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
Thomas Quella David Ridout Chenyan Wu Paul ZinnJustin
We are part of the Pure mathematics group.
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 classical symmetric spaces. With an appendix by Dennis Stanton. Represent. Theory 26 (2022), 10971144.
 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 2023 Semester 2
Organisers: Dougal Davis, Kari Vilonen, Ting Xue
Place and time: Peter Hall 162, Thursdays 3:15 – 5:00pm
Please contact one of the organisers to be added to the mailing list.
Upcoming Seminars
October 5: Dougal Davis (University of Melbourne) Mixed Hodge modules and real groups I
The aim of this lecture series is to explain recent progress of myself and Kari Vilonen on the deep problem of determining the unitary representations of a real reductive Lie group. Our main theorem is that unitary representations are completely governed by Hodge theory on the flag variety (originally conjectured by Schmid and Vilonen over 10 years ago). In these lectures, which are supposed to be accessible to students, I will explain our results (and a bit about the proofs), as well as some of the necessary background on real groups, Hodge theory and BeilinsonBernstein localisation.
Reference: D. Davis and K. Vilonen, Unitary representations of real groups and localization theory for Hodge modules.
In the first lecture, I will give an introduction to the unitary representation theory of real groups, focusing on the example of . I will try to give some idea of the shape of what is known in general, and how our work on Hodge theory enters this picture.
October 12: Dougal Davis (University of Melbourne) Mixed Hodge modules and real groups II
In this second lecture, which is a geometric interlude, I will give an introduction to the general theory of mixed Hodge modules. I will recall some basic results in classical Hodge theory, explain how Hodge modules are supposed to generalise this, and touch on some of the most important technical aspects for our story.
October 19: Dougal Davis (University of Melbourne) Mixed Hodge modules and real groups III
The aim of this third lecture is to state the main results from our recent paper. The starting point is the classical BeilinsonBernstein localisation theorem, which relates Lie algebra representations to Dmodules on the flag variety. I will briefly recall how this goes and explain how the statements can be refined from Dmodules to mixed Hodge modules. I will also explain the full conjecture of Schmid and Vilonen (which is still a conjecture) and the closely related Hodgetheoretic criterion for unitarity (which we have proved).
October 26: Dougal Davis (University of Melbourne) Mixed Hodge modules and real groups IV
In this fourth and final lecture, I will explain some of the ingredients in the proofs of our main theorems.
Past Seminars
September 14: Arun Ram (University of Melbourne) Introduction to Hessenberg varieties and relations to polynomials from combinatorics
Learning seminar on Hessenberg varieties, talk 3
I will define Hessenberg varieties and give some examples. Then I will explain some of the “combinatorial” results (ShareshianWachs, BrosnanChow, AbeHoriguschiet al, PrecupSommers) about the cohomology of Hessenberg varieties coming from bsubmodules of the adjoint representation. I will endeavour to explain the structural/conceptual relationship between this “noninteresting” setting and the “interesting” setting that Yau Wing has introduced us to. As Kari explained, the “interesting” setting is when the motive of the Hessenberg variety shows features that do not appear in the Springer resolution and the Grothendieck simultaneous resolution. For this talk I will move away from the Springer resolution and the Grothendieck simultaneous resolution but I will stick to the very fascinating and amazing “noninteresting” setting.
September 7:
Learning seminar on Hessenberg varieties, talk 2
Yau Wing Li (University of Melbourne) On the cohomology of Fano varieties and the Springer correspondence, continued
Dougal Davis and Kari Vilonen (University of Melbourne) Purity of equivalued affine Springer fibers
We will make some comments on the paper of the same title by Goresky, Kottwitz and MacPherson.
August 31: Yau Wing Li (University of Melbourne) On the cohomology of Fano varieties and the Springer correspondence
Learning seminar on Hessenberg varieties, talk 1
I will discuss the paper with the same title by TsaoHsien Chen, Kari Vilonen and Ting Xue. The main result is a computation of the cohomology of the Fano varieties of kplanes inside a smooth intersection of quadrics. The computation works by realising these as certain Hessenberg varieties and relating their cohomology to Springer theory for symmetric spaces by a Fourier transform.
Reference: T.H. Chen, K. Vilonen, T. Xue, On the cohomology of Fano varieties and the Springer correspondence.
August 17: Ryo Fujita (Kyoto University) Isomorphisms among quantum Grothendieck rings and their cluster theoretical interpretation
Quantum Grothendieck ring in this talk is a oneparameter deformation of the Grothendieck ring of the monoidal category of finitedimensional modules over the quantum loop algebras, endowed with the canonical basis consisting of simple (q,t)characters. In the case of type ADE, thanks to Nakajima’s geometric theory of quiver varieties, these simple (q,t)characters are known to compute the qcharacters of simple modules (via the analog of KazhdanLusztig algorithm) and enjoy some positivity properties. In this talk, we discuss a collection of isomorphisms between the quantum Grothendieck ring of type BCFG and that of “unfolded” type ADE, which respect the canonical bases. They are applied to verify the same positivity properties in type BCFG and the analog of KazhdanLusztig conjecture for several new cases. We also discuss their cluster theoretical interpretation, which particularly yields nontrivial birational relations among the (q,t)characters of different types. This is a joint work with David Hernandez, Sejin Oh, and Hironori Oya.
July 20: Shigenori Nakatsuka (University of Alberta) Recent progress in the dualities of Walgebras
Place and time: Peter Hall 162, 3:15pm – 5:00pm
Walgebras provide a rich family of vertex algebras parametrized by simple Lie algebras and their nilpotent orbits. They are vertex algebraic analogue of associative algebras, called the finite Walgebras, interpolating the enveloping algebras and their centers. The principal Walgebras enjoy a nontrivial isomorphism called the FeiginFrenkel duality, which can be seen as the upgrade of the isomorphism between the centers for the Langlands dual simple Lie algebras. In this talk, I will explain the recent progress on our understanding of the dualities of Walgebras motivated by physics and some consequences for their representation theory.
July 27: Wille Liu (Academia Sinica) Translation functors for trigonometric double affine Hecke algebras
Place and time: Peter Hall 162, 3:15pm – 5pm
Double affine Hecke algebras were introduced by Cherednik around 1995 as a tool to study the Macdonald polynomials. The trigonometric double affine Hecke algebras (TDAHA), degenerate version of the former, have also been found related to several other areas. In this talk, I will be focusing on specific aspects of the representation theory of the TDAHA.
Given a root system, the TDAHA depends on a family of complex parameters . Given two families of parameters and such that takes values in , there exists an equivalence of derived categories of the corresponding TDAHA: , called translation functor. After a brief introduction to the TDAHA, I will talk about a construction of translation functors.
August 3: Drazen Adamovic (University of Zagreb) On the semisimplicity of the category KL for affine vertex algebras at collapsing levels
Place and time: Peter Hall 162, 3:15pm – 4:15pm
In this talk, we will report on recent results on the representation theory of the simple affine vertex algebra at collapsing levels. When is an admissible rational number and is a Lie algebra, then each –module in the category is completely reducible by a result of Arakawa. But it turns out that the KazhdanLusztig category of –modules can be also semisimple for nonadmissible levels . We will present an approach which uses the representation theory of minimal affine algebras and the quantum Hamiltonian reduction functor. We proved this in joint papers with V. Kac, P. Moseneder Frajria, P. Papi and O. Perse that is semisimple when the simple affine algebra is rational or when is collapsing level for . This result enables us to prove the complete reducibility of modules in for some nonadmissible levels.
In the case when is a Lie superalgebra, the analysis of the category is more delicate than in the Lie algebra case, since in the super case can contain indecomposable modules. But even in this case we have a complete reducibility result for collapsing levels.
August 9: Anne Dranowski (USC) *** Extra seminar: note unusual day (Wednesday)! ***
Place and time: Peter Hall 162, 3:15pm – 4:15pm
gl(2) webs, foams and spectra
Webs were introduced by Kuperberg to construct sl(3) link invariants and foams were introduced by Khovanov to construct sl(3) link homology. Blanchet’s oriented and {1,2}labeled version of Khovanov’s foam category fixed the functoriality of Khovanov’s categorification over the integers. Via the extra info encoded in the labels it also offered a sl to gl upgrade. Relying on Blanchet’s foams and Howe duality, Lauda, Queffelec and Rose realized foams as representations of . In another direction, Lawson, Lipshitz and Sarkar constructed a stable homotopy refinement of Khovanov homology. Joint work in progress with Guo, Lauda and Manion pieces together these results to construct spectral bimodules for . In his talk we review what’s known, as well as what’s suggested, and how you might prove it.
August 10: ChengChiang Tsai (Academia Sinica) Wavefront sets for padic groups and graded Springer theory
Place and time: Peter Hall 162, 3:15pm – 5pm
For an irreducible admissible representation of a padic reductive group there is the notion of its wavefront set, which is a set of nilpotent orbits that describes the asymptotic behavior of the character near the identity. By a theorem of MoeglinWaldspurger, the set describes the least degenerate Whittaker models, which are double generalizations of local components of Fourier expansions for modular forms.
In this talk, we explain how a significant part of the determination of wavefront sets can be translated into Lietheoretic questions regarding generalizations of Kostant sections, which can be related to graded Springer theory. This last viewpoint has led us to discover surprising behaviors of wavefront sets for padic groups that do not occur for realgroups, disproving some longstanding conjectures.
(We do not assume any familiarity with graded Springer theory, or even Springer theory.)
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 19/26 Kevin Fergusson The unitary and symmetric groups
Oct 12 Ali Khalili The complexification of a compact group
Oct 4 Cancelled
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 
Oct 12, 2022  The complexification of a compact group  [S] 12  Ali Khalili 
Oct 19/26, 2022  The unitary groups and the symmetric groups  [S] 13  Kevin Fergusson 
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
Anne Dranowski (USC), August 911, 2023
Ryo Fujita (U Kyoto), August 824, 2023
ChengChiang Tsai (Academia Sinica), August 618, 2023
Wille Liu (Academia Sinica), July 26August 18, 2023
Peter Fiebig (FriedrichAlexanderUniversität) Apr 30May 6, 2023
Alexandre Minets (University of Edinburgh), April 2330, 2023
Xuhua He (Chinese University of Hong Kong), April 1218, 2023
Dennis Gaitsgory (MPIM), April 517, 2023
Alistair Savage (University of Ottawa), March 710 (TBC), 2023
Changlong Zhong (State University of New York at Albany), Feb 28Mar 28, 2023
Ben Webster (University of Waterloo), Jan 1620, 2023
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.