Workshop on“Geometry and Representation Theory”
January 16 – 27, 2017
organized byTomoyuki Arakawa (RIMS, U of Kyoto), Karin Baur (U of Graz), Victor Kac (MIT, Boston),
Anne Moreau (U of Poitiers)
Week 1January 16 – 20, 2017

• Monday, January 16, 201708:30 – 09:00 Opening & Registration
09:00 – 10:00 Pramod Achar
Title: Introduction to affine Grassmannians and the geometric Satake equivalence I.
Abstract: The goal of these lectures will be to start from scratch and learn what the affine Grassmannianof a reductive group is, and how its geometry is related to the representation theory of the Langlands
dual group. I will emphasize the examples of GLn and SL2 throughout. Here is an outline of the threelectures:
Lecture 1: definitions; lattice model for GLn ; orbits.
Lecture 2: convolution product; Lusztig’s q-analogue of the weight multiplicity.
Lecture 3: geometric Satake equivalence; MV cycles.
10:05 – 11:05 Guido Pezzini
Title: Geometry and combinatorics of spherical varieties I.
Abstract: In these lectures we will introduce spherical varieties and discuss some of their basic properties;
we will also introduce related combinatorial objects and see how they govern the geometry of suchvarieties. The topics will be:
1. Definitions, examples, first properties.
2. Local structure theorem.
3. The Luna-Vust theory of embeddings.
4. Spherical roots and the multiplication of regular functions.
11:05 – 11:35 coffee / tea break
11:35 – 12:35 Ivan Mirkovic
Title: Generalized affine Grassmannians I.
Abstract: Loop Grassmannians appear in the geometric Langlands program as a geometrization of Hecke
operators in number theory. We will reconstruct the loop Grassmannain of a reductive group in terms of
the loop Grasmmannian of its Cartan subgroup. This actually provides more spaces of a similar kind.
Talk 1. Commutative case and Class Field Theory.
Talk 2. More “loop Grassmannians” via local spaces.
Talk 3. Drinfeld’s formulation of interesting subspaces in terms of classifying pairs.
112:35 – 14:30 lunch break
14:30 – 15:30 Pavel Etingof
Title:I Introduction to D-modules I.
Abstract: I will give a brief introduction to the theory of algebraic D-modules, with an emphasis on thenotions to be used in D. Gaitsgory’s minicourse. I will review the following topics (time permitting):
1. Definition of the category of D-modules on a smooth variety.
2. Equivalence between left and right D-modules.
3. Kashiwara’s theorem and D-modules on singular varieties.
4. *-direct and !-inverse images, adjunctions between them for open embeddings and proper morphisms, base change.
5. Singular support, Gabber’s theorem.
6. Holonomic D-modules.
7. Verdier duality.
8. Formalism of six functors.
9. Beilinson-Bernstein localization theorem.
15:30 – 15:50 break
15:50 – 16:45 Pierre Baumann
Title: Bases for tensor products and geometric Satake correspondence.
Abstract: Let G be a connected reductive group over C. The geometric Satake correspondence, due to
Lusztig, Ginzburg, Beilinson-Drinfeld, and Mirković-Vilonen, asserts that the Langlands dual G∨ of G
can be obtained by Tannakian reconstruction from a category of perverse sheaves on the affine Grassmannian of G. Specifically, tensor products of simple representations of G∨ are obtained as the intersection
homology of convolution varieties. One may then define bases of these tensor products, indexed by algebraic cycles defined by Braverman-Gaitsgory, Gaussent-Littelmann-Nguyen, and Goncharov-Shen. We
will explain that these bases share many properties with Lusztig’s dual canonical bases, thereby generalizing previous results by Fontaine-Kamnitzer-Kuperberg. This is joint work with S. Gaussent and P.
Littelmann.
• Tuesday, January 17, 2017
09:00 – 10:00 David Hernandez
Title: R-matrices, affine quantum groups and applications I.
Abstract: R-matrices are solutions of the quantum Yang-Baxter equation. At the origin of the theoryof quantum groups, they can be interpreted as intertwining operators in representation theory. After reviewing standard constructions from quantum affine algebras, we will present recent development in thetheory. Maulik-Okounkov gave a general geometric construction of stable basis and R-matrices. In another direction, monoidal categorifications of cluster algebras have been established using R-matrices tocategorify Fomin-Zelevinsky mutations relations. If time allows, we will also discuss recent advanceson transfer matrices derived from R-matrices, which give new informations on corresponding quantum
integrable systems as well as on the ODE/IM correspondence seen in the context of affine opers.
Here is a tentative plan for the lectures :1. R-matrices, algebraic and geometric constructions.
2. Categorification of mutation relations.
3. Transfer matrices and spectra of quantum integrable models.
4. Langlands duality and affine opers.
10:05 – 11:05 Pavel Etingof
Title: Introduction to D-modules II.
Abstract: see lecture I.
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11:05 – 11:35 coffee / tea break
11:35 – 12:35 Pavel Etingof
Title: Introduction to D-modules III.
Abstract: see lecture I.
12:35 – 14:30 lunch break
14:30 – 15:30 Guido Pezzini
Title: Geometry and combinatorics of spherical varieties II.
Abstract: see lecture I.
15:30 – 15:50 break
15:50 – 16:45 Maria Gorelik
Title: Duflo-Serganova functor for affine Lie superalgebras.
Abstract: The Duflo-Serganova functor introduced by Dulfo and Serganova in arXiv:math/0507198 assigns to a Lie superalgebra g another Lie superalgebra gx of a smaller defect and provides a functor fromthe g-modules to the gx -modules. In my talk I will present some results on Duflo-Serganova functor foraffine Lie superalgebras. This is a joint work in progress with Vera Serganova.
• Wednesday, January 18, 2017
09:00 – 10:00 Pramod Achar
Title: Introduction to affine Grassmannians and the geometric Satake equivalence II.
Abstract: see lecture I.
10:05 – 11:05 David Hernandez
Title: R-matrices, affine quantum groups and applications II.
Abstract: see lecture I.
11:05 – 11:35 coffee / tea break
11:35 – 12:35 Dennis Gaitsgory
Title: Chiral algebras I.
Abstract:
1. Local theory of chiral algebras.
(a) Definition(b) Commutative chiral algebras(c) Lie-* algebras and chiral universal envelopes(d) Basic examples: Kac-Moody, Heisenberg.
(e) Example: chiral algebra of differential operators
2. Factorization approach and chiral homology(a) Ran space(b) Factorization algebras and equivalence with chiral algebras(c) Chiral homology(d) Sample calculations of chiral homology: the commutative case, univ. env. case.
3. Applications to the geometric Langlands theory:(a) Localization of Kac-Moody representations on BunG .
(b) The case of the chiral algebra of differential operators.
(c) The integrable quotient of the Kac-Moody chiral algebra.
(d) The chiral algebra responsible for the quantum Langlands correspondence.
12:35 – 14:30 lunch break
14:30 – 15:30 Short talks by students: Sylvain Carpentier (Integrable operators), Ryo Fujita (Tiltingmodules in affine highest weight categories), Thorge Jensen (The p-canonical basis of Hecke algebras3and p-cells), Nilamsari Kusumastuti (Kostant principal filtration and paths in weight lattices), Yang
Ruotao (Another definition of singular support for D modules).
15:30 – 15:45 break
15:45 – 16:45 Short talks by students: Léa Bittmann (Baxter relations and quantum Grothendieckrings), Francesco Allegra (Generators for W -algebras of type A), Tomasz Przezdziecki (The Suzuki
functor at the critical level), Leonardo Patimo (The Neron-Severi Lie algebra of a Schubert Variety),
Auguste Hebert (Gindikin-Karpelevich finiteness for Kac-Moody groups over local fields).
• Thursday, January 19, 2017
09:00 – 10:00 David Hernandez
Title: R-matrices, affine quantum groups and applications III.
Abstract: see lecture I.
10:05 – 11:05 Victor Batyrev
Title: Spherical varieties and arc spaces I.
Abstract: Arc spaces of algebraic varieties turn out to be very useful for defining some topological invariants of algebraic varieties via a motivic integration on their smooth birational models. The arc spaces ofspherical homogeneous spaces have been used implicitly in the Luna-Vust theory (1983) of equivariant
valuations and spherical embeddings. The purpose of my talks is to explain my recent joint results withAnne Moreau on arc spaces of spherical embeddings. These results allow in particular to derive a combinatorial formula for computing Betti numbers of an arbitrary smooth projective spherical variety.
11:05 – 11:35 coffee / tea break
11:35 – 12:35 Dennis Gaitsgory
Title: Chiral algebras II.
Abstract: see lecture I.
12:35 – 14:30 lunch break
14:30 – 15:30 Ivan Mirkovic
Title: Generalized affine Grassmannians II.
Abstract: see lecture I.
15:30 – 15:50 break
15:50 – 16:45 Alexander Premet
Title: Classification of the maximal subalgebras of exceptional Lie algebras in good characteristicAbstract: Let G be an exceptional simple algebraic group over an algebraically closed field of characteristic p > 3. If G is of type E8 suppose that p > 5. In my talk based on a joint work with David Stewart
I will describe the maximal Lie subalgebras of the Lie algebra of G.
• Friday, January 20, 2017
09:00 – 10:00 Victor Batyrev
Title: Spherical varieties and arc spaces II.
Abstract: see lecture I.
10:05 – 11:05 Pramod Achar
Title: Introduction to affine Grassmannians and the geometric Satake equivalence III.
Abstract: see lecture I.
11:05 – 11:35 coffee / tea break
11:35 – 12:35 Ivan Mirkovic
Title: Generalized affine Grassmannians III.
Abstract: see lecture I.
12:35 – 14:30 lunch break
14:30 – 15:30 Dennis Gaitsgory
Title: Chiral algebras III.
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Abstract: see lecture I.
15:30 – 15:50 break
15:50 – 16:45 Daniel Juteau
Title: Support of the spherical representation of the rational Cherednik algebra.
Abstract: I will give a very simple criterion, obtained in collaboration with Stephen Griffeth, to determine
the support of the spherical representation of a general rational Cherednik algebra (associated to anycomplex reflection group, and any parameters), assuming some conjectures on Hecke algebras which
are known for Coxeter groups but not for all complex reflection groups. In any case, we obtain a much
simpler proof of Etingof’s criterion for Coxeter groups (which itself generalized a result of Varagnoloand Vasserot for Weyl groups with equal parameters), and a simpler criterion for the infinite series thanwhat was previously known.
All talks take place at the ESI, Boltzmann Lecture Hall!

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