Future Directions in Representation Theory December 4 8, 2017 Abercrombie Business School Lecture Theatre 1170 The University of Sydney

Transcription

1 Future Directions in Representation Theory December 4 8, 2017 Abercrombie Business School Lecture Theatre 1170 The University of Sydney Abstracts for Invited Speakers Jonathan Brundan (University of Oregon), Heisenberg and Kac Moody categorification Abstract: In type A representation theory, categorical actions of two different monoidal categories, the Heisenberg category and the Kac Moody 2-category, are uniquitous. I hope to include them both in this talk, and say something about the connection between the two theories. Charlotte Chan (University of Michigan), Towards a p-adic Deligne Lusztig theory Abstract: The seminal work of Deligne and Lusztig on the representations of finite reductive groups has influenced an industry studying parallel constructions in the same theme. In 1979, Lusztig proposed a conjectural analogue of Deligne Lusztig theory for p-adic groups. In this talk, we will discuss new cohomological techniques that allow one to make progress in Lusztig s program by studying similar constructions for unipotent groups.

2 Emily Cliff (University of Illinois Urbana-Champaign), Factorization structures on the cohomology of the Hilbert scheme of points of a surface Abstract: Nakajima and Grojnowski showed that the cohomology of the Hilbert scheme of points of a complex surface X has the structure of an irreducible representation of a Heisenberg Lie algebra. It follows from work of Frenkel Lepowsky Meurman that it also has the structure of a vertex algebra, the so-called Heisenberg vertex algebra. Reinterpreting these results in geometric terms, we obtain a family of sheaves which form a Heisenberg factorization algebra over any smooth curve C. In short, we started with geometric objects (the surface and its Hilbert scheme), did some algebraic constructions, and eventually produced new geometric objects (sheaves on complex curves C, with factorization structures). The goal of this project is to use the geometry of the Hilbert scheme to construct the factorization algebras directly. In this talk we will show how the Hilbert scheme can be used to construct factorization spaces over C with the desired properties, but then explain why the naive linearization process produces a different factorization algebra than the one resulting from the work of Nakajima and Grojnowski. We also discuss a different method of linearizing which may produce the Heisenberg factorization algebra. No prior knowledge of Hilbert schemes, vertex algebras, or factorization algebras will be assumed. Kevin Coulembier (University of Sydney), Tensor ideals, Deligne categories and Lie supergroups Abstract: Twenty years ago, Deligne introduced several universal tensor categories in terms of Brauer diagrams. By their universality, they admit tensor functors to the categories of modules over the classical Lie supergroups. It actually turns out that all information one could possibly desire on the representation theory of these supergroups is concealed in the Deligne categories. One of the challenges in extracting the information is trying to understand all tensor ideals. In this talk, I will present some general techniques which allow to obtain a description of the tensor ideals in Deligne categories from the recently obtained classification of ideals in their Grothendieck rings. Time permitting, I will show how this leads to an alternative proof for the second fundamental theorem of invariant theory à la Lehrer and Zhang. 2

3 Olivier Dudas (Université Paris Diderot), Cohomology of Deligne Lusztig varieties: a change of heart Abstract: The Deligne Lusztig varieties are algebraic varieties whose cohomology yields representations of finite reductive groups (such as SL n (q), Sp 2n (q),..., E 8 (q)). They were originally introduced by Deligne and Lusztig in 1976 to study the ordinary (complex) representations, but were later shown to be of considerable interest for studying modular representations (in positive characteristic). In this talk I will explain how one can use the cohomology complexes of these varieties to define various t-structures on the derived category of representations of finite reductive groups. I will mention several conjectures on the existence of Morita and perverse derived equivalences between the hearts of these t-structures. Michael Ehrig (University of Sydney), Relative cellular algebras Abstract: In this talk we discuss the notion of relative cellular algebras, an idempotent adapted generalization of cellular algebras. We will discuss how such algebras relate to ordinary cellular algebras and which results of the theory of cellular algebras can be obtained for the relative case. Furthermore we will look at a list of examples that satisfy these properties and finally discuss some open problems concerning this notion. Inna Entova-Aizenbud (Ben Gurion University), Deligne categories and complexes of representations of symmetric groups Abstract: Let V be a finite-dimensional (complex) vector space, and Sym(V ) be the symmetric algebra on this vector space. We can consider the multiplication map Sym(V ) V V as a complex of GL(V )-representations of length 2. In this talk, I will describe how tensor powers of the above complex define interesting complexes of representations of the symmetric group S n, which were studied by Deligne in the paper La catégorie des représentations du groupe symétrique S t, lorsque t n est pas un entier naturel. I will then explain how computing the cohomology of these complexes helps establish a relation between the Deligne categories and the representations of S, which are two natural settings for studying stabilization in the theory 3

4 of finite-dimensional representations of the symmetric groups. This is a joint project with D. Barter and Th. Heidersdorf. Jessica Fintzen (Institute for Advanced Study/University of Michigan/ University of Cambridge), Representations of p-adic groups Abstract: The building blocks for representations of p-adic groups are called supercuspidal representations. I will survey what is known about the construction of supercuspidal representations, mention questions that remain mysterious until today, and explain some recent developments. Peter Hochs (University of Adelaide), K-theory, fixed points and characters Abstract: Let G be a semisimple Lie group. For an L 1 -function f on G, convolution by f is a bounded operator on L 2 (G). The reduced C -algebra of G is the closure of the set of these convolution operators in the operator norm. This algebra encodes all tempered representations of G. Because it is a C -algebra, a natural invariant to study is its K-theory. Much is known about such K-theory groups, but it is still a challenge to extract explicit representation theoretic information from them. In work with Hang Wang, we define trace maps based on orbital integrals, that can be used to read off the characters of discrete series representations on elliptic elements from K- theory classes. An advantage of using K-theory is that it is a natural setting for index theory. Using K-theory, we generalise Atiyah Segal Singer s fixed point formula for equivariant indices from compact groups and manifolds to noncompact ones. Using the trace maps just mentioned, we show that Harish-Chandra s character formula for discrete series representations is a special case of this generalised fixed point formula. Future directions in this area are to examine what other concrete representation theoretic information can be obtained from K-theory, and how this can be combined with index theory to prove results in representation theory. Anthony Licata (Australian National University), Artin groups and categorical actions Abstract: In the last 15 years, one part of representation theory that has been actively developed is the categorification of representations of quantum groups. To produce such a categorification involves constructing an explicit 4

5 action of a monoidal (or 2-) category on another additive category which is hopefully of independent mathematical interest. One important consequence of such constructions is that they induce categorical actions of braid and Artin groups (in type A, for instance, the braid group actions which arise from categorification are at the heart of the interaction between representation theory and link homology theories). A question which remains largely open, however, is what these categorical actions of groups might tell us about the groups themselves. The goal of this talk will be to explain some theorems and conjectures about Artin groups that should be able to be answered by studying existing categorical actions of them. Gunter Malle (Technische Universität Kaiserslautern), On Brauer s k(b) conjecture Abstract: In his 1954 address at the International Congress, Richard Brauer posed a list of fundamental problems in representation theory of finite groups, many of which are still open. One among them is his conjecture on the number k(b) of irreducible complex characters in a p-block B of a finite group: k(b) should be at most equal to the order of a defect group of B. We show that p-blocks of finite quasi-simple groups do not occur as a minimal counterexample for any prime p 5, and we obtain partial results for p 3. We also determine the precise number of irreducible characters in unipotent blocks of classical groups for odd primes. Peter McNamara (University of Queensland), Geometric extension algebras Abstract: Geometric extension algebras are convolution algebras in Borel Moore homology, or equivalently sheaf-theoretic Ext algebras. Interesting examples include KLR algebras, algebras related to Schur algebras, category O and Webster algebras. We discuss how geometric parity vanishing properties are equivalent to representation-theoretic properties of these algebras. Some applications to the theory of KLR algebras will be discussed if time permits. 5

6 Hiraku Nakajima (Research Institute for Mathematical Sciences, Kyoto), Quiver gauge theories and Kac-Moody Lie algebras Abstract: Quiver gauge theories give two types of algebraic symplectic varieties, which are called quiver varieties and Coulomb branches respectively. The first ones were introduced by the speaker in 1994, and their homology groups are representations of Kac Moody Lie algebras. The second ones were introduced by the speaker and Braverman, Finkelberg in The two types of varieties are very different (e.g., dimensions are different), but are expected to be related in rather mysterious ways. As an example of mysterious links, I would like to explain a conjectural realization of Kac Moody Lie algebra representations on homology groups of Coulomb branches, which the speaker proves in affine type A. It nicely matches with geometric Satake correspondence for usual finite dimensional complex simple groups and loop groups. Arun Ram (University of Melbourne), Combinatorics of level 0 representations Abstract: Recent work of Kato Naito Sagaki Feigin Makyedonskyi has provided an improved understanding of the combinatorics of integrable level 0 representations of the affine Lie algebra. In particular, there is a connection to the Schubert calculus of the semi-infinite flag variety from which Kato Naito Sagaki prove an analogue of the Pieri Chevalley formula for the semi-infinite flag variety (line bundle multiplied with a Schubert class). The connection to crystals in this new formula (analogous to that in the Pieri Chevalley formula of Pittie Ram) provides a geometric interpretation of (a part of) the path model which was used by Ram Yip to give a formula for Macdonald polynomials. Simon Riche (Université Clermont Auvergne), Modular Koszul duality patterns in representation theory Abstract: The importance of Koszul duality in (geometric) representation theory was highlighted in the 1990 s in a celebrated paper of Beilinson Ginzburg Soergel. In this talk we will present a number of occurrences of similar ideas in representation theory of reductive algebraic groups over fields of positive characteristic, which have emerged in the recent years in joint work with various people (including Pramod Achar, Carl Mautner, Shotaro Makisumi and Geordie Williamson). The main difference with the 6

7 characteristic-0 case is the importance of certain objects characterized by some parity conditions (first introduced by Juteau Mautner Williamson). Laura Rider (University of Georgia), An Iwahori Whittaker model for the Satake category Abstract: The geometric Satake equivalence gives a topological incarnation of the representation theory of a connected, reductive algebraic group over any field. This description uses so-called spherical perverse sheaves on the affine Grassmannian. In my talk, I ll discuss an Iwahori Whittaker model for this category. This model takes advantage of a cellular stratification of the affine Grassmanian, and as a result, allows for some nice applications of the equivalence. This work is joint with Roman Bezrukavnikov, Ivan Mirković, and Simon Riche. Peng Shan (Tsinghua University), On equivariant cohomology of Calogero Moser spaces Abstract: I will report a recent joint work with Cédric Bonnafé in which we compute equivariant cohomology of smooth Calogero Moser spaces using representation theory of rational Cherednik algebras. We also give an application to the computation of equivariant cohomology of related symplectic resolutions. Monica Vazirani (University of California Davis), The Springer representation of the DAHA Abstract: Building on the work of Calaque Enriquez Etingof, Lyubashenko Majid, and Arakawa Suzuki, Jordan constructed a functor from quantum D-modules on the special linear group to representations of the double affine Hecke algebra (DAHA) in type A. Our preliminary findings show when we input the so-called quantum Springer sheaf at parameter kn the output is roughly a k-thickened version of the regular representation of S N. Part of this work is defining what the quantum Springer sheaf is, and through Jordan s functor, understanding its structure. Further, we gain a greater understanding of the category of strongly equivariant quantum D-modules. This is joint work with David Jordan. 7

8 Ben Webster (University of Waterloo/Perimeter Institute for Theoretical Physics), Symplectic duality and KLR algebras Abstract: Symplectic duality is the (admittedly, vaguely-named) idea that certain symplectic singularities come in dual pairs. These singularities have associated algebras constructed via deformation quantization: for example, the universal enveloping algebra to the nilcone and Cherednik algebras to T V/G for a complex reflection group G on a vector space V. A great deal of evidence suggests that the category O s of these varieties are Koszul dual, and I ll discuss a proof of this fact for certain examples arising from physics (called Higgs and Coulomb branches). This proof passes through a combinatorial description of these categories, given by a version of KLR algebras in the case of the dual pair given by Nakajima quiver varieties and the slices between Schubert varieties in the affine Grassmannian. Ting Xue (University of Melbourne), Springer correspondence for symmetric spaces Abstract: The Springer correspondence relates nilpotent orbits in the Lie algebra of a reductive algebraic group to irreducible representations of the Weyl group. We develop a Springer theory in the case of symmetric spaces using Fourier transform, which relates nilpotent orbits in this setting to irreducible representations of Hecke algebras of various Coxeter groups with specified parameters. This in turn gives rise to character sheaves on symmetric spaces, which we describe explicitly in the case of classical symmetric spaces. A key ingredient in the construction is the nearby cycle sheaves associated to the adjoint quotient map. We will also touch on possible connections of our work with affine Springer fibers and rational Cherednik algebras. The talk is based on joint work with Kari Vilonen and partly based on joint work with Misha Grinberg and Kari Vilonen. Yaping Yang (University of Melbourne), A geometric construction of affine quantum groups Abstract: We will talk about a construction of affine quantum groups using cohomological Hall algebras in the setting of a generalized cohomology theory. The representations are given by the corresponding cohomology of Nakajima quiver varieties. In my talk, I will explain in detail two examples. 8

9 1. We use the Morava K-theory to construct a family of new quantum groups parametrized by a prime number and a positive integer. Those quantum groups are expected to be related to Lusztig s 2015 reformulation of his conjecture from 1979 on character formulas for algebraic groups over a field of positive characteristic. 2. We use the equivariant elliptic cohomology to establish a sheafified elliptic quantum group for any symmetric Kac-Moody Lie algebra. The rational sections give the algebra of elliptic R-matrix. I will also explain the relation between the sheafified elliptic quantum group and a global loop Grassmannian over an elliptic curve. This talk is based on my joint work with Gufang Zhao. Xinwen Zhu (California Institute of Technology), S-operators via the categorical trace Abstract: S-operators were introduced by V. Lafforgue, and later generalized in my joint work with Liang Xiao. They act on the cohomology of Shimura varieties and moduli of Shtukas. I will explain how to obtain them naturally via the categorical trace of the Satake category. If time permits, I will discuss possible generalizations. 9