Sophus Lie 2013 – Lyon – June, 27-29
Program and Abstracts

Thursday 27th. (afternoon)
1:50 - 2:50 PM Nicolas Perrin (Bonn)
Some properties of the cominuscule quantum K-theory
Abstract. (Joint work with A. Buch, P.E. Chaput and L. Mihalcea) In
this talk, I will recall some classical properties of the quantum cohomologyof homogeneous spaces (finitness of the product, positivity, existence ofinvertibles) and discuss their extensions to quantum K-theory.
3:00 - 4:00 PM Peter Bürgisser (Paderborn)
The Challenges of Geometric Complexity Theory
Abstract. It is a remarkable fact that two prominent problems of algebraic complexity theory, the permanent versus determinant problem andthe tensor rank problem, can be restated as explicit orbit closure problems. This offers the potential for proving lower complexity bounds byrelying on methods from algebraic geometry and representation theory.
This basic idea for the tensor rank problem goes back to work by Volker
Strassen from the mid eighties. It leads to challenging problems regarding the irreducible representions of symmetric groups over the complex
numbers (tensor products and plethysms).
In the first part of the talk, we will present the general framework andexplain some negative results. Then we will move on to outline somerecent progress for proving lower bounds on the border rank of the matrix
multiplication tensor. This is achieved by the explicit construction ofhighest weight vectors vanishing on the (higher secant) varieties of tensors
of border rank at most r.
4:30 - 5:30 PM Olivier Mathieu (CNRS, Lyon 1)
On the classification of ZN -graded Lie algebras (joint work with K. Iohara).
Abstract. Set Λ = ZN . We consider Λ = ZN -graded Lie algebras L =⊕λ∈Λ Lλ such that each homogenous component Lλ is one dimensional,
and we assume that L is graded simple.
Around 1978, V.G. Kac conjectured the classification of such Lie algebras
in the case N = 1 (proved by the author in 1983). In the 90’s, I.M. Gelfand

1and A. Kirillov raise the question of the classification of such Lie algebras
in the case N ≥ 2, but without providing an explicit list.
In this talk, we will explain how to solve Gelfand-Kirillov question. Altough the setting of the question is very abstract, it turns out that theLie algebras occuring in the list are very concrete. Most of them are connected with the Lie algebras of symbols of twisted PDO on the circle. The
remaining Lie algebras are classified by using Jordan algebras theory.

Friday 28th. (morning session)
8:50 - 9:50 AM Andrei Jaikin-Zapirain (Madrid)
The action of the absolute Galois group on quasiplatonic curves and Beauvillesurfaces.
Abstract. We prove that the absolute Galois group acts faithfully on
quasiplatonic curves and on Beauville surfaces. We also show that forevery automorphism σ of Q different from the identity and the complex
conjugation, there exists a Beauville surface S such that the fundamentalgroups of S and S σ are not isomorphic.
One of the main steps in the proof is to show that an automorphism of a
profinite completion of a triangle group of hyperbolic type fixing all theopen normal subgroups should be inner.
10:00 - 11:00 AM Nikolay Nikolov (Imperial college, London)
Abstract quotients of compact groups I
11:30 - 12:30 AM Dan Segal (Oxford)
Abstract quotients of compact groups IIAbstract (for I and II). Generalizing earlier work on subgroups of finite
index in profinite groups, we show that a compact group cannot have aninfinite finitely generated quotient (as abstract group). We will discuss
some of the ideas leading up to this.

Friday 28th. (afternoon session)
1:50 - 2:50 PM Michela Varagnolo (Cergy)
Categorifications and cyclotomic rational double affine Hecke algebras
Abstract. I will speak about an equivalence between the category O
for CRDAHA’s and a subcategory of an affine parabolic category O of2

type A. I will discuss two applications, that is a proof of a conjecture ofRouquier on the dimension of simple modules of CRDAHA’s and a proofof a conjecture of Chuang and Miyachi on the Koszulity of the category
O for CRDAHA. It is a joint work with Rouquier, Shan, Vasserot.
3:00 - 4:00 PM Ghilain Fourier (Köln)
Posets and Schur positivity
Abstract. We introduce a partial order on tuples of dominant weights forclassical simple Lie algebras. There are several conjectures on the modules
corresponding to the weights, that come along with this partial order. We
will report on their relation and on the process being made in proving
them. The main target of this project, a conjecture on generators andrelations for fusion products of current algebras, will be recalled and its
implication to the conjectures about the partial order will be emphasized.
4:30 - 5:30 PM Michel Brion (Grenoble)
Associative structures on algebraic varieties.
Abstract. The objects of this talk are the structures of algebraic semigroup on a given algebraic variety X. We will determine all such structures
when X is complete, in terms of certain abelian varieties associated with X.
Reference: M.B., On algebraic semigroups and monoids, arXiv:1208.0675(v3)

Saturday 29th. (morning session)
8:50 - 9:50 AM Geordie Williamson (MPIM, Bonn)
Hodge structures and Soergel bimodules
Abstract. I will explain joint work with Ben Elias in which we use Soergel
bimodules to give an algebraic proof of the Kazhdan-Lusztig conjecture,a character formula for simple highest weight modules over a semi-simple
Lie algebra. In the proof ideas from higher algebra, categorification andHodge theory (in particular work of de Cataldo and Migliorini) play a
crucial role. If time permits I will explain how essentially everything that
one expects from Hodge theory carries over to arbitrary Soergel bimodules,
except for the existence of an integral lattice.
10:00 - 11:00 AM Daniel Juteau (Caen)
Generalized Springer correspondence modulo ` for GLn
Abstract. For a reductive group G, the Springer correspondence is an injection from irreducible representations of the Weyl group W to the simple3

G-equivariant perverse sheaves on the nilpotent cone of the Lie algebra.
However, in general not all simple perverse sheaves in this way, which
led Lusztig to define a generalized Springer correspondence, involving theprocess of inducing cuspidal perverse sheaves from Levi subgroups (theclassical correspondence being the case of a maximal torus). In the caseof GLn , though, nothing new arises in this way.
In my thesis I studied a modular Springer correspondence, where one
takes modular representations of the Weyl group and perverse sheaves
with positive characteristic coefficients. In this talk I will explain themodular version of the generalized Springer correspondence in the caseof GLn . In the modular case there is something new, namely there is acuspidal perverse sheaf supported by the regular nilpotent orbit when n isa power of `. This is joint work with Pramod Achar, Anthony Hendersonand Simon Riche.
11:30 - 12:30 PM Wolfgang Soergel (Freiburg)
Motivic construction of graded representation categories
Abstract. I want to explain how the motivic six-functor-formalism ofCisinsky and Deglise allows to construct some of these curious gradedversions of representation theories in a very natural and geometric way.

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