University of Massachusetts, Amherst • October 9–11, 2015

7:00pm–9:00pm

Amber Russell (Butler)

and the organizers

and the organizers

Background on nilpotent orbits, Springer fibers, (generalized) Springer correspondence, and related topics. (Only graduate students, recent PhDs, and former presidents of the AMS, please.) Food will be provided.

8:30am–9:20am

Registration

9:20am

Welcoming remarks

9:30am–10:30am

Xuhua He (Maryland)

In this talk, we will explain a new relation between the Borel subalgebras and the nilpotent elements. Let us consider the space of functions on $sl_2(\mathbb F_q)$ spanned by the characteristic functions of the cosets of Borel subalgebras. Then the restriction map gives a bijection from this space to the space of all functions on the nilcone of $sl_2(\mathbb F_q)$.

10:30am

Coffee break

11:00am–12:00pm

George Lusztig (MIT)

Let G be a connected reductive group over an algebraically closed field k and let \tilde G be the set of pairs consisting of an element g of G and a Borel subgroup containing g. Springer's resolution is the obvious map p from \tilde G to G. One of its main properties is that the direct image of the constant sheaf (and other irreducible local systems) under p is a perverse sheaf up to shift. We will try to discuss other examples where this kind of property holds. One such example arises from the study of generalized Springer correspondence. Other examples arise in connection with the group of points of G over the ring k[\e]/(\e^r).

12:00pm–2:00pm

Lunch

2:00pm–3:00pm

Cheng-Chiang Tsai (MIT)

We propose a conjectural stratification for arbitrary (regular semisimple) affine Springer fibers, allowing us to understand the cohomology of affine Springer fibers in terms of that of generalizations of Hessenberg varieties of Goresky, Kottwitz and MacPherson. If time permits, we will describe some consequent endoscopic phenomenon for the cohomology of Hessenberg varieties.

3:00pm–4:00pm

Mark Reeder (Boston College)

About 50 years ago Steinberg showed that the singularity of an element g in a reductive group G is measured by the dimension of the variety B(g) of Borel subgroups of G containing g. A tamely ramified Langlands parameter is essentially such an element g, with torsion semisimple part. In this talk I will show how Adjoint Swan conductors may be regarded as arithmetic analogues of dim B(g), and how both are combined in the Local Langlands Conjecture.

4:00pm

Coffee break

4:30pm–5:30pm

Zhiwei Yun (Stanford)

Bezrukavnikov, Kazhdan and Varshavsky have proposed a geometric construction of certain elements in the stable Bernstein center of a group over a local function field. To prove the stability of the elements they propose, they need a compatibility result relating two actions on the cohomology of generalized affine Springer fibers. In joint work in progress with Varshavsky we attempt to prove this compatibility using the group version of Hitchin fibration, especially the work of A. Bouthier. I will explain what generalized Springer fibers are, how these two actions are defined, and why the problem is related to the Hitchin fibration.

9:00am–10:00am

Laura Rider (MIT)

Lusztigâ€™s generalized Springer correspondence relates perverse sheaves on the nilpotent cone to Weyl and relative Weyl group representations. In my talk, I will give a description of the equivariant derived category of sheaves on the nilpotent cone.

I'll also discuss two key components of the proof: establishing formality of some dg-algebra using Deligne's theory of weights and a basis described by Lusztig in the compactly supported cohomology of a generalized Steinberg variety. This is joint work with Amber Russell.

10:00am

Coffee break

10:30am–11:30am

Carl Mautner (UC Riverside)

(joint with Tom Braden) Let G be a complex reductive group and N denote its nilpotent cone. The category of G-equivariant perverse sheaves with coefficients in a field k on N plays an important role in Springer theory. When G = GL

11:30am–12:30pm

Ting Xue (Helsinki)

12:30pm–2:15pm

Lunch

2:15pm–3:15pm

Ivan Mirković (UMass Amherst)

Beilinson and Drinfeld noticed that two spaces related to semiinfinite Grassmannians and to loop Grassmannians the quaimap spaces and zastava spaces, have a modular interpretation as moduli of finitely supported maps from a curve to a stack. By writing down this observation systematically one notices that this extends to “all” subspaces of loop Grassmannians that have been used in representation theory.

- Pramod Achar (Louisiana State)
- Asilata Bapat (Chicago)
- Dori Bejleri (Brown)
- Tom Braden (UMass Amherst)
- Mark Andrea de Cataldo (Stony Brook)
- Harrison Chen (Berkeley)
- Mark Colarusso (Milwaukee)
- Zhijie Dong (UMass Amherst)
- Matt Douglass (North Texas)
- Theodosios Douvropoulos (Minnesota)
- Sam Evens (Notre Dame)
- Jessica Fintzen (Harvard)
- Katie Gedeon (Oregon)
- William Graham (Georgia)
- Sam Gunningham (Austin)
- Xuhua He (Maryland)
- Jim Humphreys (UMass Amherst)
- Mee Seong Im (West Point)
- Xin Jin (Northwestern)
- Ben Johnson (UMass Amherst)

- Dongkwan Kim (MIT)
- Donald King (Northeastern)
- Scott Larson (Oklahoma State)
- Trang Le (Smith)
- Jennifer Li (UMass Amherst)
- Penghui Li (Berkeley)
- Yiqiang Li (Buffalo)
- George Lusztig (MIT)
- Carl Mautner (Riverside)
- Sean McAfee (Utah)
- George McNinch (Tufts)
- Ivan Mirković (UMass Amherst)
- Emily Norton (Kansas State)
- Alexei Oblomkov (UMass Amherst)
- Cornelius Pillen (South Alabama)
- Mark Reeder (Boston College)
- Vic Reiner (Minnesota)
- Laura Rider (MIT)
- Beth Romano (Boston College)
- Seth Rothschild (Tufts)

- Amber Russell (Butler)
- Leonard Scott (Virginia)
- Eric Sommers (UMass Amherst)
- Benjamin Strasser (Minnesota)
- Changjian Su (Columbia)
- Alex Takeda (Berkeley)
- Cheng-Chiang Tsai (MIT)
- Julianna Tymoczko (Smith)
- Michael Viscardi (MIT)
- David Vogan (MIT)
- Robin Walters (Northeastern)
- Daping Weng (Yale)
- Ting Xue (Helsinki)
- Yaping Yang (UMass Amherst)
- Zhiwei Yun (Stanford)
- Zhuohui Zhang (Rutgers)
- Gufang Zhao (UMass Amherst)
- Huijun Zhao (Northeastern)
- Roger Zierau (Oklahoma State)