Tomoyuki Arakawa, Nilpotent orbits and minimal models of W-algebras.

In 1991 Frenkel, Kac and Wakimoto conjectured the existence and the construction of rational W-algebras associated principal nilpotent elements. In 2008 Kac and Wakimoto studied the rationality of W-algebras associated with other nilpotent elements and gave a conjectural list of rational W-algebras. In my talk I will prove the modular invariance of the trace functions of the modules over these W-algebras in the case when there are vertex operators algebras (that is, when they are Z_+-gradable) and prove the (generalized) version of their conjecture in some cases, including all cases of principal nilpotent elements, all cases in type A, and the ADE subregular cases.

Michel Duflo, On Frobenius Lie subalgebras of sl(n).

Frobenius Lie algebras are Lie algebras which have an open coadjoint orbit. I will present results (obtained in collaboration with Rupert Yu) on the Frobenius parabolic or biparabolic subalgebras of the Lie algebra sl(n,C).

Pavel Etingof, Cherednik algebras and torus knots.

The Cherednik algebra B(c,n), generated by symmetric polynomials and the quantum Calogero-Moser Hamiltonian, appears in many areas of mathematics. It depends on two parameters - the coupling constant c and number of variables n. I will talk about representations of this algebra, and in particular about a mysterious isomorphism between the representations of B(m/n,n) and B(n/m,m) of minimal functional dimension. This symmetry between m and n is made manifest by the fact that the characters of these representations can be expressed in terms of the colored HOMFLY polynomial of the torus knot T(m/d,n/d), where d=GCD(m,n). I will also talk about generalizations to other roots systems, as well as applications to computing characters of simple equivariant D-modules on the nilpotent cone of sl(n) and the Cohen-Macaulay property of the algebra of deformed power sums (a conjecture of Sergeev and Veselov). The talk is based on my joint work with E. Gorsky and I. Losev.

Giovanni Felder, Derived representation schemes and combinatorial identities.

Representation schemes parametrize representations of associative algebras on a given vector space. I will review a derived version of this theory, due to Berest, Khachatryan and Ramadoss, and present simple examples, such as the algebra of polynomials in two variables, featuring phenomena that are visible in computer experiments, and only partly understood mathematically. I will present some conjectures that lead to new combinatorial identities, partly proven and partly still conjectural. (Based on joint work with Y. Berest and A. Ramadoss and with Y. Berest, A. Patotski, A. Ramadoss and T. Willwacher)

Antonio Giambruno, Polynomial identities and growth.

I will give an overview of the results regarding a growth function associated to the polynomial identities satisfied by an algebra in characteristic zero. Whereas the associative case is quite understood, for Lie algebras many different scattered phenomena appear.

Maria Gorelik, Affine generalized root systems.

We suggest a notion of root system which generalizes systems of real roots of symmetrizable Kac-Moody Lie superalgebras. Particular examples of our notion are the generalized root systems of V. Serganova, as well as the affine root systems of I. Macdonald and the "symmetric" infinite root systems introduced by R. Moody and A. Pianzola. This is a joint work with Ary Shaviv.

Daniel Juteau, Modular Generalized Springer Correspondence.

For a reductive group G, the Springer correspondence is an injection from irreducible representations of the Weyl group W to the simple G-equivariant perverse sheaves on the nilpotent cone of the Lie algebra (or the unipotent variety of the group). However, in general not all simple perverse sheaves arise in this way. This led Lusztig to define a generalized Springer correspondence, involving the process of inducing cuspidal perverse sheaves from Levi subgroups. The classical correspondence is the part coming from a maximal torus. In the case of the general linear group, 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 the modular version of the generalized Springer correspondence. This is joint work with Pramod Achar, Anthony Henderson and Simon Riche.

Shrawan Kumar, Positivity in T-equivariant K-theory of flag varieties associated to Kac-Moody groups.

Let X=G/B be the full flag variety associated to a symmetrizable Kac-Moody group G. Let T be the maximal torus of G. The T-equivariant K-theory of X has a certain natural basis defined as the dual of the structure sheaves of the Schubert varieties. We show that under this basis, the structure constants are polynomials with nonnegative coefficients.

Paolo Papi, Covariants in the exterior algebra of a simple Lie algebra.

For a simple complex Lie algebra g we study the space of invariants A = (Λg⊗g)^g (which describes the isotypic component of type g in Λg) as a module over the algebra of invariants (Λg)^g. As main result we prove that A is a free module, of rank twice the rank of g, over the exterior algebra generated by all primitive invariants in (Λg)^g with the exception of the one of highest degree. We will also discuss generalizations of this result and some related problems. Joint project with C. De Concini and C. Procesi (and partly with P. Moseneder Frajria).

Alexander Premet, Maximal subalgebras of exceptional Lie algebras over fields of good characteristic.

The first important result on maximal subalgebras of exceptional Lie algebras over complex numbers was proved by Morozov in the 40s and a complete list of maximal subalgebras was obtained by Dynkin in the 50s. The problem of classifying maximal connected subgroups of exceptional algebraic groups over fields of characteristic p>0 was solved in a series of papers by Seitz, Testerman and Liebeck-Seitz. In my talk, based on a joint work with David Stewart, I'm going to report on the current state of the classification problem of maximal subalgebras of exceptional Lie algebras L over fields of prime characteristic.

Claudio Procesi, Fundamental algebras and invariant theory.

A rather general equivalence in associative algebras is PI equivalence, two algebras are PI equivalent if they satisfy the same polynomial identities.

I will discuss the special case of finite dimensional algebras, in which one can reconstruct special representatives in a PI equivalence class, by the appearance of algebras of invariants in a rather indirect and unexpected way.

Vera Serganova, Deligne's categories and Lie superalgebras.

Deligne's categories are universal tensor categories generated by one object and parametrized by the dimension of this object. When the parameter is not an integer, these categories are semisimple, when it is an integer the categories are not abelian. We suggest a construction of an abelian envelope of the Deligne category using representation theory of algebraic supergroups.

Alexey Sevastyanov, A proof of De Concini-Kac-Procesi conjecture and Lusztig's partition.

In 1992 De Concini, Kac and Procesi observed that isomorphism classes of irreducible representations of a quantum group at odd primitive root of unity m are parameterized by conjugacy classes in the corresponding algebraic group G. They also conjectured that the dimensions of irreducible representations corresponding to a given conjugacy class O are divisible by m^{1/2 dim O}. In this talk I shall outline a proof an improved version of this conjecture and derive some important consequences of it related to q-W algebras.

A key ingredient of the proof are transversal slices S to the set of conjugacy classes in G. Namely, for every conjugacy class O in G one can find a special transversal slice S such that O intersects S and dim O=codim S. The construction of the slice utilizes some new combinatorics related to invariant planes for the action of Weyl group elements in the real reflection representation. The condition dim O=codim S is checked using some new mysterious results by Lusztig on intersection of conjugacy classes in algebraic groups with Bruhat cells.

Michela Varagnolo, On the center of quiver Hecke algebras.

I will speak about another way to decategorify, using traces and center instead of the Grothendieck group. In the case of the the so-called minimal categorification (constructed using the KLR algebras) this kind of construction produces representations of a current algebra. As an application we can compute the cohomology ring of some Nakajima quiver varieties. It is a joint work with Peng Shan and Eric Vasserot.

Eric Vasserot, Categorical representations, Cherednik algebras and finite groups.

We’ll first review some basic facts on representations in categories, including some application to the representation category of Cherednik,algebras. Then we’ll explain some new applications (in progress) to the modular representation category of finite unitary groups.

Michèle Vergne, Some qualitative and quantitative properties of Kronecker coefficients.