# Workshop on geometric representation theory

Place
Kyoto University Department of Mathematics 3rd building room 110, Oiwake Kitashirakawa Sakyo Kyoto JAPAN, Campus map is here, and the 3rd building is number 5. (RIMS is number 7.)
Date
September 24 (Monday) -- September 28 (Friday), 2012.
Organizer
Noriyuki Abe, and Syu Kato (Department of Mathematics, Kyoto University, contact e-mail address: )
in collabolation with Ryosuke Kodera, Takuma Hayashi, Tatsuyuki Hikita, Hideyuki Hosaka, Kohei Yahiro, and Shintaro Yanagida
Support
Limited support might be available for young researchers. So please contact the organizer in case you want.

This workshop is supported by the Kyoto Univeristy Global COE Program Fostering top leaders in mathematics - broadening the core and exploring new ground", Japan Society for the Promotion of Science Grand-in-Aid for Young Scientists (B) 23740014, Japan Society for the Promotion of Science Grant-in-Aid for Scientific Research (B) 23340005.

## Speakers include

• Rina Anno (Pittsburgh) 2 talks
• Alexander Braverman (Providence) 3 talks
• Jochen Heinloth (Essen) 3 talks
• Emanuel Letellier (Caen) 3 talks
• Hiraku Nakajima (Kyoto) 1 talk
• Kevin McGerty (London) 3 talks
• Yuji Tachikawa (Tokyo) 1 talk
• Michela Varagnolo (Paris) 1 talk
• Eric Vasserot (Paris) 2 talks

## Program (tentative)

9:30--10:3010:45--11:4513:30--14:3014:45--15:4516:00--17:00
24Heinloth IAnno IBraverman ITachikawa
25Vasserot IVasserot IIVaragnoloHeinloth IIBraverman II
26Letellier IMcGerty I
27KatoHeinloth IIIBraverman IIILetellier IIMcGerty II
28Varagnolo IIAnno IILetellier IIIMcGerty IIINakajima

The registration will be started at 09:30 in September 24.

TBA is to be announced.

## Accomodation, Banquet information, and so on

• Programs: to be determined in a week, starts from morning Sep/24, and closes afternoon Sep/28.
• Discussion Room: Rooms 108 & 109 are reserved for discussion (9:00--18:00)
• Banquet will be at Tuesday 18:00--20:00 at Science Seminar House located behind the 3rd building (no.10 in the above map). The fee is expected to be 4000--5000 yen
• Wednesday afternoon is free: There will be excursion to Fushimi-Inari shrine.

## Titles and Abstracts

### Rina Anno: Bridgeland's stability conditions.

I will give an introductory lecture on Bridgeland's stability conditions, based on his works arXiv:math/0212237v3 (Stability conditions on triangulated categories) and arXiv:math/0508257v2 (Stability conditions and Kleinian singularities). I will go over the proof of the main theorem in the latter work, since this is how one utilizes an action of the braid group with generators being reflections on the level of K-theory, to construct a submanifold of the space of stability conditions that will contain the "good" t-structure we are interested in.

### Rina Anno: A thin stringy moduli space for Slodowy slices after arXiv:1108.1563 (joint work with R. Bezrukavnikov and I. Mirkovic).

The space of stability conditions on an triangulated category T (t-structures with some additional data) has a topology, and thus may be thought of as the moduli space of t-structures. They key property here is that such a space has a natural map to the complexified dual to K-theory $K(T)\otimes C$, and this map is a local isomorphism. We describe a submanifold in the space of stability conditions for the derived category of coherent sheaves on a Slodowy slice, which covers a submanifold in $K(T)\otimes C$. The braid group action on the category restricts to the affine Weyl group action on this submanifold (the exponent of $H^2(X)$), and acts on the group of deck transformations of the cover. The t-structures involved are Bezrukavnikov's exotic t-structures.

### Alexander Braverman: Towards the theory of automorphic forms for affine Kac-Moody groups

• Talk 1: Spherical and Iwahori Hecke algebras for affine Kac-Moody groups. Affine Satake isomorphism and relation to Cherednik algebras.
• Talk 2: Explicit form of the affine Satake isomorphism (affine version of the Macdonald formula for spherical function). Geometric Satake isomorphism and its relation to geometry of quiver varieties and Uhlenbeck spaces.
• Talk 3: Eisenstein series for affine Kac-Moody groups over functional fields. Discussion of possible applications in number theory.

### Jochen Heinloth: Automorphic sheaves and rigid local systems

According to the geometric Langlands programme G-local systems on a curve C are related to certain eigensheaves on the stack of Bundles (for the dual group) on the curve. Since these spaces of bundles are not easy to describe it is usually hard to give an explicit description of automorphic sheaves. Motivated by work of Gross and Frenkel, examples of such sheaves were found in joint work with B.C. Ngô and Z. Yun. From these sheaves one obtains a uniform geometric construction of (cohomologically) rigid local systems on the punctured projective line. After explaining some background motivating the problem, I would like to explain the construction and how it was found. If time permits I'd also like to indicate some more recent applications of this strategy and explain some open problems related to the construction.

### Kevin McGerty: Quantization of symplectic singularities and localization

There has been considerable interest recently in how one might generalize the classical localization theory of Beilinson-Bernstein for flag varieties to a wider class of spaces. Examples such as the Hilbert scheme of points in the plane suggest that the deformation quantization of symplectic varieties provides a good context in which to formulate such generalizations. I will discuss joint work with Tom Nevins on localization for such varieties which arise as GIT quotients and some related questions.

### Syu Kato: An algebraic study of extension algebras

We present a simple condition which guarantees a geometric extension algebra to behave like a variant of the quasi-hereditary algebra. In particular, standard modules of the affine Hecke algebras of type $\mathsf{BC}$, and the quiver Schur algebras are shown to satisfy the Brauer-Humphreys type reciprocity and the semi-orthogonality property. In addition, we present a new criterion of purity of weights in the geometric side.

### Hiraku Nakajima: Quiver varieties and tensor products

We define a family of homomorphisms on a collection of convolution algebras associated with quiver varieties, which gives a kind of coproduct on the Yangian associated with a symmetric Kac-Moody Lie algebra. We study its property using perverse sheaves.

### Emmanuel Letellier: Representations of quivers and the character ring of GL_n(q)

In these talks we will discuss connections between several polynomials, namely Kac polynomials (which counts the number of absolutely indecomposable representations of a quiver over a finite field), motivic Donaldson-Thomas invariants attached to a quiver (with potential 0), and multiplicities of unipotent characters of GL_n(q) in tensor products of unipotent characters. We will prove the non-negativity of the coefficients of these polynomials using Nakajima's construction of Weyl groups action on cohomology of quiver varieties. Recall that positivity for Kac polynomials was conjectured by Kac in 1982, positivity for DT-invariants was conjectured by Kontsevich-Soibelman and first proved by Efimov in 2011, and positivity for multiplicities in tensor products of unipotent characters of GL_n(q) was observed by Hiss-Lubeck-Mattig in 2003 for n<9 with the help of a computer. We will also investigate the degree, give a criterion for these polynomials to be non-zero (in the case of Kac polynomials this is known since 1982), and finally discuss about their (q,t)-deformation. This work is partially joint with Tamas Hausel and Fernando Rodriguez-Villegas.

### Yuji Tachikawa: Nilpotent orbits and supersymmetric gauge theories

Various properties of nilpotent orbits are now known to govern many aspects of four-dimensional supersymmetric gauge theories. Combined with information on the gauge theory side already known to physicists, this relation can sometimes lead to new mathematical conjectures concerning nilpotent orbits. One such conjecture, about a graded polynomial ring assigned to each nilpotent orbit, will be described in detail. (Based on the preprint http://arxiv.org/abs/1203.2930 with Chacaltana and Distler.)

### Michela Varagnolo: Categorifications and CRDAHA

The talk is based on a joint work with P.Shan, R.Rouquier and E.Vasserot. Some years ago in a joint work with Vasserot, we state a conjecture which links a Schur subcategory of a parabolic affine category O and the category O of a Cyclotomic Rational Double Affine Hecke Algebra. This conjecture implies both the dimension conjecture by Rouquier and the level-rank conjecture by Chuang and Miyachi. It is now a theorem and I will sketch an idea of the proof.

### Eric Vasserot: AGT conjecture and Hall algebras (with O. Schiffmann)

Hall algebras of curves are relatively well understood. We'll explain their structure and their relations with some new algebras which acts on the moduli spaces of instantons.