebra coming from a semi-simple finite dimensional complex Lie algebra. Lusztig has defined a canonical basis B for the minus part of U- of U. It has the remarkable property that one gets a basis of each highest-weight irreducible U-module V, with highest weight vector v, as the set of all bv which are not 0, as b varies in B. It is not known how to give the elements b explicitly, although there are algorithms.

For each reduced expression of the longest word in the Weyl group, Lusztig has defined a PBW basis P of U-, and for each b in B there is a unique p(b) in P such that b = p(b) + a linear combination of p' in P where the coefficients are in qZ[q]. This is much easier in the simply laced case. I show that the set of p(b)v, where b varies in B and bv is not 0, is a basis of V, and I can explicitly exhibit this basis in type A, and to some extent in types B, C, D.

It is known that B and P are crystal bases in the sense of Kashiwara. I will define Kashiwara operators, and briefly describe Kashiwara's approach to canonical bases, which he calls global bases. I show how one can calculate the Kashiwara operators acting on P, in types A, B, C, D, using tableaux of Kashiwara-Nakashima.

We describe the representation theory of loop groups in

terms of K-theory and noncommutative geometry. For any simply

connected compact Lie group G and any positive integer level l we

consider a natural noncommutative universal algebra whose 0th K-group

can be identified with abelian group generated by the level l

positive-energy representations of the loop group LG.

Moreover, for any of these representations, we define a spectral

triple in the sense of A. Connes and compute the corresponding index

pairing with K-theory. As a result, these spectral triples give rise

to a complete noncommutative geometric invariant for the

representation theory of LG at fixed level l. The construction is

based on the supersymmetric conformal field theory models associated

with LG and it can be generalized, in the setting of conformal nets,

to many other rational chiral conformal field theory models including

loop groups model associated to non-simply connected compact Lie

groups, coset models and the moonshine conformal field theory. (Based

on a joint work with Robin Hillier)

Abstract: n-homological algebra was initiated by Iyama

via his notion of n-cluster tilting subcategories.

It was turned into an abstract theory by the definition

of n-abelian categories (Jasso) and (n+2)-angulated categories

(Geiss-Keller-Oppermann).

The talk explains some elementary aspects of these notions.

We also consider the special case of an n-representation finite algebra.

Such an algebra gives rise to an n-abelian

category which can be "derived" to an (n+2)-angulated category.

This case is particularly nice because it is

analogous to the classic relationship between

the module category and the derived category of a

hereditary algebra of finite representation type.

Firstly, we will discuss how the category of strict polynomial functors can be endowed with a monoidal structure, including concrete calculations. It is well-known that the above category is equivalent to the category of modules over the Schur algebra. The so-called Schur functor in turn relates the category of modules over the Schur algebra to the category of representations of the symmetric group which posseses a monoidal structure given by the Kronecker product. We show that the Schur functor is monoidal with respect to these structures.

Finally, we consider the right and left adjoints of the Schur functor. We explain how these can be expressed in terms of one another using Kuhn duality and the central role the monoidal structure on strict polynomial functors plays in this context.

We will briefly review the notions of Dirac cohomology and of $A_{\mathfrak{q}}(\lambda)$ modules of real reductive groups, and recall a formula for the Dirac cohomology of an $A_{\mathfrak{q}}(\lambda)$ module. Then we will discuss to what extent an $A_{\mathfrak{q}}(\lambda)$ module is determined by its Dirac cohomology. This is joint work with Jing-Song Huang and David Vogan.

I will explain how to relate the center of a cyclotomic quiver Hecke algebras to the cohomology of Nakajima quiver varieties using a current algebra action. This is a joint work with M. Varagnolo and E. Vasserot.