May 24-27, 2016 University of Virginia

Main talks: Clark 108 | |||
---|---|---|---|

Tuesday, May 24 | Wednesday, May 25 | Thursday, May 26 (Brian Parshall Day) |
Friday, May 27 |

9:00-10:00: Lusztig 10:15-11:15: Morse 11:30-12:30: Achar |
8:30-9:30: Carlson 9:50-10:50: Brundan 11:10-12:10: Kolb |
8:30-9:30: Drupieski 9:50-10:50: Nakano 11:10-12:10: Wang 12:10: Conference photo |
8:30-9:30: Williamson 9:50-10:50: Achar 11:10-12:10: discussion (w/ Rider) |

Lunch | Lunch | Lunch | |

2:00-3:00: Williamson 3:20-4:20: discussion (w/ Rider) |
1:40-2:40: Shan 3:00-4:00: Bao |
1:40-2:40: Friedlander 3:00-4:00: Du 4:20-4:50: Ko |

Tuesday, May 24 | Wednesday, May 25 | ||
---|---|---|---|

Clark 102 | Clark 108 | Clark 102 | Clark 108 |

4:40-5:00:Matherne 5:10-5:30: Norton 5:40-6:00: Makisumi |
4:40-5:00: Vella 5:10-5:30: Lai 5:40-6:00: Zhu |
4:20-4:40: Nandakumar 4:50-5:10: Simental-Rodriguez 5:20-5:40: Chen 5:50-6:10: Im |
4:20-4:40: Clark 4:50-5:10: Nguyen 5:20-5:40: Ganev 5:50-6:10: Reeks |

**Pramod Achar **(LSU): * I. Flag varieties in representation theory*

Abstract: I will explain how the algebraic topology of the (ordinary, finite-dimensional) flag variety is related to category O for a semisimple Lie algebra (characteristic-0 case) and to Soergel's "modular category O" (positive characteristic). Along the way, we will try to get used to the idea of looking at perverse sheaves or parity sheaves for representation-theoretic information. (No prior knowledge of perverse/parity sheaves is necessary.)

* II. Geometric local Langlands duality*

Abstract: I will give a sketch of the (characteristic-zero) "geometric local Langlands duality program," which was carried out by Bezrukavnikov and numerous collaborators between 1998 and 2012. A central theme of this program is that algebraic notions for G should be related to topological notions for G^\vee. The philosophy of this program serves as the inspiration for some current directions in positive-characteristic geometric representation theory. I will try to place the statements from Geordie's talks in this context.

** Geordie Williamson **(Sydney/MPI): * Tilting modules and the Hecke category*

Abstract: I will introduce the Hecke category (this is a explicitly presented diagrammatic category, but also has realisations as Soergel bimodules or as parity sheaves on the flag variety as in Pramod's lectures). I will explain a conjecture (with Simon Riche) which implies that the Hecke category for the affine Weyl group controls the representation theory of reductive algebraic groups. The conjecture is a theorem for G = GL_n.

** Laura Rider** (MIT/Georgia) will lead 2 discussion
sections on the minicourses.

Notes from discussion 1

Fact sheet from discussion 2

Exercise sheet from discussion 2

** Huanchen Bao **(Maryland): *Canonical bases arising from quantum symmetric pairs*

Abstract: A quantum symmetric pair consists of a quantum group and its coideal subalgebra. The coideal subalgebra is a quantum analog of the fixed point subalgebra of the enveloping algebra with respect to certain involution. In this talk, we shall describe the construction of (i-)canonical bases on the modified coideal subalgebras and their modules for all quantum symmetric pairs of finite type. This is joint work with Weiqiang Wang.

** Jonathan Brundan **(Oregon): *Type A blocks of super category
O*

Abstract: We establish an equivalence between so-called type A blocks of category O for the Lie superalgebra $q_n(\C)$ and integral blocks of category O for an associated general linear Lie superalgebra. This implies the truth of the Kazhdan-Lusztig conjecture for such type A blocks of $q_n(\C) as formulated by Cheng, Kwon and Wang. The proof depends on (1) an isomorphism theorem of Kang, Kashiwara and Tsuchioka involving quiver Hecke superalgebras, and (2) the uniqueness of certain tensor product categorifications established by Losev and Webster. (Joint work with N. Davidson.)

** Jon Carlson **(UGA): *Tensor products and Hopf structure
*

Abstract: Suppose that $G$ is an elementary abelian $p$-group of rank $r$,
and that $k$ is a field of characteristic $p$. The group algebra $kG$ is
isomorphic to the enveloping algebra of a restricted commutative Lie
algebra of dimension $r$. So the two have the same representation theory.
However, the tensor products of modules depends on the Hopf algebra
structures which are different. In this lecture we discuss some results
that resolve the differences for the tensor products of some canonically
defined modules. This is joint work with Srikanth Iyengar.

** Christopher Drupieski **(DePaul): *Support varieties for Lie superalgebras and finite graded group schemes*

Abstract: Following the pioneering work of Quillen in the 1970s, Carlson, Avrunin and Scott, Friedlander and Parshall, Jantzen, and others made much progress in the 1980s studying the cohomology and representation theory of finite groups and restricted Lie algebras by way of their associated cohomological support varieties. Later, many of these methods and results were generalized first to infinitesimal group schemes by Suslin, Friedlander, and Bendel, and then to arbitrary finite group schemes by Friedlander and Pevtsova. In this talk I will discuss some results and conjectures concerning how some of the aforementioned methods and results can be generalized to restricted (and non-restricted) graded finite-dimensional Lie algebras and to certain finite graded group schemes. This is joint work with Jonathan Kujawa.

** Jie Du **(UNSW): *q-Schur algebras and integral quantum affine gl_n*

Drinfeld's new realisation of the quantum loop algebra of gl_n does not offer naturally a Lusztig type form. We will show in this talk how affine q-Schur algebras and the Ringel-Hall algebras of cyclic quivers are used to reconstruct this quantum affine gl_n. In particular, we will show that the integral structures of both q-Schur algebras and Hall algebras give rise naturally to a Lusztig form for the quantum group. Further applications to the integral modified quantum groups and their canonical bases will also be discussed. This is joint work with Qiang Fu.

** Eric Friedlander **(USC): *Support
varieties for linear algebraic groups*

Abstract: We present the beginnings of a theory of support varieties for certain
linear algebraic groups over an algebraically closed field of positive
characteristic. For such an algebraic group $G$ and a rational $G$-module
$M$, we consider the ind-variety of 1-parameter subgroups $V(G)$ and the
subvariety $V(G)_M \subset V(G)$. This theory satisfies some of the
standard properties enjoyed by support varieties for finite groups (and,
more generally, for finite group schemes). We introduce two interesting
classes of rational $G$-modules, those that are mock injective and those
that are mock trivial. We also construct a companion theory
$V^{coh}(G)_M \subset V^{coh}(G)$ of cohomological support varieties
based on the rational cohomology of $G$. This leads us to the computation
of the "continuous rational cohomology" of certain unipotent algebraic groups.

** Stefan Kolb **(Newcastle): *Universal K-matrix for coideal subalgebras*

In this talk I will introduce the notion of a universal K-matrix for a coideal subalgebra of a quasitriangular Hopf algebra. A universal K-matrix provides representations of the braid group of type B_n on the n-fold tensor product of any representation of the Hopf algebra. An important class of coideal subalgebras for quantum groups is provided by G. Letzter's theory of quantum symmetric pairs. I will review this theory and show that there exists a universal K-matrix for any quantum symmetric pair coideal subalgebra of finite type. This result is based on joint work with Martina Balagovic.

** George Lusztig **(MIT): *Odd vanishing properties arising from nilpotent orbits attached to
Z/m-graded Lie algebras*

This talk is based on a joint work with Zhiwei Yun.
The variety of nilpotent elements in a graded component of a Z/m-graded
simple Lie algebra is a union of finitely many orbits for the action of
a certain reductive group. One can consider the intersection cohomology
complex of the closure of one such orbit with coefficients in a certain
local system. One can show that the cohomology sheaves in odd degree of
such
a complex are zero.

** Jennifer Morse **(Drexel/UVA): *Combinatorics of affine Schubert calculus*

Abstract: We will discuss a long-standing open combinatorial problem concerning
the basis of Macdonald symmetric functions and how we were led to
representatives which simultaneously enable computation in quantum,
affine and equivariant Schubert calculus.

** Daniel Nakano **(UGA): * Irreducibility of Weyl modules over fields of arbitrary characteristic *

Abstract: In the representation theory of split reductive algebraic groups, the
following is a well known fact: for every minuscule weight, the Weyl module with that
highest weight is irreducible over every field. The adjoint representation of E_8 is also
irreducible over every field. Recently, Benedict Gross conjectured that these two examples
should be the only cases where the Weyl modules are irreducible over arbitrary fields. In this
talk I will present our proof of Gross' suggested converse to these statements, i.e., that if a Weyl module is
irreducible over every field, it must be either one of these, or trivially constructed from one of these.
My coauthors will be revealed during my talk.

** Peng Shan **(Paris-Sud): *Categorification of coideal algebras*

Abstract: Certain coideal subalgebras of the quantized enveloping algebra
of sl_n arise naturally as Schur-Weyl dual to Hecke algebras of type B.
They admit a theory of canonical bases. In this talk, we explain how to
categorify the coideal algebras using a 2-category analogue to
Khovanov-Lauda-Rouquier's categorification of U_q(sl_n), and some
applications of this 2-category. This is a joint work with Huanchen Bao,
Weiqiang Wang and Ben Webster.

**Jian-pan Wang **(ECNU): *Classification of Irreducible Representations of q-Schur Superalgebras at a Root of Unity*

Abstract:
This talk will give a complete classification of irreducible representations of q-Schur Superalgebra SF(m|n,r) (for m + n greater than or equal to r) at a primitive lth root of unity (l odd) over an algebraically closed field F in both the case char F = 0 and the case char F = p > 0 with (l , p) = 1. (joint work with Jie Du and Haixia Gu)

**Xueqing Chen **(Wisconsin-Whitewater): *From Hall
algebras to cluster algebras*

Abstract:A homomorphism from Hall
algebra of some orbit triangulated category via derived Hall algebra
to quantum cluster algebra will be presented.

** Sean Clark **(Northeastern/MPI): *Quantum enveloping gl(m|1) and canonical bases*

Abstract:
Quantum enveloping algebras associated to Lie superalgebras have been
a subject of significant interest since the successes of the early
90's in understanding their non-super counterparts. Recently, some
progress has been made toward constructing canonical bases for
certain families of Lie superalgebras using a variety of
approaches. In this talk, I will describe a new construction of
canonical bases for the quantum enveloping algebra of gl(m|1) via
crystal bases and braid operators. I will also explain some of the
technical challenges towards generalizing this construction.

** Iordan Ganev **(Texas): *The wonderful compactification for quantum groups *

Abstract: The wonderful compactification of a group links the geometry of
the group to the geometry of its partial flag varieties, encodes the
asymptotics of matrix coefficients for the group, and captures the
rational degenerations of the group. It plays a crucial role in several
areas of geometric representation theory and related fields. In this talk,
we review several constructions of the wonderful compactification and its
relevant properties. We then introduce quantum group versions of the
wonderful compactification and its stratification by G x G orbits. A key
player in our approach is the Vinberg semigroup, which arises as a Rees
construction and forms a multi-cone over the wonderful compactification.

**Mee Seong Im **(USMA): *
Towards a categorification of infinite-dimensional modules*

Abstract: Raphael Rouquier asked a question on how one should categorify infinite-dimensional modules called Verma modules. Combining diagrammatic categories by Khovanov and Licata-Savage, Chuang-Rouquier's categorification of certain finite-dimensional modules, and the 1979 Annals of Mathematics paper by Tom Enright, we propose constructing the tensor product of certain categories, where under the direct sum decomposition of these categories, a categorification of Verma modules appears. I will give key techniques in our construction. This is joint work with Ben Cox.

**Hankyung Ko **(UVA): *
Some conjectures of Parshall and Scott*

Abstract: I will discuss certain algebraic group representations 'reduced modulo p' from irreducible representations over quantum groups at roots of unity and some conjectures made by Parshall-Scott (2014) regarding these representations.

**Chun-Ju Lai **(UVA): *Affine Hecke algebras and quantum
symmetric pairs*

Abstract: Recently, generalizing the work of
Beilinson, Lusztig, and MacPherson of finite type A, Bao, Kujawa, Li,
and Wang constructed the quantum algebras arising from partial flag
varieties of finite type B/C, altogether with their canonical
bases. These quantum algebras are coideal subalgebras of quantum
gl(n), which form quantum symmetric pairs together with quantum
gl(n). The canonical bases arising from quantum symmetric pairs was
used by Bao-Wang to formulate Kazhdan-Lusztig theory for BGG category
O. The above can be reformulated within the framework of Hecke
algebras (without geometry) by realizing the q-Schur algebras as
endomorphism algebras of a direct sum of certain q-permutation modules
for Hecke algebra. In this talk I will explain the affinization of the
theory above using (non-extended) affine Hecke algebras of type
C. This is a joint work with Zhaobing Fan, Yiqiang Li, Li Luo, and
Weiqiang Wang.

** Shotaro Makisumi **(Stanford): *Koszul duality for Soergel bimodules
*

Abstract: I will report on a Koszul (self-)duality for Soergel bimodules analogous to that of BGG category O.

** Jacob Matherne **(LSU):
*Derived geometric Satake equivalence, Springer correspondence, and small representations*

Abstract: Two major theorems in geometric representation theory are the geometric Satake equivalence and the Springer correspondence, which states:

- For G a semisimple algebraic group, we can realize Rep(G) as intersection cohomology of the affine Grassmannian for the Langlands dual group.
- For W a Weyl group, we can realize Rep(W) as intersection cohomology of the nilpotent cone.

** Vinoth Nandakumar **(Utah/Sydney): *Categorification via
blocks of modular representations for sl_m*

Abstract: We give a characteristic p analogue of [BFK]'s result categorifying representations of sl_2 using singular category O.

**Bach Nguyen **(LSU): *Non-commutative Discriminants and Poisson Geometry
*

Abstract: In this talk, we will present a general method for
computing discriminants of noncommutative algebras via Poisson
primes. It will be illustrated with the specializations of the
algebras of quantum matrices at roots of unity. If time permits, we'll
also discuss a more general case, the quantum Schubert cell
algebras. This is a joint work with Kurt Trampel, and Milen Yakimov.

** Emily Norton **(Kansas State): *BGG resolutions for Cherednik algebras*

**Mike Reeks **(UVA): *The cocenter of the Hecke-Clifford algebras*

Abstract: The degenerate affine Hecke-Clifford algebras, and the closely related family of spin affine Hecke algebras, are essential to the study of the spin representation theory of classical Weyl groups. In studying the representation theory of these algebras, it is useful to have a description of the cocenter: the quotient of the algebra by the linear subspace spanned by the commutators. In this talk, we will determine a linear basis for the cocenter of the Hecke-Clifford algebras in the classical types, adapting methods used by Ciubotaru and He to solve the corresponding problem for degenerate affine Hecke algebras.

** José Simental-Rodriguez **(Northeastern): *Harish-Chandra
bimodules for rational Cherednik algebras*

Abstract: I present a classification of irreducible fully supported Harish-Chandra bimodules for RCA's associated to any complex reflection group.

** David Vella **(Skidmore): *Nilpotent Orbits for Borel
Subgroups*

Abstract: Let B a Borel subgroup of modality
zero. Defining equations for the nilpotent B-orbits are determined, as
well as dimensions and closure ordering on the orbits. Detailed
abstract.

** Jieru Zhu **(Oklahoma): *Presenting the hyperoctahedral
Schur algebra*

Abstract: We showed a Schur-Weyl duality in the classical type B/C case and further a presentation of the hyperoctahedral Schur algebra.