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Members

Dougal Davis Yau Wing Li Peter McNamara Arun Ram

Kari Vilonen Ting Xue Yaping Yang Gufang Zhao

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, Inventiones Mathematicae, 226 (2021),139-193.
  - 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 2

Organisers: Dougal Davis, Kari Vilonen, Ting Xue

Topics: Real groups, affine Springer fibers.

Time: Thursdays 3:15pm-5pm.

Location: Peter Hall 107

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

Schedule

Oct 6 Yaping Yang (Melbourne) Affine Springer fibers and Hitchin fibers, continued

Sep 29 Qixian Zhao (Utah) Around Adams-Barbasch-Vogan

I will first explain what strong real forms and extended groups are. Then, I will explain the statement of local Langlands correspondence over $\mathbb{R}$ , how it interacts with the localization picture, and how this interaction leads to Vogan duality. I will use the $SL_2$ example to demonstrate. Definitions and statements will (hopefully) be given precisely, but there will be no proofs.

Sep 22 No seminar

Sep 15 Davood Nejaty and Linfeng Wei (Melbourne) Recap of real reductive groups and Beilinson-Bernstein localisation

Sep 8 Yaping Yang (Melbourne) Affine Springer fibers and Hitchin fibers, continued

Sep 1 Yaping Yang (Melbourne) Affine Springer fibers and Hitchin fibers,

In my talk, I will discuss a relation between the homogeneous affine Springer fibers and Hitchin fibers over the weighted projective line. I will mainly follow Section 6.6 of the paper “Geometric representations of graded and rational Cherednik algebras” by Oblomkov and Yun.

Notes, Masoud’s notes on affine Springer fibres

Aug 25 Dougal Davis (Melbourne) The local Langlands correspondence for real groups and other stories

In this talk, I will give a rough introduction to the main themes of the real groups strand of our learning seminar, with the intention that the details will be fleshed out in future talks. I will sketch the Adams-Barbasch-Vogan approach to the local Langlands correspondence for real groups, which gives a duality between Grothendieck groups of representations of a real group on one side and of perverse sheaves on a stack of Langlands parameters on the dual side. I will also explain how this category of perverse sheaves is related to coherent sheaves on a different stack of Langlands parameters by the $S^1$ -equivariance story of Ben-Zvi and Nadler, resolving the apparent clash with Matt Emerton’s talk two weeks ago. If time permits, I may also say a few words about Arthur representations and conjectures concerning them.

Aug 18 Volker Heiermann (Université d’Aix-Marseille) Affine Hecke Algebras, Intertwining Operators and Types

To put into relation a Bernstein component of the category of complex smooth representations of a p-adic reductive group and representations of Affine Hecke Algebras, one can in principle use two methods: the theory of types of Bushnell-Kutzko or the intertwining algebra of a certain projective generator of this category. From the first method, it is rather clear that unitarity is preserved, which is less immediate for the second method. The aim of the talk is to explain the situation, as time permits.

Aug 11 Matthew Emerton (Chicago) Categorical perspectives on the arithmetic Langlands program

A categorical perspective, which has long been the norm in the geometric Langlands program, has recently emerged in the arithmetic Langlands program as well. I will explain some of the ideas related to this perspective, including the Fargues–Scholze conjecture; conjectures and results (due variously to Ben-Zvi–Chen–Helm–Nadler, Hellmann, Zhu, as well as the speaker and Toby Gee) regarding fully faithful functors from categories of representations of p-adic reductive groups to categories of coherent sheaves on stacks of Langlands parameters; and conjectural descriptions of the cohomology of locally symmetric congruence quotients (e.g. Shimura varieties) in terms of these categorical ideas.

Aug 4 joint Representation Theory/Number theory seminar

Chen Wan (Rutgers) A multiplicity formula of K-types

In this talk, by using the trace formula method, I will prove a multiplicity formula of K-types for all representations of real reductive groups in terms of the Harish-Chandra character.

Jul 28 David Ridout (Melbourne) Inverse quantum hamiltonian reduction

Let g be a (complex, finite-dimensional) simple Lie algebra. Quantum hamiltonian reduction is a cohomological functor from category O for an affine vertex algebra $V_k(g)$ to the corresponding category for a W-algebra $W_k(g;f)$ . This functor depends on the orbit of a nilpotent f in g, as does the W-algebra. Unfortunately, it is only well understood for regular and minimal orbits.

A new approach to W-algebra module categories starts with Adamovic’s recent observation that there exist functors going the other way: from the W-algebra to the affine vertex algebra (or more typically, to another W-algebra). Interestingly, these functors do not naturally land in category O but in a much larger category. More interestingly, it appears that this approach will work for general nilpotent orbits.

We currently only understand these “inverse functors” in a few examples. I will attempt to explain some of what we know (and maybe some of what we believe) when $g=sl_2$ .

Past Seminars:

Representation Theory Student Seminar 2021 Semester 2 and 2022 Semester 1&2

Organisers: Ting Xue, Yaping Yang, Gufang Zhao

Student organiser (2022S2): 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: Wednesdays 11-12

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

Exercises

Schedule (2022S2)

Time: Wednesdays 11-12

Oct 4

Sep 21 Amit Ben Harim Induced representations, continued

Sep 14 Amit Ben Harim Induced representations

Sep 7 Haris Rao The Peter-Weyl theorem, continued

Aug 31 Yuhan Gai Functions on $\mathbb{R}^n$ and $S^{n-1}$

Aug 24 Cancelled

Aug 17 Haris Rao The Peter-Weyl theorem

Schedule (2022S1)

Time: Mondays 11-12

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

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

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, 2021	Examples	[S] 1	Benjamin Gerraty
Aug 11/18, 2021	$SU_2,SO_3,SL_2\mathbb{R}$	[S] 2	Eskander Salloum
Aug 25/Sep 1, 2021	Homogeneous spaces	[S] 3	Yifan Guo
Sep 8, 2021	Polar decomposition, Graham-Schmidt, Bruhat decomposition	[S] 4	Grace Yuan
Sep 22/29, 2021	Diagonalisation and maximal tori	[S] 4	Abraham Zhang
Oct 6/20/27, 2021	Smooth manifolds, tangent space, one parameter subgroups and the exponential map	[S] 5	Adam Monteleone
Nov 24/Dec1, 2021	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
Aug 17, Sep 7, 2022	The Peter-Weyl theorem	[S] 9	Haris Rao
Aug 31, 2022	Functions on $\mathbb{R}^n$ and $S^{n-1}$	[S] 10	Yuhan Gai
Sep 14/21, 2022	Induced representations	[S] 11	Amit Ben Harim
	The complexification of a compact group	[S] 12	Ali Khalili
	The unitary groups and the symmetric groups	[S] 13
	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

Andrea D’Agnolo (Padova), May 20-Jun 17, 2022

Wille Liu (MPIM), May 30-Jun 10, 2022

Cheng-Chiang Tsai (Academia Sinica), May 29-Jun 11, 2022

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

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

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

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

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

Emily Norton (MPIM), Feb 19-Mar 2, 2019

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

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

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

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

Jessica Fintzen (University of Michigan), Nov 27-Dec 3, 2017

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

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

Carl Mautner (UC Riverside), Aug 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.