# DAHA, Hall Algebras, Torus knots and Combinatorics

**DAHA, Algèbres de Hall, Noeuds Toriques et Combinatoire**

**DAHA, Hall Algebras, Torus Knots and Combinatorics**

**Orsay, 17-18-20 Nov. 2014**

**Sophie Germain, 19 Nov. 2014**

**Orateurs**

** F. BERGERON P. BOALCH B. DAVISON E. GORSKY T. HAUSEL M. McBREEN**

** M. NAZAROV A. NEGUT F. SALA P. SAMUELSON O. SCHIFFMANN B. SZENDROI**

** Programme**

** **Lundi 17 (Bât. 425, 113-115, 117-119)

- 10h ---> : O. Schiffmann
- 13h30 ---> : F. Bergeron
- 16h ---> : E. Gorsky

Mardi 18 (Bât. 430, E. LEDERER)

- 9h30 ---> : M. Nazarov
- 11h ---> : M. McBreen
- 13h30 ---> : P. Samuelson
- 16h ---> : P. Boalch

Mercredi 19 ( Univ. Paris Diderot, Bât. Sophie Germain, Amphi Turing ***)

- 10h ---> : A. Negut
- 13h30 ---> : B. Szendroi
- 16h ---> : T. Hausel

Jeudi 20 (Bât 430, E. LEDERER, Bât 425, 117-119)

- 10h ---> : B. Davison
- 13h30 ---> : F. Sala

*** : l' accès se fait au RdC, en face des ascenceurs, c.f. : http://www.math.univ-paris-diderot.fr/ufr/batsophiegermain

**Titres / Résumés**

**F. Bergeron (Montréal) : ***The combinatorial side (and more) of the elliptic hall algebra*

**P. Boalch (Orsay) : ***Wild character varieties*

The wild character varieties are a new class of symplectic/Poisson varieties that generalise the complex character varieties of Riemann surfaces. They were first defined analytically in 1999 and more recently there is a purely algebraic approach generalising the quasi-Hamiltonian framework, involving group valued moment maps. They give the simplest description of the symplectic varieties underlying the hyperkahler manifolds appearing in wild nonabelian theory. In this talk I will describe several examples, leading up to the new theory of multiplicative quiver varieties that naturally appears. If time

permits I will discuss the "fission algebras" that control these multiplicative quiver varieties, generalising the deformed multiplicative preprojective algebras of Crawley-Boevey--Shaw (examples of which contain the generalised DAHAs of Etingof-Oblomkov-Rains).

**E. Gorsky (Columbia) : ** *Rational Cherednik algebras and Hilbert schemes*

In a joint work with Pavel Etingof and Ivan Losev, we identified the characters of minimally supported representations of rational Cherednik algebras with the colored HOMFLY invariants of torus knots. I will review this relation and, in the finite-dimensional case, the conjectural relation to HOMFLY homology. In the second part of the talk (based on a joint work with Andrei Negut), I will describe an explicit sheaf on the Hilbert scheme of points on C^2 whose equivariant Euler characteristic matches the "refined HOMFLY invariant" of Aganagic-Shakirov and Cherednik. Conjecturally, a "quantization" of his sheaf is the finite-dimensional representation of Cherednik algebra.

**T. Hausel (Lausanne) :** *Towards p-adic harmonic analysis on character and quiver varieties*

First I will discuss the state of the art of arithmetic harmonic analysis on character and quiver varieties. This includes a conjecture on some sort of mixed Hodge polynomial of twisted and untwisted wild character varieties in terms of certain refined invariants of torus knots and links. This part surveys results with Villegas, Letellier, Mereb, Wong and Wyss. Then I motivate the study of p-adic harmonic analysis on such varieties. I will finish with a discussion of results of representations of quivers over Frobenius rings. This part is based on joint work with Letellier and Villegas.

**M. McBreen (Lausanne) :*** Quantum Cohomology and Intersection Cohomology*

I will discuss joint work with Nicholas Proudfoot, which relates the intersection cohomology of a singular affine hypertoric variety to the quantum cohomology of its resolution. We conjecture that a similar relation holds for all conical symplectic resolutions.

**A. Negut (Columbia) :** *Hilbert schemes, vertex operators and shuffle algebras*

This talk will pick up where Eugène Gorsky's talk will have left off. I will discuss the (gl_1) shuffle algebra and its two incarnations: geometric (Hilbert schemes) and algebraic (vertex operators). I will mention a combinatorial consequence known as "the m/n Pieri rule", which is a generalization of the work of Leclerc-Lascoux-Thibon on ribbon tableaux. The first part of the talk will be general and survey in style, then after a short break, I will discuss some of the more technical aspects of working with the shuffle algebra.

**M. Nazarov (York) :** *Lax operator for Macdonald symmetric functions*

This is a joint work with Evgeny Sklyanin. Using the Lax operator formalism, we construct a family of pairwise commuting operators such that the Macdonald symmetric functions of infinitely many variables and of two parameters $q,t$ are their eigenfunctions. We express our operators in terms of the Hall-Littlewood symmetric functions of the same variables and of the parameter $t$ corresponding to the partitions with one part only. Our expression is based on the notion of Baker-Akhiezer function.

**F. Sala (London, Ontario) :** *Sheaves on root stacks, Nakajima quiver varieties and gauge theories on A-type ALE spaces*

In the first part of the talk, I will describe a new (conjectural) relation between moduli spaces of framed sheaves on some two-dimensional root toric stacks and Nakajima quiver varieties of type the affine Dynkin diagram A^{(1)}_n. This relation yields also a new approach to the study of gauge theories on A-type ALE spaces and their Alday-Gaiotto-Tachikawa conjectures, which will be the subject of the second part of the talk.

I will describe an action of the affine Lie algebra of type gl(n+1) on the equivariant cohomology of moduli spaces of framed sheaves introduced before and its relation with the AGT conjecture in the massless case. Moreover, I will introduce Carlsson-Okounkov vertex operators associated with universal sheaves over these moduli spaces, I will characterize them and I will state state their relations with AGT conjectures in the case of gauge theories with masses.

If time permits, I will describe a relation between products of elliptic Hall algebras and the equivariant K-theory of these moduli spaces, and a possible extension of the AGT conjectures in this case.

**P. Samuelson (Toronto)** : *Topological Representation Theory*

There is a topological construction (depending on a group G) which associates an algebra S(F) to a thickened surface F x [0,1], and an S(F)-module to a 3-manifold with boundary F. The underlying vector spaces are quantizations of representations of the fundamental group into G, and multiplication is given by gluing boundary components. I will first give a survey of a few things that are known about this construction, including relations to double affine Hecke algebras and knot invariants. I then describe a recent result with Morton which (roughly) involves this construction when g = gl_\infty: the Homfly skein algebra of the torus is isomorphic to the t=q specialization of the elliptic Hall algebra.

**O. Schiffmann (Orsay)** : *Hall-Lie algebras of curves, Moduli spaces of Higgs bundles and shuffle algebras*

We will introduce the Lie algebra analog of the spherical Hall algebra of a smooth projective curve X of genus g, state (and motivate) a conjecture relating the root multiplicities of such a Lie algebra to the cohomology of the moduli space of stable Higgs bundles on one hand, and to the Poincarré series of the (spherical) cohomological / K-theoretical Hall algebra of the g-loop quiver on the other hand. We will also explain how to compute the cohomology of the moduli space of stable Higgs bundles and hence get a (conjectural) expression for the root multiplicities of the Hall-Lie algebra of X.

**B. Szendroi (Oxford) :** *Euler characteristics of Hilbert schemes of simple surface singularities*

Given a smooth surface, the generating series of Euler characteristics of its Hilbert schemes of points can be given in closed form by (a specialisation of) Goettsche's formula. I will discuss a generalisation of this formula to surfaces with rational double points. A certain representation of the affine Lie algebra corresponding to the surface singularity (via the McKay correspondence), and its crystal basis theory, play an important role in our approach. (Joint work with Adam Gyenge and Andras Nemethi, Budapest)

Avec le financement de l'ANR VARGEN, ANR-13-BS01-0001-01

Contact : Olivier.Schiffmann--@--math.u-psud.fr