# Representation theory seminar 2018

## The University of Melbourne

School of Mathematics and Statistics

Representation theory seminar 2018

Dates July. 20-Dec 1, 2018.

Organisers: Ting Xue, Yaping Yang

Time: Tuesdays 15:30-17:00.

Place: Peter Hall Building, Room 107 weekly

## Schedule

### July 24, Arik Wilbert: Exotic Springer fibers and two-boundary Temperley-Lieb algebras.

Abstract: In this talk we will study the geometry and topology of a certain family of exotic Springer fibers. These algebraic varieties appear as the fibers under a resolution of singularities of the exotic nilpotent cone which plays a prominent role in Kato’s Deligne-Langlands type classification of simple modules for multiparameter Hecke algebras of type C. We describe our results in terms of the combinatorics of the two-boundary Temperley-Lieb algebra.

### July 31, Jan De Gier: Construction of stochastic duality functions using the Hecke algebra

Abstract: We discuss a method for constructing duality observables in integrable stochastic particle processes from polynomial solutions to the quantum Knizhnik-Zamolodchikov equation. These solutions are constructed using the polynomial representation of the Hecke algebra and non-symmetric Macdonald polynomials at the degenerate points $q^kt^r = 1$.

### Aug 7, Edmund Howse (National University of Singapore): Invariants of Kazhdan–Lusztig cells

Abstract: Lusztig has described the partition of a Coxeter group W into left, right and two-sided cells with respect to a weight function. This description relies on certain equivalence relations that are calculated in the corresponding Iwahori–Hecke algebra H, and the resulting cells afford representations of both W and H.

As cells are difficult to calculate directly from the definition, invariants of cells are sought after to make it possible to determine cells purely at the level of the Coxeter group. For instance, a classical result of Kazhdan and Lusztig is that the left cells of the symmetric group are characterised by the generalised \tau-invariant.

In this talk, we discuss invariants such as the Vogan classes of Bonnafe and Geck and introduce a modified version of the right descent set. We then describe how a combination of these concepts leads to a characterisation of the left cells in type B_n with respect to two different choices of weight function.

### Title: Representations of two boundary Hecke and Temperley-Lieb algebras (TBHA and TBTL)

Abstract: I will discuss classifications, constructions and combinatorics of irreducible and standard modules of TBHA and TBTL. The TBTA is the affine Hecke algebra of type C with arbitrary “unequal” parameters. The TBTL is a quotient of the TBHA by local idempotents (for rank 2 sub root systems). The TBTL has been of interest in statistical mechanics: Heisenberg spin chains with boundaries (de Gier-Nichols). The geometry construction of TBHA-modules (Kato) for unequal parameters is via the exotic nilpotent cone.

### Title: A Borel-Weil-Bott Theorem for toroidal algebras

Abstract: I will talk about the notion of local spaces over Hilbert scheme of points on a smooth algebraic variety $M$, as a refined version of factorization spaces of Beilinson-Drinfeld. When $M$ is an algebraic surface, building up on the work of Feigin-Loktev and Chari-Pressley on local Weyl modules, as well as the work of Haiman on Hilbert schemes, an example of local space will be given. This local space parametrizes torsion free sheaves on $M$. Global sections of a tautological line bundle on this local space yield a local Weyl module of the toroidal algebra, whose characters are given by Macdonad polynomials. This is based on a work in progress in collaboration with Ivan Mirkovic and Yaping Yang, aiming to construct higher loop Grassmannians.

### Title: The elliptic part of the cohomology of moduli of Shtukas

Abstract: I’ll discuss the cuspidal cohomology of moduli of Shtukas over elliptic Langlands parameters. In this case of GL(n), this recovers L. Lafforgue’s result, and in the general case, it agrees with the Arthur-Kottwitz heuristics. The proof is based on an idea of Drinfeld’s, and completely bypasses the trace formula. Joint work with V. Lafforgue.

### Title: The cactus group and a monodromy action on crystals

Abstract: The cactus group J(g) associated to a finite-dimensional semisimple Lie algebra g can be defined in a straightforward way from its Dynkin diagram and has properties analogous to those of the braid group. Davis, Januszkiewicz, and Scott show that it is in fact the fundamental group of a certain De Concini-Procesi moduli space M(g). Using this result, we construct two actions of J(g) on any crystal associated to a representation V of g. The first is combinatorial via Schützenberger involutions, while the second comes from the monodromy action of a covering space of M(g) constructed using the representation V and a family of maximal commutative subalgebras of U(g) called the shift of argument algebras. We show that these two actions coincide.

### Title: Quantum cohomology and DAHA actions

Abstract:

I will explain a combinatorial/integrable systems model for quantum cohomology of Grassmannians and then connect it with the action of some DAHA at roots of unity. If time allows I will explain the topology behind DAHAs.

### Title: Categorification at a root of Unity

Abstract:

Being able to categorify structures where a root of unity is present is motivated by Turaev-Viro invariants of 3-manifolds. I will discuss the problem of categorifying at a root of unity. Current progress rests on the foundations of hopfological algebra, which generalises the theory of dg-algebras, which I will discuss.

### Title: Representations of Yangians and cohomology

Abstract: We'll discuss a construction (focusing on the case of Y(sl_2)) generalizing that of Maulik and Okounkov for Nakajima quiver varieties, in order to obtain general finite-dimensional representations of Yangians, their R-matrices and their fusion''. In particular for sl_2 we reproduce this way a recent formula of Mangazeev for the R-matrix.

### Title: Schubert calculus on semi-infinite flag varieties

Abstract:

I will endeavour to explain the main facets of the way that I think about Schubert calculus, focusing on the case of the semi-infinite flag variety. I will review the definition of the semi-infinite flag variety, the Schubert classes, the action of polynomials, the moment graph description, the push-pull operators and the Pieri-Chevalley formula. None of this is my work except, perhaps, a certain point of view on the subject.

### Title: Cohomological Hall algebras, vertex algebras and instantons

Abstract.

A new class of vertex operator algebras, vertex algebra at the corner, are recently introduced by Gaiotto and Rapčák, generalizing the affine W-algebra of gl_N. In my talk, I will discuss an action of this new vertex algebra on the cohomology of certain spiked instanton moduli spaces on 3CY manifold in the sense of Nekrasov. This action is naturally obtained using the cohomological Hall algebras of Kontsevich-Soibelman. This talk is based on my work in progress, in collaboration with Miroslav Rapčák, Yan Soibelman, and Yaping Yang.

### Title: A p-modular topological vertex and Z_p conformal field theories

Abstract:

I wish to outline recent work with Wee Chaimanowong in which we introduce a p-modular topological vertex that depends on p species of free bosons. We use this vertex to reproduce (known) matrix elements of primary-state vertex operators in 2D conformal field theories that are based on (conjectured) limits of the Elliptic Hall algebra, and that contain Z_p parafermions.

### Title: On unipotent representations of real classical groups

Abstract:

In this talk, I will present a recent work with Binyong Sun and Chengbo Zhu on unipotent representations of real classical groups (real symplectic groups, real orthogonal groups, quaternionic orthogonal groups or quaternionic symplectic groups). Unipotent representations are certain irreducible admissible representations characterized by their associated varieties and infinitesimal characters. They consist the unipotent L-packet in Langlands' philosophy and they are related to the quantization of nilpotent orbits. Barbasch and Vogan established the theory of special unipotent representations for complex classical groups and unitary groups. They also made conjectures for the general case, including a conjecture that unipotent representations attached to special nilpotent orbits are unitarizable. In 90's, thanks to many peoples work, it become clear that iterated theta lifting could be an effective way to construct unipotent representations of real classical groups.

In our work, we constructed all unipotent representations attached to quasi-disdistinguished nilpotent orbits utilizing algebraic and analytic properties of theta lifts. Then, the unitarity of these representations follows from the construction, thanks to Jian-shu Li, Hongyu He and Harris-Li-Sun's results on matrix coefficients integral. The construction of unipotent representations attached to a general special unipotent is working in progress (joint with Barbasch).

### Title: A bridge between p-adic and quantum group representations via Whittaker coinvariants.

Abstract: Unramified principal series representations of p-adic GL(r) and its metaplectic covers are important in the theory of automorphic forms. I will present a method of endowing the Whittaker coinvariants of such a representation with the structure of a quantum affine gl_n module (where n is the degree of the metaplectic cover). If time permits I will explain a version of this result for the symplectic group Sp(2r) (which involves coideal subalgebras) and a conjecture relating representations of p-adic and quantum groups via a Schur-Weyl duality.

### Previous seminars:

Representation theory seminar 2017