The University of MelbourneDates July. 20-Dec 1, 2018. Organisers: Ting Xue, Yaping YangTime: Tuesdays 15:30-17:00.Place: Peter Hall Building, Room 107 weeklyScheduleAbstract: 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. 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. Aug 14, Arun Ram 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. Aug 21, Gufang Zhao 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. Aug 28, Xinwen Zhu (California Institute of Technology) 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. Sep 4, Iva Halacheva Sep 11, Peter McNamara Sep 18 Oct 2 Oct 9 Oct 16 Oct 23 Oct 30 *= to be confirmed. Previous seminars: |

### Representation theory seminar 2018

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Representation theory seminar 2017