Workshop on Springer Theory
and Related Topics
University of Massachusetts, Amherst • October 9–11, 2015
Schedule and abstracts of talks
All talks will be held in Lederle Graduate Research Tower (LGRT) 1634 (16th floor).Friday, October 9
7:00pm–9:00pm
Amber Russell (Butler)
and the organizers
and the organizers
Introductory lectures on Springer theory
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.
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.
Saturday, October 10
8:30am–9:20am
Registration
9:20am
Welcoming remarks
9:30am–10:30am
Xuhua He (Maryland)
Cosets of Borel subalgebras and nilcone in sl2(Fq)
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 $sl_2(\mathbb F_q)$.
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)
Generalizations of the Springer resolution
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).
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)
Stratification of affine Springer fibers
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.
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)
Adjoint Swan Conductors
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.
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)
Actions on the cohomology of generalized Springer fibers
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.
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.
Sunday, October 11
9:00am–10:00am
Laura Rider (MIT)
Formality for the nilpotent cone and Lusztig's generalized Springer correspondence
Lusztig’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.
Lusztig’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)
Hypertoric Schur algebras
(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 = GLn, this category is equivalent to the category of representations of the Schur algebra Sk(n,n). Motivated by similarities between the geometry of hypertoric varieties and the nilpotent cone for GLn, we consider an analogous category of perverse sheaves on hypertoric varieties. We show that the resulting category is highest weight and that its Ringel dual is equivalent to the corresponding category for the Gale dual hypertoric variety.
(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 = GLn, this category is equivalent to the category of representations of the Schur algebra Sk(n,n). Motivated by similarities between the geometry of hypertoric varieties and the nilpotent cone for GLn, we consider an analogous category of perverse sheaves on hypertoric varieties. We show that the resulting category is highest weight and that its Ringel dual is equivalent to the corresponding category for the Gale dual hypertoric variety.
11:30am–12:30pm
Ting Xue (Helsinki)
Hessenberg varieties and the Springer correspondence for symmetric spaces
We consider the Springer correspondence in the case of symmetric spaces. In this setting various new phenomena occur which are not present in the classical Springer theory. For example, we obtain representations of (the Tits extension) of the braid group rather than just Weyl group representations. These representations come from cohomology of families of Hessenberg varieties. In the situations we consider the Hessenberg varieties can be interpreted as classical objects in algebraic geometry: Jacobians and moduli spaces of vector bundles on curves with extra structure, Fano varieties of k-planes in the intersection of two quadrics, etc. This is joint work with Tsao-hsien Chen and Kari Vilonen.
12:30pm–2:15pm
Lunch
2:15pm–3:15pm
Ivan Mirković (UMass Amherst)
Moduli of finitely supported maps and the geometry of loop Grassmannians
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.
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.
List of Participants
- 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)
