# Representation theory seminar 2020 Semester 1

Organisers: Gufang Zhao, Ting Xue

Time: Tuesdays 3:15-5:15pm

Location: Peter Hall Room 107 (Online from March 31)

June 9, Sasha Garbali (Melbourne)

Title: The Fock R-matrix of the quantum toroidal $\mathfrak{gl}_1$

Abstract: The quantum toroidal $\mathfrak{gl}_1$ algebra has an exciting representation theory and multiple connections to various problems in mathematical physics. An interesting direction is the study of the integrable model defined by the R-matrix on the Fock representation of quantum toroidal $\mathfrak{gl}_1$. I will focus on the problem of calculation of the R-matrix using the coproduct relation and show how this relation can be rephrased in the language of the Macdonald theory of symmetric functions.

May 26, Chenyan Wu (Melbourne)

Title: $(\chi,b)$-factors of Arthur parameters for symplectic or metaplectic group

Abstract: Let $\sigma$ be a (genuine) cuspidal automorphic representation of the symplectic or metaplectic group. Arthur defined  the global $A$-parameter of $\sigma$. Via the Waldspurger correspondence, this is also defined for the metaplectic groups by work of Gan and Ichino. When $\chi$ is a quadratic character, we give constraints on existence of $(\chi, b)$-factors in the global $A$-parameter of $\sigma$ in terms of the so-called lowest occurrence index of theta lifts to various orthogonal groups of quadratic spaces with discriminant $\chi$. We also give a refined result that relates the first occurrence index to non-vanishing of period integrals of residues of Eisenstein series associated to the cuspidal datum $\chi\otimes\sigma$.

May 19, Penghui Li (Tsinghua University)

Title: Whittaker sheaf, commuting scheme and Geometric Langlands conjecture.

Abstract: It is an long standing open problem that whether the scheme of commuting matrices is reduced. We suggest a way to tackle this problem via Langlands duality. In the talk, we briefly recall the definition of commuting scheme, and how it is related to Ben-Zvi–Nadler’s Betti Geometric Langlands (BGL) conjecture. Then we summarize recent progresses on BGL conjecture, and sketch a proof of reduceness of the ring of invariant functions on commuting schemes, based on the conjecture in genus 1. This work is based on a joint work with David Nadler.

May 12, Dongwen Liu (Zhejiang University)

Title: Langlands correspondence, local descent and algebraic wavefront set

Abstract: In this talk we introduce the local descent for classical groups, which is a refinement of the local Gan-Gross-Prasad conjecture. As an application we can determine the wavefront set of an irreducible tempered representation in terms of its L-parameters. This is a joint work in progress with Dihua Jiang and Lei Zhang.

May 5, Jack Hall (Melbourne)

Title: Coherent completeness

Abstract: If $(R,\mathfrak{m})$ is a complete local Noetherian ring, then there is a well-known equivalence of categories in commutative algebra:  f.g. $R$-modules <—–> inverse systems $\{M_n\}$, where each $M_n$ is a f.g. $R/\mathfrak{m}^n$-module. I will discuss a useful generalization of this to representation theory and equivariant geometry.

Apr 28, Ryo Fujita (Kyoto University)

Title: Singularities of R-matrices, graded quiver varieties and generalized quantum affine Schur-Weyl duality

Abstract: The R-matrices are realized as intertwining operators between tensor products of two finite-dimensional simple modules of the quantum loop algebras. They can be seen as matrix-valued rational functions in spectral parameters, whose denominators determine when the tensor product modules become reducible. In this talk, we present a simple unified formula expressing the denominators of the R-matrices between the fundamental modules of type ADE and explain its relation to the representation theory of the Dynkin quivers and the geometry of Nakajima’s graded quiver varieties. As an application, we obtain a geometric interpretation of Kang-Kashiwara-Kim’s generalized quantum affine Schur-Weyl duality functor when it arises from a family of fundamental modules.

Apr 21, Peter McNamara (Melbourne)

Title: Representations of Coxeter groups and Hecke algebras

Abstract: This talk is Peter learning to use electronic means of delivery by giving an expository talk about the mathematics in Kazhdan and Lusztig’s famous paper with the same title.

Apr 14, Dougal Davis (The University of Edinburgh)

Title: Towards quantised moduli of principal bundles on elliptic curves

Abstract: Let G be a reductive algebraic group with Lie algebra Lie(G), and let E be an elliptic curve. From these data, one can cook up three intimately related algebraic stacks: the adjoint quotients Lie(G)/G and G/G, and the stack $Bun_G$ of principal G-bundles on E. The first two are fairly accessible and well-studied. While the third is less so, many interesting constructions and theorems that work for Lie(G)/G and G/G have analogues for $Bun_G$. For example, both the Chevalley isomorphisms and Grothendieck-Springer resolutions for adjoint quotients have direct analogues for $Bun_G$.

The aim of this talk is to discuss a conjectural deformation quantisation of $Bun_G$, motivated by the existence of explicit deformation quantisations of Lie(G)/G and G/G. I will explain what kind of objects these are, give some of the properties I expect the conjectural quantisation of $Bun_G$ to have, and discuss how it should be related to known algebras in some special cases.

Apr 7, Yaping Yang (Melbourne)

Title: The Knizhnik–Zamolodchikov functor for rational Cherednik algebras

Abstract: I will talk about the category $\mathcal{O}$ of representations of the rational Cherednik algebras. I will focus on the KZ functor introduced by Ginzburg-Guay-Opdam-Rouquier in 2003, which is one of the major tools to study the category $\mathcal{O}$. It is an exact functor from the category $\mathcal{O}$ to the module category of the (finite) Iwahori-Hecke algebra, which induces an equivalence between $\mathcal{O}/\mathcal{O}_{tor}$, the quotient of $\mathcal{O}$ by the subcategory of certain torsion modules, and the category of finite-dimensional Hecke algebra-modules.

March 31, Arun Ram (Melbourne)

Title: Formulas for Macdonald polynomials

Abstract: I will review/compare and contrast some of the formulas for nonysmmetric (and symmetric) Macdonald polynomials including the Haglund-Haiman-Loehr formula, the Ram-Yip formula, and the recent formulas of de Gier-Cantini-Wheeler and Corteel-Williams-Mandelshtam. One result I’d like to highlight provides the specialisations of the Ram-Yip formula for q and t taking values 0 or infinity. I may also make some comments about the Macdonald polynomials for type $(C^\vee, C)$, which are called Koornwinder polynomials, and Macdonald polynomials for other classical (unitary, orthogonal and symplectic) types.

March 10, Gufang Zhao (Melbourne)

Title: Representation theory underlying PT-invartiants of the resolved conifold

Abstract: We consider the cohomology of the Pandharipande-Thomas moduli space of the resolved conifold. We prove that the cohomology admits an action of a shifted affine Yangian. This fact was originally conjectured by Costello from AdS/CFT consideration. In this talk, we give a heuristic proof of this fact using dynamics of D2-branes and Coulomb branch of quiver gauge theory, as well as a representation theoretical/ combinatorial proof using Kontsevich-Soibelman cohomological Hall algebras and pyramid partitions. I will also discuss different representation theory features in DT and PT-invariants, and general speculations regarding D6 and D4 branes on toric Calabi-Yau. This is based on work in progress, joint with Rapcak, Soibelman, and Yang.

March 3, Peter McNamara (Melbourne)

Title: Heisenberg categorification.