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one-day workshop onÌýMarch 18thÌýatÌýÌý(5th floor of UQaM'sÌý).Ìý

Everyone is welcome to attend. The rough subject of the workshop isÌýrepresentation theory of double-affine quantum groups, Cherednik algebras and Coulomb branches.ÌýTalks will start at 10h00 in room PK-5115. The afternoon talks will take place in room PK-5675.Ìý

Ìý

Ìý(Université de Montréal, 10h00—11h00)

Title:ÌýGeneralisations of Macdonald–Ruijsenaars operators in external fields and a subalgebra of DAHA

AbstractÌý

Firstly, I will present a construction of commuting elements in the double affine Hecke algebra (DAHA) of type GL_n whose action on symmetric polynomials leads to generalisations of Van Diejen's Macdonald–Ruijsenaars system with an external Morse potential and other new integrable q-difference operators. Secondly, as another application of this construction, I will discuss a subalgebraÌýAÌýinside the DAHA that flatly deforms the crossed product of the symmetric group with the image of the Drinfeld–Jimbo quantum group U_q(gl_n) under its q-oscillator representation. The algebraÌýAÌýreduces in the q = 1 limit to the degree zero part of the corresponding rational Cherednik algebra. The degree zero part is a flat deformation of the crossed product of the symmetric group with a quotient of the universal enveloping algebra of gl_n, and it is related to generalised Howe duality and to the Calogero–Moser integrable system in an external harmonic potential. This talk is based on a joint paper with Misha Feigin and a current work in progress.

Ìý(Harvard, 11h00—12h00)Ìý–

Title:ÌýGraded traces on quantized Coulomb branches

AbstractÌý

Higgs and Coulomb branches of quiver gauge theories form two important families of Poisson varieties that are expected to be exchanged under so-called 3D mirror symmetry. The representation theory of quantized Coulomb branches is deeply connected with the enumerative geometry of Higgs branches. One important approach to studying modules over quantized Coulomb branches is by analyzing their graded traces. Graded traces generalize the notion of characters and are closely related to the q-characters introduced by Frenkel and Reshetikhin. Any graded trace defines a solution of the D-module of graded traces introduced by Kamnitzer, McBreen, and Proudfoot.

Ìý

In this talk, I will discuss techniques that allow one to explicitly compute characters and graded traces of certain modules over quantized Coulomb branches, using the D-module of graded traces combined with analytic methods. Time permitting, I will explain how some of these results naturally appear on the Higgs side, leading to an explicit description of the D-module of graded traces for a quantized Coulomb branch via the geometry of the Higgs branch. We prove these results for ADE quivers and formulate some conjectures in the general case. Talk is based on joint works with Dinkins, Karpov, Klyuev, and Lance.

Ìý(University of Edinburgh, 13h30—14h30)

Title:ÌýRepresentations of quantum toroidal algebras

AbstractÌý

Quantum toroidal algebras are the ‘double affine’ objects within the quantum world. Their principal module category Ô is the natural toroidal analogue of the finite-dimensional modules for quantum affine algebras.

Ìý

After introducing these algebras and discussing their structure, we shall outline some recent results on their representation theory. These include a well-defined tensor product and monoidal structure on Ô, compatible with both Drinfeld polynomials and q-characters, and a meromorphic braiding by R-matrices.

Ìý

Time permitting, I’ll briefly mention work in progress with Théo Pinet exhibiting special subcategories of Ô as monoidal categorifications of cluster algebras in type A.

Ìý(MIT, 14h30—15h30)Ìý–

Title:ÌýPursuing equivariant sheaves on the double affine Grassmannian

AbstractÌý

I will explain how to view equivariant coherent sheaves on the affine Grassmannian as sheaves on an affine Grassmannian slice with a certain additional structure. Then I will speculate how one could try to generalize this to non-finite types, using the coproduct for Coulomb branches, which is to be defined. I will explain the relation of this construction to the category of Poisson sheaves, and discuss the perverse coherent t-structure on this category.

Ìý(Université de Sherbrooke, 16h00—17h00)

Title:ÌýTwisted Yangians and fixed points in the affine GrassmannianÌý

Abstract

Ìý

Yangians are infinite-dimensional quantum groups, which arise naturally as quantizations of enveloping algebras of current algebras.Ìý Via the quantum duality principle, they can also be viewed as quantizations of loop groups.Ìý This ultimately leads to a variety of other connections, including to affine Grassmannian slices and to the theory of Coulomb branches.

Ìý

Twisted Yangians are closely related algebras, which have been studied essentially the introduction of Yangians themselves, and which relate to symmetric pairs.Ìý A natural question is whether they can also be viewed as quantizations of some geometry related to loop groups.Ìý In this talk, I will discuss joint work with Kang Lu and Weiqiang Wang where we address this question.Ìý We also discuss relations to affine Grassmannian slices, potential connections to Coulomb branches, and potential generalizations such as to affine types.