Abstract: The goal of the talk is to describe a bridge between two directions in representation theory. The first is the theory of short star products and twisted traces of Etingof-Stryker. The second is the theory of quantum symmetric pairs of Letzter-Kolb based on the classification of Kac-Wang of automorphisms of Kac-Moody algebras. The central result is that every quantum symmetric subalgebra is realized via a short star product on a quantum horospherical subalgebra by a dual quantum Popov degeneration. The fact that Popov's degenerations are always short star products gives short and conceptual proofs of many results for quantum symmetric pairs that before were proved by long and diverse methods. Our methods prove that all quasi K-matrices for these quantum Kac-Moody symmetric pairs are explicitly expressible in terms of the much studied quasi R-matrices of Drinfeld and Lusztig. This is a joint work with Stefan Kolb.

Abstract: Canonical basis theory has been established for many quantum/i-quantum groups, quantum linear supergroups and their modified ones. However, the existence of such a theory for quantum queer supergroups $\mathbf U_\upsilon(\mathfrak q_n)$ is still mysterious. This is partly because no appropriate bar involution on $\mathbf U_\upsilon(\mathfrak q_n)$ is known.

A recent completion of a new realization of $\mathbf U_\upsilon(\mathfrak q_n )$ via Hecke-Clifford superalgebras and queer q-Schur superalgebras in [1] gives us a chance for possibly resolving the problem. Building on this work, we may first establish such a theory for Hecke-Clifford superalgebras and then extend it to $\mathbf U_\upsilon(\mathfrak q_n )$.

In this talk, I will start with bar involutions $\flat$ on Hecke-Clifford superalgebras $\mathscr{H}_r^c$ ($r\geq1$), and explain how to lift them to bar involutions on $\mathbf U_\upsilon(\mathfrak q_n )$. I then introduce a standard basis $\mathcal B_r(c)$ for $\mathscr{H}_r^c$ and a natural partial ordering $<$ on its index set. Unfortunately, canonical bases associated with the data $(\mathcal B_r(c),<,\flat)$ do not exist. However, either replacing $\mathcal B_r(c)$ by some new bases or twisting the order $<$ to a new order $<'$ can define canonical bases for $\mathscr{H}_r^c$. Thus, canonical bases associated with $(\mathcal B_r(c),<',\flat)$ exist. If time permits, I will mention some possible ways to extend the construction above to queer $q$-Schur superalgebras and the modified $\mathbf U_\upsilon(\mathfrak q_n )$

Abstract: In this talk, I will describe a one-parameter family of algebras that naturally encompasses the Schur algebras. These algebras are generically semisimple, they have the ordinary Schur algebras as a quotient, and their modules are naturally a subcategory of a parabolic Category O for the general linear Lie algebra.
This is joint work with Addison Day.

Abstract: Hypertoric varieties are a class of symplectic singularities and their resolutions, obtained as Hamiltonian reductions of a symplectic vector space acted on by a torus. In physics, they appear as Higgs (and Coulomb) branches of 3d N-4 supersymmetric quantum field theories with abelian gauge group.

In this talk, we construct an $\hbar$-adic sheaf of vertex operator superalgebras over a given smooth hypertoric variety. Its global sections give the A-twisted boundary of the corresponding 3d gauge theory. We use this to prove that the associated affine variety of this hypertoric vertex operator superalgebra recovers the singular hypertoric variety. This proves the 3d Higgs branch conjecture for a large class of boundary vertex operator superalgebras. In particular, these vertex operator superalgebras are quasi-lisse.

This is in contrast to the (purely even) hypertoric vertex operator superalgebras constructed previously by Kuwabara as global sections of sheaves on families of universal Poisson deformations of the hypertoric varieties. These are generally not quasi-lisse. We show that the vertex operator superalgebras defined in this paper are (fermionic) simple-current extensions of those defined by Kuwabara, and investigate the consequences for symplectic duality and characters. We observe that the latter are upgraded from partial (or false) theta functions to quasimodular forms.

This is a joint work with Andrea Ferrari and Sven Möller.

Abstract: $W$-algebras are a class of vertex algebras that are associated to a Lie (super)algebra $g$ and a nilpotent orbit in the even part of $g$. Principal $W$-algebras (i.e., the nilpotent is principal) are the best-studied examples. When $g$ is a Lie algebra, they satisfy Feigin-Frenkel duality, which is the isomorphism between the principal $W$-algebras of $g$ and its Langlands dual Lie algebra. When $g$ is simply-laced, they satisfy another duality called the coset realization, which was proven in my work with Arakawa and Creutzig in 2018. A common generalization of these dualities was conjectured in 2017 by the physicists Gaiotto and Rapcak and was proven in my work with Creutzig in 2020. I will discuss these dualities and some of their applications in representation theory, as well as some generalizations to W-algebras with supersymmetry in recent work with Creutzig, Kovalchuk, Song, and Suh.

Let $G$ be a connected simply-connected simple algebraic group over $\mathbb{C}$ and let $T$ be a maximal torus, $B\supset T$ a Borel subgroup and $K$ a maximal compact subgroup. Then, the product in the (algebraic) based loop group $\Omega(K)$ gives rise to a comultiplication in the topological $T$-equivariant $K$-ring $K_T^{\top}(\Omega(K))$. Recall that $\Omega(K)$ is identified with the affine Grassmannian $\mathcal{X}$ (of $G$) and hence we get a comultiplication in $K_T^{\top}(\mathcal{X})$. Dualizing, one gets the Pontryagin product in the $T$-equivariant $K$-homology $K^T_0(\mathcal{X})$, which in-turn gets identified with the convolution product (due to S. Kato).

Now, $ K_T^{\top}(\mathcal{X})$ has a basis $\{\xi^w\}$ over the representation ring $R(T)$ given by the ideal sheaves corresponding to the finite codimension Schubert varieties $X^w$ in $\mathcal{X}$. We make a positivity conjecture on the comultiplication structure constants in the above basis. Using some results of Kato, this conjecture gives rise to an equivalent conjecture on the positivity of the multiplicative structure constants in $T$-equivariant quantum $K$-theory $QK_T(G/B)$ in the Schubert basis.

Abstract: Semi-infinite sheaves are certain complexes of sheaves on (possibly partial) affine flag varieties of reductive algebraic groups which are equivariant with respect to the « semiinfinite Iwahori subgroup », i.e. the product of the arc group of a maximal torus and the loop group of the unipotent radical of a Borel subgroup. In this talk I’ll explain how (infinity-)categories of such complexes can be defined, explain some of their basic properties, and report on connections (partly under construction) with representations of the Langlands dual group (and its Lie algebra).
This is joint work with Pramod Achar, Gurbir Dhillon and Quan Situ.

Abstract:We consider the quiver Hecke algebra $R$ associated with a Kac-Moody Lie algebra $\mathfrak{g}$. Let ${\hbox{$R$-\hbox{gmod}}}$ denote the category of finite-dimensional graded $R$-modules, and let $\mathscr C_w$ be its full subcategory associated with a Weyl group element $w$. Let $\widetilde{{\hbox{$R$-\hbox{gmod}}}}$ and $\widetilde{\mathscr C_w}$ be their localizations, referred to as localized quantum unipotent coordinate categories.

It has been shown by M. Kashiwara and T.N. that the set of isomorphism classes of simple objects up to grading shifts, $\mathrm{Irr}(\widetilde{\mathscr C_w})$, carries a crystal structure and is isomorphic to the cellular crystal $\mathbb{B}_{\mathbf{i}}=B_{i_1}\otimes\cdots\otimes B_{i_k}$, where $\mathbf{i}=i_1\cdots i_k$ is a reduced word of $w$.

In the case where $\mathfrak g$ is of classical type and $w_0$ is the longest element of the Weyl group, this isomorphism induces functions $\varepsilon_i^*$ on $\mathbb{B}_{\mathbf{i_0}}$, where $\mathbf{i_0}$ is a reduced word of $w_0$. We provide an explicit description of the functions $\varepsilon_i^*$ and, using these, give a characterization of the unit object of $\widetilde{{\hbox{$R$-\hbox{gmod}}}}$.

This is joint work with K. Matsuura.

Abstract: The category of smooth complex representations of $p$-adic groups decomposes into Bernstein blocks and by a joint result with Adler, Mishra and Ohara from August 2024 we know that under some minor tameness assumptions each Bernstein block is equivalent to a depth-zero Bernstein block, which are the representations that correspond roughly to representations of finite groups of Lie type. This result allows to reduce a lot of problems about representations of $p$-adic groups and the Langlands correspondence to their depth-zero counterpart that is often easier to solve or already known.

In this talk we present analogous results for R-representations of $p$-adic groups where $R$ is any ring that contains all $p$-power roots of unity, a fourth root of unity and the inverse of a square-root of $p$, for example, $R$ could be an algebraically closed field of characteristic different from $p$ or the ring $\bar{\mathbb{Z}}[1/p]$. This is a joint work with Jean-François Dat. While the result is analogous to the result with complex coefficients (except for the “blocks” being “larger”), the proof is of a very different nature. In the complex setting the proof is achieved via type theory and an isomorphism of Hecke algebras, which are techniques not available for general $R$-representations. We sketch in the talk how we deal with the category of $R$-representations instead.

Abstract: Deligne--Lusztig varieties are natural subvarieties of the flag variety whose cohomology encodes (much of) the representation theory of finite groups of Lie type. In the last two decades, there have been many advances in developing a Deligne-Lusztig theory in the context of representations of $p$-adic groups. I'll discuss recent progress in this subject.

Abstract: The uniqueness of Whittaker models for the $p$-adic group $G = GL_d(F)$ relies heavily on a deep understanding of the Iwahori component of the Gelfand--Graev $G$-module. As a module over the affine Hecke algebra, this component identifies with the Kashiwara--Miwa--Stern tensor space at $n=1$, meaning its corresponding endomorphism algebra is the affine $q$-Schur algebra $\widehat{S}_q(1,d)$. In this talk, we provide a transparent description of these $p$-adic objects and their metaplectic (i.e., $n>1$) analogs via the theory of quantum wreath products. This new perspective leads to results that were previously inaccessible through standard $p$-adic methods. This is based on joint work with Alexandre Minets (Bonn) and Valentin Buciumas (Pohang).

Abstract: The Temperley-Lieb algebras form a natural tower, where moving up and down corresponds to induction and restriction. In this talk, I will describe a graphical calculus that makes these functors, and the natural transformations between them, visible. The calculus is built from a diagrammatic 2-category with generators for one-step induction and restriction, together with two-step summands coming from cup-cap idempotents. A key result is a basis theorem: morphism spaces have explicit bases indexed by bridges, a family of paths generalizing Dyck paths. This theorem shows that the graphical calculus faithfully captures the corresponding Temperley-Lieb bimodule theory. I will also discuss the decategorified picture, where standard modules are represented by homogenized Chebyshev polynomials, and idempotent completion leaves the Grothendieck ring unchanged.

Abstract:We will present a canonical basis character formula for the irreducible modules in parabolic BGG-type categories, including the category of finite-dimensional modules, for finite W-(super)algebras of type A. These categories categorify the tensor product modules of irreducible polynomial representations (and their duals) over a quantum group of type A. Moreover, the standard modules and irreducible modules in these categories categorify the standard basis and Lusztig's dual canonical basis in the tensor product modules.

This is joint work with Shun-Jen Cheng.