Home page
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
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 padic 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, Ktheory 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 Dmodules from the microlocal point of view. His research on real groups, joint with Schmid, includes the proof of the BarbaschVogan 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 KnizhnikZamolodchikov equations, Cohomological Hall algebras, and vertex algebras associated to toric CalabiYau 3folds.
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

 A full list of each member’s publications can be found on their individual webpages.

 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 KhovanovLaudaRouquier Algebras III: Symmetric Affine Type. arXiv. Math. Z. 287 (2017), no. 12, 243–286.
 Peter J. McNamara, Finite Dimensional Representations of KhovanovLaudaRouquier 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), 257283.
 A. Kleshchev, A. Mathas, and Arun Ram, Universal Specht modules for cyclotomic Hecke algebras , arXiv1102.3519, Proc. London Math. Soc. (3) 105 (2012) 12451289.
 P. Diaconis and Arun Ram, A probabilistic interpretation of the Macdonald polynomials, arXiv1007.4779, The Annals of Probability 40 (2012) Vol. 40 No. 5, 18611896.
 Roman Bezrukavnikov and Kari Vilonen, Koszul Duality for Quasisplit Real Groups, arXiv:1510.08343, Inventiones Mathematicae, 226 (2021),139193.
 Masaki Kashiwara and Kari Vilonen, Microdifferential systems and the codimensionthree conjecture. Ann. of Math. (2) 180 (2014) no. 2, 573620.
 Wilfried Schmid and Kari Vilonen, Hodge theory and unitary representations of reductive Lie groups. Frontiers of mathematical sciences, 397420, Int. Press, Somerville, MA, 2011, arXiv
 Kari Vilonen and Ting Xue, Character sheaves for symmetric pairs, arXiv:1806.02506
 TsaoHsien Chen, Kari Vilonen and Ting Xue, Springer correspondence for the split symmetric pair in type A , Compos. Math. 154 (2018), no. 11, 24032425. arXiv
 TsaoHsien 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), 515533. 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, 10291074.

Representation theory seminar 2022 Semester 2
Organisers: Dougal Davis, Kari Vilonen, Ting Xue
Topics: Real groups, affine Springer fibers.
Time: Thursdays 3:15pm5pm.
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 AdamsBarbaschVogan
I will first explain what strong real forms and extended groups are. Then, I will explain the statement of local Langlands correspondence over , how it interacts with the localization picture, and how this interaction leads to Vogan duality. I will use the 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 BeilinsonBernstein 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 AdamsBarbaschVogan 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 equivariance story of BenZvi 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’AixMarseille) Affine Hecke Algebras, Intertwining Operators and Types
To put into relation a Bernstein component of the category of complex smooth representations of a padic reductive group and representations of Affine Hecke Algebras, one can in principle use two methods: the theory of types of BushnellKutzko 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 BenZvi–Chen–Helm–Nadler, Hellmann, Zhu, as well as the speaker and Toby Gee) regarding fully faithful functors from categories of representations of padic 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 Ktypes
In this talk, by using the trace formula method, I will prove a multiplicity formula of Ktypes for all representations of real reductive groups in terms of the HarishChandra character.
Jul 28 David Ridout (Melbourne) Inverse quantum hamiltonian reduction
Let g be a (complex, finitedimensional) simple Lie algebra. Quantum hamiltonian reduction is a cohomological functor from category O for an affine vertex algebra to the corresponding category for a Walgebra . This functor depends on the orbit of a nilpotent f in g, as does the Walgebra. Unfortunately, it is only well understood for regular and minimal orbits.
A new approach to Walgebra module categories starts with Adamovic’s recent observation that there exist functors going the other way: from the Walgebra to the affine vertex algebra (or more typically, to another Walgebra). 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 .
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 1112
Location: Peter Hall 107 and Zoom (please contact the organisers for zoom link)
Exercises
Schedule (2022S2)
Time: Wednesdays 1112
Oct 4
Sep 21 Amit Ben Harim Induced representations, continued
Sep 14 Amit Ben Harim Induced representations
Sep 7 Haris Rao The PeterWeyl theorem, continued
Aug 31 Yuhan Gai Functions on and
Aug 24 Cancelled
Aug 17 Haris Rao The PeterWeyl theorem
Schedule (2022S1)
Time: Mondays 1112
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 3:154:15pm
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, GrahamSchmidt
Sep 1 Yifan Guo Homogeneous spaces, continued. Notes
Aug 25 Yifan Guo Homogeneous spaces Notes
Aug 18 Eskander Salloum , continued. Notes
Aug 11 Eskander Salloum
Aug 4 Benjamin Gerraty Examples
Date  Topics  Reference  Speaker 
Aug 4, 2021  Examples  [S] 1  Benjamin Gerraty 
Aug 11/18, 2021  [S] 2  Eskander Salloum  
Aug 25/Sep 1, 2021  Homogeneous spaces  [S] 3  Yifan Guo 
Sep 8, 2021  Polar decomposition, GrahamSchmidt, 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 PeterWeyl theorem  [S] 9  Haris Rao 
Aug 31, 2022  Functions on and  [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 BorelWeil 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
Andrea D’Agnolo (Padova), May 20Jun 17, 2022
Wille Liu (MPIM), May 30Jun 10, 2022
ChengChiang Tsai (Academia Sinica), May 29Jun 11, 2022
Dougal Davis (University of Edinburgh), Mar 30Apr 3, 2020
Xinwen Zhu (Caltech), Nov 24Dec 1, 2019
ChengChiang Tsai (Stanford University), Nov 19Dec 8, 2019
Geordie Williamson (Sydney), Oct 2628, 2019
Bill Casselman (University of British Columbia), Oct 2427, 2019
Emily Norton (MPIM), Feb 19Mar 2, 2019
Luca Migliorini (University of Bologna), Dec 39, 2018
ChengChiang Tsai (Stanford University), Nov 3Dec 1, 2018
Xinwen Zhu (Caltech), Aug 2330, 2018
Anthony Henderson (Sydney), Aug 2124, 2018
Jessica Fintzen (University of Michigan), Nov 27Dec 3, 2017
Sam Raskin (University of Texas at Austin) , Sep 1222, 2017
Takuro Mochizuki (RIMS, Kyoto University), Sep 415, 2017
Carl Mautner (UC Riverside), Aug 1626, 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 coorganising (with Masoud Kamgarpour and Peng Shan) the MATRIX workshop Frontiers in Representation Theory, 1425 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 CalabiYau 3folds” at the GRT at Home seminar on 23 June, 2020.
 Nora Ganter, Yaping Yang, and Gufang Zhao coorganized with Daniel Berwick Evans and Theo JohnsonFreyd the workshop on elliptic cohomology and physics 2528 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 1620 2019, Yaping Yang and Gufang Zhao coorganised 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 FebMar 2019. During the week of February 25March 1, 2019, Yaping Yang coorganised 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.