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Members

Affiliated Members

Jan de Gier Nora Ganter Alex Ghitza Christian Haesemeyer Jack Hall

Thomas Quella David Ridout Chenyan Wu Paul Zinn-Justin

Peter McNamara‘s research is in categorical and geometric representation theory. He has worked on a range of topics including quantum groups and their categorifications, perverse sheaves and Schubert varieties, algebraic combinatorics and p-adic groups.

Arun Ram’s research is in the area of Combinatorial Representation Theory. Topics he has worked on include tableaux combinatorics, crystals, diagram algebras, Lie theory, quantum groups, Hecke algebras, Schubert calculus, K-theory and cohomology of flag varieties and affine flag varieties. Current projects include constructions of representations of quantum affine algebras and a study of the combinatorics of double affine Hecke algebras and Macdonald polynomials.

Kari Vilonen‘s research is in the areas of real groups, the Langlands program, and related algebraic geometry. He has worked on several aspects of the geometric Langlands program and on more foundational questions on perverse sheaves and D-modules from the microlocal point of view. His research on real groups, joint with Schmid, includes the proof of the Barbasch-Vogan conjecture and a conjectural theory of Hodge structures on representations of real groups.

Ting Xue‘s research is in the areas of representation theory and algebraic groups. She has worked on questions related to geometry of nilpotent orbits and Springer theory, including small or bad characteristics. She is also interested in combinatorics arising from representation theory.

Yaping Yang‘s research lies in the area of Lie algebras, geometric representation theory, quantum groups, and the related geometry and topology. Her current work includes Knizhnik-Zamolodchikov equations, Cohomological Hall algebras, and vertex algebras associated to toric Calabi-Yau 3-folds.

Gufang Zhao‘s research lies at the interface between algebraic geometry and representation theory. More specifically, he has been working on projects concerning derived category of coherent sheaves, oriented cohomology theories of algebraic varieties, and their applications in representation theory. He is also fond of varieties of local systems and instantons, quantum integrable systems, and related aspects in mathematical physics.

Selected papers

- - M. Lanini and P. J. McNamara, Singularities of Schubert varieties within a right cell. arXiv2003.08616.
  - P. J. McNamara, Representation Theory of Geometric Extension Algebras, arXiv:1701.07949.
  - P. J. McNamara. Representations of Khovanov-Lauda-Rouquier Algebras III: Symmetric Affine Type. arXiv. Math. Z. 287 (2017), no. 1-2, 243–286.
  - Peter J. McNamara, Finite Dimensional Representations of Khovanov-Lauda-Rouquier Algebras I: Finite Type, arXiv. J. Reine Angew. Math. 707 (2015), 103–124.
  - D. George, Arun Ram, J. Thompson and R. Volkas, Symmetry breaking, subgroup embeddings and the Weyl group, arXiv1203.1048, Physical Review D 87 105009 (2013) [14 pages]
  - Z. Daugherty, Arun Ram and R. Virk, Affine and degenerate affine BMW algebras: The center, arXiv1105.4207, Osaka J. Math 51 (2014), 257-283.
  - A. Kleshchev, A. Mathas, and Arun Ram, Universal Specht modules for cyclotomic Hecke algebras , arXiv1102.3519, Proc. London Math. Soc. (3) 105 (2012) 1245-1289.
  - P. Diaconis and Arun Ram, A probabilistic interpretation of the Macdonald polynomials, arXiv1007.4779, The Annals of Probability 40 (2012) Vol. 40 No. 5, 1861-1896.
  - Roman Bezrukavnikov and Kari Vilonen, Koszul Duality for Quasi-split Real Groups, arXiv:1510.08343, Under revision for Invent. Math.
  - Masaki Kashiwara and Kari Vilonen, Microdifferential systems and the codimension-three conjecture. Ann. of Math. (2) 180 (2014) no. 2, 573-620.
  - Wilfried Schmid and Kari Vilonen, Hodge theory and unitary representations of reductive Lie groups. Frontiers of mathematical sciences, 397-420, Int. Press, Somerville, MA, 2011, arXiv
  - Kari Vilonen and Ting Xue, Character sheaves for symmetric pairs, arXiv:1806.02506
  - Tsao-Hsien Chen, Kari Vilonen and Ting Xue, Springer correspondence for the split symmetric pair in type A , Compos. Math. 154 (2018), no. 11, 2403-2425. arXiv
  - Tsao-Hsien Chen, Kari Vilonen and Ting Xue, On the cohomology of Fano varieties and the Springer correspondence, With an appendix by Dennis Stanton. Adv. Math. 318 (2017), 515-533. arXiv.
  - Ting Xue, Combinatorics of the Springer correspondence for classical Lie algebras and their duals in characteristic 2. Adv. Math. 230 (2012) no. 1, 229–262.
  - Marc Levine, Yaping Yang, and Gufang Zhao, Algebraic Elliptic cohomology theory and flops 1, appendix by Joël Riou. Mathematische Annalen volume 375, pages 1823–1855 (2019).
  - Miroslav Rapcak, Yan Soibelman, Yaping Yang, and Gufang Zhao, Cohomological Hall algebras, vertex algebras and instantons. Communications in Mathematical Physics volume 376, pages 1803–1873 (2020)
  - Yaping Yang, and Gufang Zhao, The cohomological Hall algebras for a preprojective algebra. Proc. Lond. Math. Soc. 116, 1029-1074.

Representation theory seminar 2022, Semester 1

Organisers: Kari Vilonen, Ting Xue, Yaping Yang

Topics: This semester our learning seminar will focus on real groups. On alternate weeks we will have research talks by the members of our group.

Time: Thursdays 3:15pm-5pm.

Location: Peter Hall 101 and Zoom

Please contact one of the organisers to be added to the mailing list.

Schedule

Jun 9 Arun Ram Is there a Kac-Moody-like presentation of toroidal algebras?

(Evan Williams Theatre)

Abstract: Ion-Sahi have pointed to a Coxeter like presentation of the double affine Artin group (DAArt). I will explain how this presentation could be discovered from a matrix representation of the double affine Weyl group (DAWG) which naturally exhibits the action of $SL_2(\mathbb{Z})$ (acting on the DAWG) by automorphisms. The position of the Heisenberg group inside the DAWG is clearly visible in this representation. The Coxeter-like presentation uses three affine Dynkin diagrams of the same type glued together along the common finite Dynkin diagram and a single additional “superglue” relation. I wonder if these results could be extended to provide a Kac-Moody-like presentation of quantum toroidal algebras.

Jun 2 Peter Hall 213

3-4pm Wille Liu (Max-Planck-Institut für Mathematik) Trigonometric Knizhnik-Zamolodchikov functor and affine Hecke algebra

Abstract: The Knizhnik–Zamolodchikov (KZ) equations arised from the theory of Wess-Zumino-Witten conformal blocks on the Riemann sphere associated with a Lie group. Inspired by the Schur–Weyl duality, Cherednik introduced in the 90s a version of KZ equations for graded affine Hecke algebras. The operators (called Dunkl operators) that show up in the KZ equations can be conveniently organised into an associative algebra, called trigonometric double affine Hecke algebra (DAHA).

In this talk, I will explain certain representation-theoretic aspect of the trigonometric DAHA with emphasis on its KZ equations.

4-5pm Cheng-Chiang Tsai (Academia Sinica) Harmonic analysis of p-adic groups and affine Springer theory

Abstract: In this presentation we give a survey about less well-known connections between harmonic analysis of p-adic groups and affine Springer theory. One upshot is the so-called “homogeneity property” in harmonic analysis which describes the asymptotic behavior of the (co)homology affine Springer fiber as the dimension increases. The other upshot is the theory of endoscopy, which features global methods in number theory, has been developed a lot on the side of p-adic groups since the 80s, and occasionally serves as heuristic for graded and affine Springer theory.

May 26 Travis Scrimshaw (Osaka City University) Probability measures from representation theory

Howe duality for the general linear group $GL_n$ can be described as the fact that the biregular representation of $GL_n$ , where it acts on itself on the right and left, decomposes into a multiplicity free sum of $(GL_n \times GL_n)$ representations. In terms of characters, this yields the Cauchy identity. By dividing both sides by the product, we obtain the famous Schur (probability) measure on partitions. In this talk, we will examine a similar measure constructed from skew Howe duality and discuss the relationship with Krawtchouk polynomials, a family of classical orthogonal polynomials in the Askey scheme. No prior background knowledge will be assumed. This is based on joint work with Anton Nazarov and Olga Postnova.

May 19 Gufang Zhao Shifted symplectic structures, mapping stacks, and virtual fundamental classes

The first half of the talk is an informal review of shifted symplectic stacks following Pantev, Toën, Vaquié, and Vezzosi. Familiar examples in representation theory will be discussed. The second half follows the work of Oh and Thomas to introduce virtual fundamental classes for -1 and -2-shifted symplectic stacks. I will also discuss my work in progress joint with Yalong Cao, where we apply this construction to study quantum cohomology of some moduli spaces.

May 12 Qixian Zhao (Utah) Reducibility of standard representations: Examples. Notes

I will rephrase the irreducibility criterion in algebraic terms (i.e. without exponentials), demonstrate the proof of the irreducibility criterion explicitly on concrete examples (SU(2,1), SL(3,R) and SL(2,R)), and answer some questions raised in previous talks along the way.

May 5 Chenyan Wu Theta correspondence and poles of Eisenstein series

Let $\pi$ be a cuspidal automorphic representation of a classical group. First I will define two invariants attached to $\pi$ , namely, the lowest occurrence of $\pi$ in the theta correspondence and the location of the maximal pole of an Eisenstein series built from $\pi$ and a character. Then I will show a relation between the two invariants and talk about an implication of this result on certain global Arthur packets.

Apr 28 Qixian Zhao (Utah) Reducibility of standard representations, continued Notes

Apr 14 Peter McNamara Sheaves behaving badly

Given a complex algebraic variety, we construct a canonical sheaf with mod p coefficients that tells us something about the geometry of the singularities and what possible resolutions they have. We discuss how badly these sheaves behave for Schubert varieties. This is motivated
by the desire to study parity sheaves in geometric representation theory.

Apr 7 Qixian Zhao (Utah) Reducibility of standard representations Notes

I will present the irreducibility criterion for standard K-equivariant D-modules in the linear case, mostly based on the argument in Hecht-Milicic-Schmid-Wolf. I will present the statement and an outline/idea of proof, and (if time permits) compute the SU(2,1) example.

Apr 7 Kari Vilonen Real groups

I will finish the geometrization of Langlands parameters and will introduce the Arthur parameters. I will also make some general remarks about representations leading to the talk of Qixian.

Mar 31 Yaping Yang Quantum groups at roots of 1

I will start with Lusztig’s quantum groups at roots of unity and explain the quantum Frobenius homomorphism and the Steinberg tensor product theorem. I will then talk about a family of quantum groups associate to Morava E-theories. I will also explain the quantum Frobenius homomorphisms among these quantum groups constructed by Gufang and myself. The main ingredient in constructing these Frobenii is the transchromatic character map of Hopkins, Kuhn, Ravenal, and Stapleton. This is based on my joint work with Gufang Zhao.

Mar 24 Kari Vilonen Real groups Notes

I will briefly discuss the Fourier transform and nearby cycles in response to questions asked during Ting’s talk on March 17. The main point of this lecture is to explain how to view Langlands parameters for real groups geometrically. This was first explained in a book by Adams, Barbasch, Vogan and in a slightly different formulation in a paper by Adams and du Cloux.

Mar 17 Ting Xue Character sheaves and Hecke algebras

We discuss character sheaves in the setting of graded Lie algebras. Via a nearby cycle construction irreducible representations of Hecke algebras of complex reflection groups at roots of unity enter the description of character sheaves. Recent work of Lusztig and Yun relates character sheaves to irreducible representations of trigonometric double affine Hecke algebras. We will explain the connection between the work of Lusztig-Yun and our work, and discuss some conjectures arising from this connection. If time permits, we will discuss applications to cohomology of Hessenberg varieties and affine Springer fibres. This is based on joint work with Kari Vilonen and partly with Tsao-Hsien Chen and Misha Grinberg.

Mar 10 Kari Vilonen Real groups Notes

This is the first talk in learning seminar on real groups. In this talk I will give a broad outline of the state of representation theory of real groups. I will also discuss possible future research directions.

Mar 3 Organisational meeting

Past Seminars:

Representation Theory Student Seminar 2021 Semester 2 and 2022 Semester 1

Organisers: Ting Xue, Yaping Yang, Gufang Zhao

Student organisers (2022S1): Weiying Guo and Davood Nejaty

Overview

This is a learning seminar on representation theory and related topics, aimed at 3rd year undergraduate students, MSc students, as well as PhD students in mathematics or related fields.

In the learning seminar, participants are expected to learn a topic based on the reference given, and present their work to the other participants. The audience are expected to ask questions and make comments during the presentation. Discussions tangential to the topics are welcome. The idea is for all participants to actively learn and discuss together.

This semester the topic will be Lie groups and their representations. We will follow the chapter on Lie groups by Graeme Segal in the reference below.

Reference: Carter, Roger; Segal, Graeme; Macdonald, Ian, Lectures on Lie groups and Lie algebras. With a foreword by Martin Taylor. London Mathematical Society Student Texts, 32. Cambridge University Press, Cambridge, 1995.

Time: Mondays 11-12

Location: Peter Hall 107 and Zoom (please contact the organisers for zoom link)

Exercises

Schedule (2022S1)

May 30 Beaudon Anasson Maximal compact subgroups

May 23 Zhongtian Chen Compact groups and integration, continued

May 16 Zhongtian Chen Compact groups and integration

May 9 Kevin Fergusson Fourier series and Representation theory

May 2 Davood Nejaty Review, continued

Grace Yuan Bruhat decomposition Notes

Apr 11 Organisational meeting and review of previous topics

Schedule (2021S2)

Time: Wednesdays 3:15-4:15pm

Location: Zoom (please contact the organisers for link)

Dec 1 Linfeng Wei Lie’s theorems, continued. Notes

Nov 24 Linfeng Wei Lie’s theorems

Oct 27 Adam Monteleone Smooth manifolds, tangent space, one parameter subgroups and the exponential map, continued. Notes

Oct 20 Adam Monteleone Smooth manifolds, tangent space, one parameter subgroups and the exponential map, continued.

Oct 6 Adam Monteleone Smooth manifolds, tangent space, one parameter subgroups and the exponential map

Sep 29 Abraham Zhang Diagonalisation and maximal tori, continued. Notes

Sep 22 Abraham Zhang Diagonalisation and maximal tori

Sep 8 Grace Yuan Polar decomposition, Graham-Schmidt

Sep 1 Yifan Guo Homogeneous spaces, continued. Notes

Aug 25 Yifan Guo Homogeneous spaces Notes

Aug 18 Eskander Salloum $SU_2,SO_3,SL_2\mathbb{R}$ , continued. Notes

Aug 11 Eskander Salloum $SU_2,SO_3,SL_2\mathbb{R}$

Aug 4 Benjamin Gerraty Examples


Date	Topics	Reference	Speaker
Aug 4	Examples	[S] 1	Benjamin Gerraty
Aug 11/18	$SU_2,SO_3,SL_2\mathbb{R}$	[S] 2	Eskander Salloum
Aug 25/Sep 1	Homogeneous spaces	[S] 3	Yifan Guo
Sep 8	Polar decomposition, Graham-Schmidt, Bruhat decomposition	[S] 4	Grace Yuan
Sep 22/29	Diagonalisation and maximal tori	[S] 4	Abraham Zhang
Oct 6/20/27	Smooth manifolds, tangent space, one parameter subgroups and the exponential map	[S] 5	Adam Monteleone
Nov 24/Dec1	Lie’s theorems	[S] 5	Linfeng Wei
May 9/2022	Fourier series and Representation theory	[S] 6	Kevin Fergusson
May 16, 23/2022	Compact groups and integration	[S] 7	Zhongtian Chen
May 30/2022	Maximal compact subgroups	[S] 8	Beaudon Anasson
	The Peter-Weyl theorem I	[S] 9	Haris Rao
	The Peter-Weyl theorem II	[S] 9	Haris Rao
	Functions on $\mathbb{R}^n$ and $S^{n-1}$	[S] 10	Yuhan Gai
	Induced representations	[S] 11
	The complexification of a compact group	[S] 12
	The unitary groups and the symmetric groups	[S] 13	Weiying Guo
	The Borel-Weil theorem	[S] 14	Ali Khalili
References	[S] Graeme Segal, Lie groups, in Carter, Roger; Segal, Graeme; Macdonald, Ian, *Lectures on Lie groups and Lie algebras.* With a foreword by Martin Taylor. London Mathematical Society Student Texts, 32. Cambridge University Press, Cambridge, 1995.

Past seminar

Special lectures by Arun Ram summer 2021/22

Student seminar 2020S2 and 2021S1

Dougal Davis (University of Edinburgh), Mar. 30, 2020-Apr. 3, 2020

Xinwen Zhu (Caltech), November 24-Dec 1, 2019

Cheng-Chiang Tsai (Stanford University), Nov 19-Dec 8, 2019

Geordie Williamson (Sydney), October 26-28, 2019

Bill Casselman (University of British Columbia), Oct 24-27, 2019

Emily Norton (MPIM), February 19-March 2, 2019

Luca Migliorini (University of Bologna), December 3-9, 2018

Cheng-Chiang Tsai (Stanford University), Nov 3-Dec 1, 2018

Xinwen Zhu (Caltech), August 23-30, 2018

Anthony Henderson (Sydney), August 21-24, 2018

Jessica Fintzen (University of Michigan), November 27-December 3, 2017

Sam Raskin (University of Texas at Austin) , September 12-22, 2017

Takuro Mochizuki (RIMS, Kyoto University), September 4-15, 2017

Carl Mautner (UC Riverside), August 16-26, 2017

Kari Vilonen was awarded a highly prestigious ARC Laureate Fellowship on Real groups and the Langlands program.
Peter McNamara gave a talk on June 25, 2020 at the informal Friday seminar at U Sydney.
Nora Ganter, Peter McNamara, Yaping Yang and Gufang Zhao are co-organising (with Masoud Kamgarpour and Peng Shan) the MATRIX workshop Frontiers in Representation Theory, 14-25 February 2022.
Ting Xue will be a speaker at AMSI Winter School 2020, New directions in representation theory, University of Queensland. (Postponed.)
Yaping Yang gave a talk titled “Cohomological Hall algebras and perverse coherent sheaves on toric Calabi-Yau 3-folds” at the GRT at Home seminar on 23 June, 2020.
Nora Ganter, Yaping Yang, and Gufang Zhao co-organized with Daniel Berwick Evans and Theo Johnson-Freyd the workshop on elliptic cohomology and physics 25-28 May 2020.
Arun Ram gave a performance of “Mendelssohn Salon 1828” with pianist Michael Leslie on
12 March 2020 at Tempo Rubato.
From Dec 2019 to Feb.29 2020, Yaping Yang and Gufang Zhao visited the Kavli Institute for the Physics and Mathematics of the Universe (IPMU), Japan. During the visit, Gufang Zhao gave a seminar talk titled “Cohomological Hall algebras and their representation theories” at the Mathematics and String Theory Seminar at IPMU.
The week of December 16-20 2019, Yaping Yang and Gufang Zhao co-organised the workshop on “Geometric Representation Theory and Quantum Field Theory” together with Hiraku Nakajima (IPMU), Peng Shan (Tsinghua), Wenbin Yan (Tsinghua) at TSIMF, Sanya, China.
Yaping Yang and Gufang Zhao visited the Perimeter Institute for Theoretical Physics, Waterloo, Canada during Feb-Mar 2019. During the week of February 25-March 1, 2019, Yaping Yang co-organised the workshop “Cohomological Hall algebras in Mathematics and Physics” at Perimeter Institute (with Kevin Costello (PI) and Yan Soibelman (KSU)).
Yaping Yang received an ARC Discovery Early Career Award (DE 190101231) in Dec 2018.
Gufang Zhao received an ARC Discovery Early Career Award (DE 190101222) in Dec 2018.