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.
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.
- 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.
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.
Videos (Thanks to Omar!)
Ron Donagi's talk on Hitchin’s system and Geometric Langlands, String math conference, Hamburg, 25 July, 2017.
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.
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.