# Representation theory seminar 2017

## The University of Melbourne

School of Mathematics and Statistics

Representation theory seminar 2017

**Dates** Aug. 7-Sep. 30.

**Organisers**: Ting Xue, Yaping Yang

**Time:** Thursdays 15:30-17:30. (2 hours!!)

**Place:** Peter Hall Building, __Room 107 __weekly,

Exception: Aug. 31: __Belz Room.__

### About topics:

The topics are broad. It may cover the general ideas in representation theory around affine Springer fibers, Hitchin fibrations, rational Cherednik algebras, orbital integrals, Macdonald polynomials, and etc. We will try to focus on what's known, and what's not known, how we think about it, and our personal point of views.

### Schedule

Aug. 10, Yaping Yang, An introduction to rational Cherednik algebras. Lecture notes.

Aug. 17, Ting Xue, Affine Springer fibers. Lecture notes

Aug. 24, Paul Norbury: Moduli space of Higgs bundles. Lecture notes.

Aug. 31, Arun Ram: Parking functions, the Shi arrangement and Macdonald polynomials. Lecture notes.

- Abstract: A parking function is a sequence (b1, ..., bn) of positive integers which, when rearranged in increasing order (a1 ≤ a2 ≤ ... ≤ an), is such that ai≤ i. I will first convert parking functions to elements of the affine Weyl group which correspond to regions of the Shi hyperplane arrangement and bases of a module for the rational Cherednik algebra (or double affine Hecke algebra). As explained, for example, in papers of Varagnolo-Vasserot and Oblomkov-Yun, this module can be realized as the cohomology (or K-theory) of an affine Springer fiber. These bases are closely connected to Macdonald polynomials. Goresky-Kotwitz-Macpherson explain how to chop up the affine Springer fiber into tractable pieces indexed by the (generalised) parking functions (paving by Hessenbergs). I'll start by drawing the pictures and then explain how to read the connections off the picture.

Sep. 7, Kari Vilonen: Springer fibers and Hitchin fibers. Lecture notes

Sep. 14, Vinoth Nandakumar: [BMR]-localization for Lie algebras in positive characteristic, and Lusztig's conjectures. Lecture notes

Abstract: This is an expository overview of Bezrukavnikov-Mirkovic-Rumynin's localization theory, which gives derived equivalences between representation categories of Lie algebras in positive characteristic, and coherent sheaves on Springer fibers. The resulting "exotic t-structures" can also be defined geometrically using an action of the affine braid group, and are used to prove Lusztig's conjectures that the classes of the simple modules give a canonical basis in the Grothendieck group of a Springer fiber. For the case of a two-row nilpotent in type A, in joint work with Rina Anno and David Yang, we use [BMR]'s techniques in conjunction with Cautis-Kamnitzer's tangle categorification results to give combinatorial formulae for the dimension of the irreducible modules.

Sep. 21, Omar Foda: The infinite-dimensional representations of Schiffmann-Vasserot SH^c algebra. Lecture notes

Sep. 28. Michael Wheeler: Kostka-Foulkes polynomials and integrable vertex models. Lecture notes

### Videos (Thanks to Omar!)

Ron Donagi's talk on Hitchin’s system and Geometric Langlands, String math conference, Hamburg, 25 July, 2017.

https://lecture2go.uni-hamburg.de/l2go/-/get/v/21917

Andrei Negut's talk on W-algebras, moduli of sheaves on surfaces, and AGT, String-Math 2017 held at Hamburg University, July 24-28, 2017.

https://www.youtube.com/watch?v=OfUnY8nY1C8

### References

Bezrukavnikov: Seminar notes on Hitchin fibers and affine Springer fibers. Here.

Nadler: The Geometric Nature of the Fundamental Lemma, Bull. Amer. Math. Soc. 49 (2012), 1--50. available here.

Etingof, Ma: Lecture notes on Cherednik algebras, available here.

Goresky, Kottwitz, MacPherson: Homology of affine Springer fibers in the unramified case. Duke Math. J. 121 (2004), no. 3, 509–561. available at arXiv:math/0305144

Goresky, Kottwitz, MacPherson: Purity of equivalued affine Springer fibers. Represent. Theory 10 (2006), 130–146. available here.

Kazhdan, Lusztig: Fixed point varieties on affine flag manifolds. *Israel J. Math.* 62 (1988), no. 2, 129–168.

Oblomkov, Yun: Geometric representations of graded and rational Cherednik algebras. *Adv. Math.* 292 (2016), 601–706. available here.

Schiffmann, Vasserot: Cherednik algebras, W-algebras and the equivariant cohomology of the moduli space of instantons on A^2. *Publ. Math. Inst. Hautes Études Sci. 118 (2013), *213–342. available here.

Yun: Lectures on Springer theories and orbital integrals, available here.