Oberseminar Representation Theory, Sommer Term 2017

Catharina Stroppel email
Daniel Tubbenhauer email

Where and when?



Ruslan Maksimau,
Title: Higher level affine Hecke algebras

This is a joint work with Catharina Stroppel.
KLR algebras were introduced by Khovanov-Lauda and Rouquier to categorify quantum groups. For each weight $\Lambda$, the KLR algebra $R$ has a special quotient $R^\Lambda$ (called cyclotomic quotient) that categorifies the simple module $L(\Lambda)$ over a quantum group. It is proved by Brundan--Kleshchev and Rouquier that the cyclotomic quotient $R^\Lambda$ is isomorphic to some similar quotient of the affine Hecke algebra.
Ben Webster has defined new algebras (called tensor product algebras) that generalize KLR algebras. The cyclotomic quotients of the tensor product algebras categorify tensor products of simple modules over a quantum group. However, tensor product algebras have no known analogue from the Hecke side.
In my talk I introduce such an analogue called “higher level affine Hecke algebra”. This new algebra contains the usual affine Hecke algebra. This algebra also has a special quotient that is isomorphic to the cyclotomic quotient of the tensor product algebra.

Iuliya Beloshapka,
Title: Irreducible representations of finitely generated nilpotent groups

At ICM 2010 Parshin conjectured that irreducible complex representations of finitely generated nilpotent groups are monomial if and only if they have finite weight. This was previously known to be true for finite nilpotent groups and for unitary irreducible representations of connected nilpotent Lie groups (A.A. Kirillov and J. Dixmier). We prove Parshin's conjecture in full generality. We also show that for a wide class of induced representations Schur's lemma is equivalent to irreducibility of a representation. The talk is based on joint work with S.Gorchinskiy.

Hankyung Ko,
Title: Cohomological difference between quantum groups and algebraic groups

I will introduce a category that arises from the difference between $\mathrm{Rep}(G)$, the category of rational representations for an algebraic group $G$ in characteristic p, and representations for the corresponding quantum group at a $p$-th root of unity. The cohomology of the latter is well understood via the Kazhdan-Lusztig theory, while it is not so for $\mathrm{Rep}(G)$. Our category can be thought of as cohomological complement of the Kazhdan-Lusztig theory in $\mathrm{Rep}(G)$. I discuss translation functors in this category and, as an application, give some information on the irreducible characters for $G$ when the Lusztig character formula does not hold.

Paolo Papi,
Title: Covariants in the exterior algebra of a simple Lie algebra

For a simple complex Lie algebra \(\mathfrak{g}\) we study the space of invariants $A=(\bigwedge\mathfrak{g}\otimes\mathfrak{g})^{\mathfrak{g}}$ (which describes the isotypic component of type \(\mathfrak{g}\) in $\bigwedge\mathfrak{g}$) as a module over the algebra of invariants $(\bigwedge\mathfrak{g})^{\mathfrak{g}}$. As main result (joint with C. De Concini and C. Procesi) we prove that $A$ is a free module, of rank twice the rank of \(\mathfrak{g}\), over the exterior algebra generated by all primitive invariants in $(\bigwedge\mathfrak{g})^{\mathfrak{g}}$ with the exception of the one of highest degree. We will also discuss recent develpments, such as a conjectural result on a natural Lie superalgebra structure on A and connections with a conjecture of Reeder on covariants of small representations

Stéphane Gaussent,
Title: A Macdonald formula for Kac-Moody groups

In this talk, I will report on a joint work with Nicole Bardy-Panse and Guy Rousseau. The Macdonald formula that will be discussed is the one giving the image of the Satake isomorphism between the spherical Hecke algebra and the algebra of W-invariant functions on the coweight lattice of a maximal torus in a Kac-Moody group over a local field. To establish the formula, on the one hand, we use the action of the affine Iwahori-Hecke algebra defined via its Bernstein-Lusztig presentation. On the other hand, we compute the image of the Satake isomorphism using Hecke paths in the standard apartment of the masure associated to the situation. The masure is a generalization of the Bruhat-Tits building.

Olivier Schiffmann,
Title: Talk canceled!

Dmitry Kaledin,
Title: Non-commutative Hodge-to-de Rham degeneration, for complexes and for spectra

I will review my recent proof of the non-commutative Hodge-to-de Rham Degeneration Conjecture of Kontsevich and Soibelman, and discuss how it could possibly be related to Topological Hochschild Homology.

Matthew Hogancamp,
Title: Khovanov-Rozansky homology and $q,t$ Catalan numbers

I will discuss a recent proof of the Gorsky-Oblomkov-Rasmussen-Shende conjecture for $(n,nm+1)$ torus knots (even more recently this was extended to arbitrary positive torus knots by Mellit), which generally expresses the Khovanov-Rozansky homology of torus knots in terms of representations of rational DAHA. The proof is based off of a computational technique introduced by myself and Ben Elias, using complexes of Soergel bimodules which categorify certain Young symmetrizers. We will summarize this technique and indicate how it results in a remarkably simple recursion which computes the knot homologies in question.

Gwyn Bellamy,
Title: Graded algebras admitting a triangular decomposition

The goal of this talk is to describe the representation theory of finite dimensional graded algebras $A$ admitting a triangular decomposition (in much the same flavour as the enveloping algebra of a semi-simple Lie algebra admits a triangular decomposition). The examples to keep in mind are restricted rational Cherednik algebras, restricted enveloping algebras and hyperalgebras. We exploit the fact that the category of graded modules for such an algebra is a highest weight category. This allows us to prove two key results. First that the degree zero part $A_0$ of the algebra is cellular, and secondly a canonical subquotient of our highest weight category provides a highest weight cover of $A_0\text{-}\mathrm{mod}$.
This is based on joint work with U. Thiel.

Laura Rider,
Title: Formality for the nilpotent cone and the generalized Springer correspondence

The Springer correspondence attaches to each irreducible representation of the Weyl group some geometric information (in the form of perverse sheaves) from the nilpotent cone. In my talk, I'll give a brief introduction to the Springer correspondence, and then explain mixed/derived versions of the correspondence. As time allows, I'll also discuss Lusztig's generalized Springer correspondence and recent progress towards mixed/derived versions of the generalized Springer correspondence.

Ivan Mirković,
Title: TBA

Bea Schumann,
Title: Combinatorics of canonical bases and cluster duality

In this talk we explain how Lusztig's and the string parametrization of the canonical basis of a representation of a simple, simply-laced algebraic group arises from the tropicalizations of a potential function on a cluster variety in the setup of Gross-Hacking-Keel-Kontsevich. In the type A situation, the explicit form of such potential functions written in certain torus coordinates is intimately related to the explicit description of crystal operations on Lusztig data.

Valentin Buciumas,
Title: Hecke algebra modules from representations of $p$-adic groups and quantum groups

I will present a general class of modules of the affine Hecke algebra and show how such modules arise in a number of settings involving representations of $p$-adic groups and $R$-matrices for quantum groups. An example I will focus on is the space of Whittaker functionals on a $p$-adic (metaplectic) group which can be endowed with a Hecke algebra module structure. In type A this module is isomorphic to a “natural” module coming from the theory of quantum groups. Time permitting, I will discuss an application in giving a new algebraic proof of the Casselman-Shalika formula. This talk is based on joint work with B. Brubaker, D. Bump, and S. Friedberg.