# Workshop Programme

## for period 22 - 26 June 2009

### Representation Theory and Lie Theory

22 - 26 June 2009

Timetable

Monday 22 June | ||||

08:30-09:55 | Registration | |||

09:55-10:00 | Welcome - Ben Mestel | |||

10:00-11:00 | Brundan, J (Oregon) |
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Blocks of the general linear supergroup | Sem 1 | |||

I will relate the endomorphism algebra of a minimal projective generator for a block of the general linear supergroup to a limiting version of Khovanov's diagram algebra. One consequence is that blocks of the general linear supergroup are Koszul, in the same spirit as classical work of Beilinson, Ginzburg and Soergel on blocks of the BGG category O for a semisimple Lie algebra. This is joint work with Catharina Stroppel. |
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11:00-11:30 | Coffee and Posters | |||

11:30-12:30 | Opdam, EM (Amsterdam) |
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Spectral transfer category of affine Hecke algebras | Sem 1 | |||

We introduce a notion of a ``spectral transfer morphism'' between affine Hecke algebras. Such a spectral transfer morphism from H_1 to H_2 is not given by an algebra homomorphism from H_1 to H_2 but rather by a homomorphism from the center Z_2 of H_2 to the center Z_1 of H_1 which is required to be ``compatible'' in a certain way with the Harish-Chandra \mu-functions on Z_1 and Z_2. The main property of such a transfer morphism is that it induces a correspondence between the tempered spectra of H_1 and H_2 which respects the canonical spectral measures (``Plancherel measures''), up to a locally constant factor with values in the rational numbers. The category of smooth unipotent representations of a connected split simple p-adic group of adjoint type G(F) is Morita equivalent to a direct sum R of affine Hecke algebras. It is a remarkable fact that R admits an essentially unique ``spectral transfer morphism'' to the Iwahori-Matsumoto Hecke algebra of G. This fact offers a new perspective on Reeder's classification of unipotent characters for exceptional split groups which works in the general case, leading to an alternative approach to Lusztig's classification of unipotent characters of G(F). |
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12:30-13:30 | Lunch at Wolfson Court | |||

14:00-15:00 | Achar, P (Louisiana State) |
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Staggered sheaves | Sem 1 | |||

Let X be a variety endowed with an action of an algebraic group G acting with finitely many orbits. "Staggered sheaves" are certain complexes of G-equivariant coherent sheaves on X, generalizing the "perverse coherent sheaves" of Deligne and Bezrukavnikov. They form an abelian category that has many remarkable algebraic properties resembling those of l-adic perverse sheaves. In particular, this category is quasi-hereditary and admits a mixed structure. If time permits, I will describe some small examples. Some of these results are joint work with David Treumann. |
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15:00-15:30 | Tea and Posters | |||

15:30-16:30 | Toledano-Laredo, V (Northeastern) |
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Stability conditions and Stokes factors | Sem 1 | |||

I will explain how the wall-crossing formulae studied by D. Joyce in the context of an abelian category A can be understood as Stokes phenomena for a connection on the Riemann sphere having an irregular singularity at 0 and values in the Ringel-Hall Lie algebra of A. This allows one to interpret Joyce's holomorphic generating functions as defining an isomonodromic family of such connections on the space of stability conditions of A. This is a joint work with Tom Bridgeland. |
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16:30-17:30 | Rouquier, R (Oxford) |
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Higher representations: geometry and tensor structures | Sem 1 | |||

We will discuss the geometrical realisation of simple 2-representations of symmetric Kac-Moody algebras and their tensor products on one hand, and the algebraic construction of tensor products on the other hand. |
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17:30-18:30 | Wine reception and Poster Session | |||

18:45-19:30 | Dinner at Wolfson Court |

Tuesday 23 June | ||||

09:00-10:00 | Ariki, S (Kyoto) |
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Graded q-Schur algebras | Sem 1 | |||

Just 10 years ago, the decomposition matrix theorem for cyclotomic Hecke algebras was generalized to the decomposition matrix theorem for the q-Schur algebra by Varagnolo and Vasserot. This year, Brundan and Kleshchev proved graded analogue of the decomposition matrix theorem for cyclotomic Hecke algebras. Hence it is natural to give graded analogue of the decomposition matrix theorem for the q-Schur algebra. This may be done by defining appropriate setting for the graded version, and following ideas of Hemmer and Nakano, and Leclerc. |
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10:00-11:00 | Friedlander, E (Southern California) |
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Investigating $kG$-modules using nilpotent operators | Sem 1 | |||

This is a report of on-going work with Jon Carlson, Julia Pevtsova, and Andrei Suslin. Our object of study is the representation theory of $kG$ where $G$ is a finite group scheme. Following Quillen's early work, first invariants involve cohomology and cohomological support varieties. These have interpretations in terms of 1-parameter subgroups and $\pi$-points. Finer invariants arise from considering the Jordan type of nilpotent operators, leading to local Jordan types, generalized support varieties, and algebraic vector bundles on projective varieties. |
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11:00-11:30 | Coffee and Posters | |||

11:30-12:30 | Fiebig, P (Freiburg) |
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Lusztig's conjecture as a moment graph problem | Sem 1 | |||

To any root system we associate a labelled, partially ordered graph and a sheaf theory on the graph with coefficients in an arbitrary field k. An extension property then leads to the definition of a certain universal class of sheaves, the Braden-MacPherson sheaves. We formulate a conjecture about the multiplicity of their stalks. This conjecture implies Lusztig's conjecture on the irreducible characters of the simply connected algebraic group over k associated to the root system. Finally we list the proven instances of the conjecture. |
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12:30-13:30 | Lunch at Wolfson Court | |||

14:00-15:00 | Nakano, D (Georgia) |
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Atypicality, complexity and module varieties for classical Lie superalgebras | Sem 1 | |||

Let ${\frak g}={\frak g}_{\bar 0}\oplus {\frak g}_{\bar 1}$ be a classical Lie superalgebra and ${\mathcal F}$ be the category of finite dimensional ${\frak g}$-supermodules which are semisimple over ${\frak g}_{\bar 0}$. In this talk we investigate the homological properties of the category ${\mathcal F}$. In particular we prove that ${\mathcal F}$ is self-injective in the sense that all projective supermodules are injective. We also show that all supermodules in ${\mathcal F}$ admit a projective resolution with polynomial rate of growth and, hence, one can study complexity in $\mathcal{F}$. If ${\frak g}$ is a Type I Lie superalgebra we introduce support varieties which detect projectivity and are related to the associated varieties of Duflo and Serganova. If in addition ${\frak g}$ has a (strong) duality then we prove that the conditions of being tilting or projective are equivalent. |
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15:00-15:30 | Tea and Posters | |||

15:30-16:30 | Bernstein, J (Tel Aviv) |
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On Casselman-Wallach globalization theorem | Sem 1 | |||

About 20 years ago B.Casselman and N.Wallach proved a remarkable theorem about representations of real reductive groups. Namely they proved that any Harish Chandra module $M$ for a group $G$ can be uniquely extended to a smooth Frechet representation $V$ of this group (they called this procedure "the canonical globalization"). Unfortunately their proof is quite involved. For this reason their result is not that well known as it should be. For the same reason some natural generalizations of this result (e.g. how to make such globalization in a family) could not be studied. In my lecture I will describe a new approach to this problem developed in a joint work by B. Kroetz and myself that is technically simpler and allows to investigate some related phenomena. |
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18:45-19:30 | Dinner at Wolfson Court |

Wednesday 24 June | ||||

09:00-10:00 | Guralnick, RM (Southern California) |
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Derangements in Finite and Algebraic Groups | Sem 1 | |||

A derangement is a fixed point free permutation. We will consider transitive actions of finite and algebraic groups and the prevalence of derangements. We will discuss the Boston-Shalev conjecture and related problems about conjugacy classes as well as various applications. |
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10:00-11:00 | Srinivasan, B (Illinois at Chicago) |
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Quadratic unipotent blocks of finite general linear, unitary and symplectic groups | Sem 1 | |||

Let G be a finite reductive group. The (complex) irreducible characters of G were classified by Lusztig. They fall into Lusztig series E(G, (s)) where (s) is a semisimple class in a dual group G*. If a character is in E(G, (1)) it is a unipotent character. Let l be a prime different from the characteristic of the field of definition of G. A block of l-modular characters containing unipotent characters is a unipotent block. Unipotent blocks of finite reductive groups have been intensely investigated by many authors over many years. In particular, correspondences such as perfect isometries (in the sense of Michel Broue) or derived equivalences have been established between unipotent blocks of two general linear groups, a finite reductive group and a "local" subgroup, and so on. In this talk we will focus on recent work which shows that if we enlarge the set of unipotent blocks to the set of "quadratic unipotent blocks" for some choices of l some surprising results "across types" are obtained. In particular, correspondences are obtained between blocks of a unitary group and a symplectic group, and between blocks of a general linear group and a symplectic group. It would appear that quadratic unipotent blocks are a natural generalization of unipotent blocks in classical groups. |
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11:00-11:30 | Coffee and Posters | |||

11:30-12:30 | Juteau, D (CNRS) |
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Perverse sheaves and modular representation theory | Sem 1 | |||

I will talk about some relationships between modular representation theory and perverse sheaves with positive characteristic coefficients. |
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12:30-13:30 | Lunch at Wolfson Court | |||

15:00-15:30 | Tea and Posters | |||

18:45-19:30 | Dinner at Wolfson Court |

Thursday 25 June | ||||

09:00-10:00 | Premet, A (Manchester) |
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Modular Lie algebras and the Gelfand-Kirillov conjecture | Sem 1 | |||

In my talk I am going to discuss the current status of the Gelfand-Kirillov conjecture (from 1966) on the structure of the Lie field of a finite dimensional complex simple Lie algebra |
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10:00-11:00 | Lehrer, GI (Sydney) |
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Non-commutative invariant theory | Sem 1 | |||

We define a (non-commutative) quantum analogue of the coordinate ring of a finite dimensional module of a quantum group, which in many (but not all) cases is a flat deformation of the classical coordinate ring. We prove a quantum analogue of the first fundamental theorem of invariant theory in a form which generalises the classical cases. This is joint work with R. Zhang and H. Zhang. |
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11:00-11:30 | Coffee and Posters | |||

11:30-12:30 | Vasserot, E (Université Paris 7 - Denis-Diderot) |
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Canonical bases and KLR-algebras | Sem 1 | |||

We'll explain how Khovanov-Lauda-Algebras categorify the canonical basis of the negative part of the quantum enveloping algebra, and we'll give some motivation for such constructions which come from Cherednik algebras. |
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12:30-13:30 | Lunch at Wolfson Court | |||

14:00-15:00 | Lam, T (Harvard) |
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Total positivity, Schubert positivity, and geometric Satake | Sem 1 | |||

Let G be a complex simple simply-connected algebraic group. A theorem proved independently by Ginzburg and Peterson states that the homology H_*(Gr_G) of the affine Grassmannian of G is isomorphic to the ring of functions on the centralizer X of a principal nilpotent in the Langlands dual G^\vee. There is a notion of total positivity on X, using Lusztig's general definitions, and there is also a notion of Schubert positivity, using Schubert classes of Gr_G. We connect the two notions using the geometric Satake correspondence. In addition, we give an explicit parametrization of the positive points of X. This is joint work with Konstanze Rietsch, generalizing work of hers in type A. |
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15:00-15:30 | Tea and Posters | |||

15:30-16:30 | Cherednik, I (UNC Chapel Hill and RIMS) |
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Difference spherical and Whittaker functions | Sem 1 | |||

The definition of q,t-spherical functions was sugested by the speaker (reduced root systems) and Stokman (the C-check-C case). They generalize the classical spherical functions, the basic hypergeometric function and the p-adic spherical functions. Recently, their systematic algebraic and analytic theory was started including q-generalizations of the Harish-Chandra asymtotic formula and the Helgason-Johnson description of bounded spherical functions, as well as the theory of the q-Whittaker functions. The latter are related to the Givental-Lee theory, the IC-theory of the affine flag varieties and are expected to have connections with the quantum geometric Langlands program. |
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19:30-23:00 | Conference dinner at Christ's College (Dining Hall) |

Friday 26 June | ||||

09:00-10:00 | Soergel, W (Albert-Ludwigs-Universität Freiburg) |
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Graded version of tensoring with finite dimensional representations | Sem 1 | |||

I want to discuss how to lift the functors of tensoring with a finite dimensional representation to the graded representation categories and what these functors correspond to under Koszul duality |
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10:00-11:00 | Kac, V (Massachusetts Institute of Technology) |
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On classification of Poisson vertex algebras | Sem 1 | |||

11:00-11:30 | Coffee and Posters | |||

11:30-12:30 | Losev, IV (Massachusetts Institute of Technology) |
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One-dimensional representations of W-algebras | Sem 1 | |||

Premet conjectured that any (finite) W-algebra has a one-dimensional representation. The goal of this talk is to explain results of the speaker towards this conjecture. We will start giving a sketch of proof for the classical Lie algebras. Then we explain a reduction to rigid nilpotent elements using a parabolic induction functor. Finally, we will explain how using the Brundan-Goodwin-Kleshchev category O one can try to describe one-dimensional representations of W-algebras associated to rigid elements in exceptional Lie algebras. |
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12:30-13:30 | Lunch at Wolfson Court | |||

14:00-15:00 | Bonnafe, C (Franche-Comté) |
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Compactification of Deligne-Lusztig varieties | Sem 1 | |||

Joint work with Raphaël Rouquier: we give an explicit construction of the normalization of the Bott-Samelson-Demazure compactification of Deligne-Lusztig varieties in their classical étale covering (with a finite torus as Galois group). We retrieve an old result of Deligne and Lusztig about the local monodromy around the divisors of this compactification. |
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15:00-15:30 | Tea | |||

15:30-16:30 | Broué, M (Institut Henri Poincaré) |
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Deligne-Lusztig varieties, Braid groups and cyclotomic Hecke algebras | Sem 1 | |||

18:45-19:30 | Dinner at Wolfson Court |