Abstracts of Talks

The character topological field theory

David Ben-Zvi

In it will discuss joint work with David Nadler. We construct parts of a new three dimensional topological quantum field theory, which organizes representation theories associated to a complex reductive group, including Lusztig's character sheaves, Harish-Chandra modules for real forms of the group, and conjecturally the mixed Hodge theory of character varieties of the group. The character theories for Langlands dual groups are equivalent, leading to a collection of dualities for the objects listed above.

Reference

Supports of irreducible spherical representations of rational Cherednik algebras of finite Coxeter groups

Pavel Etingof

I will explain how to determine the support of the irreducible spherical representation (i.e., the irreducible quotient of the polynomial representation) of the rational Cherednik algebra of a finite Coxeter group for any value of the parameter c. In particular, this allows one to determine for which values of c this representation is finite dimensional. This generalizes a result of Varagnolo and Vasserot, arXiv:0705.2691, who classified finite dimensional spherical representations in the case of Weyl groups and equal parameters (i.e., when c is a constant function). Our proof is based on the Macdonald-Mehta integral and the elementary theory of distributions.

Parity Sheaves (based on joint work with C. Mautner and G. Williamson)

Daniel Juteau

The Kostka polynomials make the transition between Schur functions and Hall-Littlewood polynomials. Lusztig also interpreted them in terms of the stalks of intersection cohomology sheaves on the nilpotent cone of the general linear group (going through the affine Grassmannian and the affine flag manifold, so that they are also expressed in terms of Kazhdan-Lusztig polynomials).

If one considers positive characteristic coefficients, the intersection cohomology sheaves are not so well behaved. For example, their stalks do not necessarily satisfy a parity vanishing. However, one can consider the indecomposable complexes which satisfy a parity vanishing for their stalks and costalks, and it turns out that they are still classified by partitions. They are the direct summands of direct images of constant sheaves on resolutions of partial flag varieties. If the characteristic is large enough, they are the intersection cohomology complexes. In general, their stalks yield generalizations of Kostka polynomials (which are combinations of the classical ones), depending on the characteristic.

In our article, we work in a more general setting. Under certain conditions which are often satisfied in "representation theoretic situations" (including the above mentioned nilpotent cone, Kac-Moody Schubert varieties like the affine Grassmannian, and also toric varieties), the indecomposable constructible complexes having a parity vanishing for stalks and costalks are parametrized by pairs consisting of an orbit and an irreducible local system (just like simple perverse sheaves, in the equivariant setting). In the case of a semismall "even" resolution, we express the multiplicities of the parity sheaves in the direct image of the constant sheaf as the ranks modulo p of certain intersection forms (appearing in the work of de Cataldo and Migliorini). This gives a measure of the failing of the decomposition theorem with positive characteristic coefficients. On the affine Grassmannian, parity sheaves correspond to tilting modules under the geometric Satake correspondence, when the characteristic is large enough (greater than h + 1, where h is the Coxeter number, is enough).

Vertex operators in Donaldson-Thomas theory

Nagao Kentaro

I will introduce Okounkov-Reshetikhin-Vafa type vertex operators to compute the generating function of Donaldson-Thomas type invariants of a small crepant resolution of a toric Calabi-Yau 3-fold. The commutator relation of the vertex operators gives the wall-crossing formula of Donaldson-Thomas type invariants.

Reference

Intersection cohomology on character/quiver varieties and the character ring of finite general linear groups

Emmanuel Letellier

Here we show that the "generic" part of the character ring of finite general linear groups can be described in terms of intersection cohomology of character and quiver varieties.

Geometry of character sheaves

David Nadler

The aim of this talk will be to explain concrete geometric pictures of field theoretic phenomena appearing in joint work with David Ben-Zvi on character sheaves.

Quantum cohomology of the Hilbert scheme of points on An-resolutions

Alexi Oblomkov

We determine the two-point invariants of the equivariant quantum cohomology of the Hilbert scheme of points of surface resolutions associated to type An singularities. The operators encoding these invariants are expressed in terms of the action of the affine Lie algebra gl(n+1) on its basic representation. Assuming a certain nondegeneracy conjecture, these operators determine the full structure of the quantum cohomology ring. A relationship is proven between the quantum cohomology and Gromov-Witten/Donaldson-Thomas theories of An x P1. The talk is based on the joint paper with Davesh Maulik.

Counting points of character varieties over finite fields

Fernando Rodrigues-Villegas

I will give an overview of the results on the topic of the title obtained in collaboration with T. Hausel and E. Letellier. The varieties in question are those parameterizing representations of the fundamental group of a punctured Riemann surface to GL_n with values in prescribed generic semisimple conjugacy classes at the punctures.

The results are best expressed as a specialization of a generating series involving the Macdonald polynomials. We conjecture that the full generating series actually gives the mixed Hodge polynomials of the varieties. We prove that taking the pure part of these polynomials, which amounts to a different specialization of the generating series, actually gives the number of points of an associated quiver variety over finite fields.

Hall algebras of curves and Langlands duality

Olivier Schiffmann

The geometric Langlands duality principle predicts an equivalence of (derived) categories between the category of coherent sheaves over the moduli space of rank r local systems over a curve X and the category of D-modules over the moduli space of rank r vector bundles over X. Using the theory of Hall algebras, we give a version of this equivalence, at the level of Grothendieck groups, for local systems in the formal neighborhood of the trivial local system. This is valid in fact for any curve and any group. We will illustrate this with the example of elliptic curves : this particular case draws a bridge between Haiman's construction of Macdonald polynomials using the K-theory of Hilbert schemes and a similar construction based on Eisenstein series for elliptic curves due to E. Vasserot and myself.

k-shape functions and cohomology of the affine Grassmannian

Mark Shimozono

k-Schur functions were invented by Lapointe, Lascoux, and Morse, due to their apparent triangularity (over N[q,t]) with modified Macdonald polynomials. At t=1 the k-Schur (resp. dual k-Schur) functions represent Schubert homology (resp. cohomology) classes for the affine Grassmanian of SL_{k+1}, as proved by Lam. We introduce a new family of symmetric functions called (dual) k-shape functions, which should be viewed as effective cohomology classes in the affine Grassmannian. We give a combinatorial decomposition of dual k-shape functions into Schubert classes (dual k-Schur functions). Dualizing, we obtain the explicit expansion of a k-1-Schur function into k-Schur functions. Iterating this, we obtain an explicit positive expansion of k-Schur functions into Schur functions, which is related to the positivity result of Assaf and Billey. We conjecture that the above expansion coefficients agree with the expansion into irreducibles, of Chen's graded symmetric group modules.
This is joint work with Thomas Lam, Luc Lapointe, and Jennifer Morse.

Gelfand-Tsetlin bases via Laumon spaces

Sasha Tsymbaliuk

I will talk about Laumon spaces and some aspects of representation theory coming through them. Laumon spaces are natural geometric moduli spaces, which can be as well be defined through quivers. Analogously to H. Nakajima's construction it turns out not to be useless to consider the sum of equivariant cohomologies or K groups. The natural geometric "bases of fixed points" coincides with a Gelfand-Tsetlin bases in those cases. One goes further if deals with a generalized affine Laumon spaces. In these cases we get a representations of affine yangians and quantum toroidal algebras. This gives a motivation to call the bases of fixed points in this generalized case as quantum Gelfand-Tsetlin bases.

The talk will be based on

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