# Lie theory seminar

## Department of Mathematics, University of California Riverside

### Organizers

 Vyjayanthi Chari vyjayanc at ucr.edu Wee Liang Gan wlgan at math.ucr.edu Jacob Greenstein jacobg at ucr.edu Carl Mautner carlm at ucr.edu

## Upcoming talks

 May 22, 2018 1-2 p.m. Surge 284 Kayla Murray Thesis defense May 24, 2018 1-2 p.m. Surge 284 Léa Bittmann (Université Paris-Diderot, France) Asymptotics of standard modules of quantum affine algebras Abstract. Finite dimensional representations of quantum affine algebras have a behavior similar to the Kazhdan-Lusztig conjecture. Nakajima showed that characters of standard modules can be expressed as linear combinations of characters of simple modules of "lower weight". Moreover, the coefficients are non-negative and can be expressed as evaluation of some polynomials. In our work, we want to obtain the same type of results for the category $\mathscr O$ of representations of a Borel subalgebra (introduced by Hernandez-Leclerc). In this talk, we will present the motivation behind the definition of our asymptotical standard modules, as well as an idea for their construction in the case where $\mathfrak{g}=\hat{\mathfrak{sl}}_2$ (upported by the ERC Grant Agreement no. 647353 "Qaffine").

## Recent talks

### Spring 2018

 April 3, 2018 1-2 p.m. Surge 284 You Qi (Caltech) On the center of small quantum groups Abstract. We will report some recent progress on the problem of determining the centers of small quantum groups at a root of unity. This is joint work with A. Lachowska.

### Winter 2018

 January 23, 2018 1-2 p.m. Surge 284 Jens Eberhardt (UCLA) Category $\mathcal{O}$ and Mixed Geometry. Abstract. Many questions about the representation theory of a complex semisimple Lie group can be understood in terms of the category $\mathcal{O}(\mathfrak{g})$ associated to its Lie algebra. In analogy, Soergel constructed a modular category $\mathcal{O}(G)$ of representations of a reductive algebraic group $G$ over a field in characteristic $p$, which was recently used by Williamson to construct counterexamples to Lusztig's conjecture ("Williamson's Torsion Explosion"). Both categories are intimately related to the mixed geometry of the flag variety. In characteristic $0$, categories of certain mixed $\ell$-adic sheaves, mixed Hodge modules or stratified mixed Tate motives provide geometric versions of the derived graded category $\mathcal{O}(\mathfrak{g})$ (Beilinson, Ginzburg, Soergel and Wendt). Using the work of Soergel, we prove analogous statements in characteristic $p$. First, we construct an appropriate formalism of "mixed modular sheaves", using motives in equal characteristic. We then apply this formalism to construct a geometric version of the of the derived graded modular category $\mathcal{O}(G)$. (This is joint work with Shane Kelly). January 30, 2018 1-2 p.m. Surge 284 Karina Batistelli (CIEM - CONICET, Universidad Nacional de Córdoba, Argentina) Quasifinite highest weight modules of the "orthogonal" and "symplectic" types Lie subalgebras of the matrix quantum pseudodifferential operators Abstract. In this talk we will characterize the irreducible quasifinite highest weight modules of some subalgebras of the Lie algebra of $N\times N$ matrix quantum pseudodifferential operators. In order to do this, we will first give a complete description of the anti-involutions that preserve the principal gradation of the algebra of $N\times N$ matrix quantum pseudodifferential operators and we will describe the Lie subalgebras of its minus fixed points. We will obtain, up to conjugation, two families of anti-involutions that show quite different results when $n = N$ and $n < N$. We will then focus on the study of the "orthogonal" and "symplectic" type subalgebras found for case $n = N$, specifically the classification and realization of the quasifinite highest weight modules. February 6, 2018 1-2 p.m. Surge 284 Matheus Brito (Universidade Federal do Paraná, Brazil) BGG resolutions of prime representations of quantum affine $sl_{n+1}$. Abstract. We study the family of prime representations of quantum affine $sl_{n+1}$ introduced in the work of Hernandez and Leclerc which are defined by using an $A_n$ quiver. We show that such representations admit a BGG-type resolution where the role of the Verma module is played by the local Weyl module. This leads to a closed formula (the Weyl character formula) for the character of the irreducible representation as an alternating sum of characters of local Weyl modules.

### Fall 2017

 October 17, 2017 1-2 p.m. Surge 284 Neal Livesay Simple affine roots, lattice chains, and parahoric subgroups Abstract. In the representation theory of $\mathrm{GL}_n(\mathbb{C})$, there are correspondences between sets of simple roots, flags in $\mathbb{C}^n$, and parabolic subgroups, that are well-behaved with respect to actions by $\mathrm{GL}_n(\mathbb{C})$. These objects index a simplicial complex $\mathcal{B}(\mathrm{GL}_n(\mathbb{C}))$, called the building of $\mathrm{GL}_n(\mathbb{C})$. There is an affine version of the building, called the Bruhat-Tits building, whose simplices correspond to sets of simple affine roots, lattice chains, and parahoric subgroups. The primary goal of this expository talk is to give lots of low rank examples (i.e., for $\mathrm{SL}_2$, $\mathrm{SL}_3$, and $\mathrm{Sp}_4$) for each of these correspondences. If time permits, I will describe the relation between the Bruhat--Tits building and a well-behaved class of filtrations on the loop algebra called Moy-Prasad filtrations. These filtrations will be used in my next talk to define a geometric analogue of fundamental strata (originally developed by C. Bushnell for $p$-adic representation theory). October 24, 2017 1-2 p.m. Surge 284 Neal Livesay Simple affine roots, lattice chains, and parahoric subgroups (cont.) November 3, 2017 12:10-1 p.m. Surge 268 Ivan Loseu (Northeastern University) Deformations of symplectic singularities and Orbit method Abstract. Symplectic singularities were introduced by Beauville in 2000. These are especially nice singular Poisson algebraic varieties that include symplectic quotient singularities and the normalizations of orbit closures in semisimple Lie algebras. Poisson deformations of conical symplectic singularities were studied by Namikawa who proved that they are classified by the points of a vector space. Recently I have proved that quantizations of a conical symplectic singularity are still classified by the points of the same vector spaces. I will explain these results and then apply them to establish a version of Kirillov's orbit method for semisimple Lie algebras. November 7, 2017 1-2 p.m. Surge 284 Bach Nguen (Louisiana State University) Noncommutative discriminants via Poisson geometry and representation theory Abstract. The notion of discriminant is an important tool in number theory, algebraic geometry and noncommutative algebra. However, in concrete situations, it is difficult to compute and this has been done for few noncommutative algebras by direct methods. In this talk, we will describe a general method for computing noncommutative discriminants which relates them to representation theory and Poisson geometry. As an application we will provide explicit formulas for the discriminants of the quantum Schubert cell algebras at roots of unity. If time permits, we will also discuss this for the case of quantized coordinate rings of simple algebraic groups and quantized universal enveloping algebras of simple Lie algebras. This is joint work with Kurt Trampel and Milen Yakimov. November 14, 2017 1-2 p.m. Surge 284 Ethan Kowalenko Kazhdan-Lusztig polynomials and Soergel bimodules Abstract. Over the summer, I learned about Soergel bimodules at the MSRI from Ben Elias and Geordie Williamson, who were able to use these bimodules to solve a conjecture about Kazhdan-Lusztig (KL) polynomials for an arbitrary Coxeter system $(W,S)$. The aim of these talks will be two-fold. First, I want to define the KL polynomials. This will involve looking in the Hecke algebra of a Coxeter system, and showing how the KL polynomials describe one basis of the Hecke algebra in terms of another. The second goal will then be to describe the category of Soergel bimodules, and to show how they can be used to prove that the coefficients of the KL polynomials are non-negative. November 16, 2017 1-2 p.m. Surge 284 Ethan Kowalenko Kazhdan-Lusztig polynomials and Soergel bimodules (cont.)

### Spring 2017

 May 2, 2017 1-2 p.m. Surge 284 Daniele Rosso Exotic Springer Fibers Abstract. The Springer resolution is a resolution of singularities of the variety of nilpotent elements in a semisimple Lie algebra. It is an important geometric construction in representation theory, but some of its features are not as nice if we are working in Type $C$ (Symplectic group). To make the symplectic case look more like the Type $A$ case, Kato introduced the exotic nilpotent cone and its resolution, whose fibers are called the exotic Springer fibers. We give a combinatorial description of the irreducible components of these fibers in terms of standard Young bitableaux. This is joint work with Vinoth Nandakumar and Neil Saunders. May 16, 2017 1-2 p.m. Surge 284 Daniele Rosso A graphical calculus for the Jack inner product on symmetric functions Abstract. Starting from a graded Frobenius superalgebra $B$, we consider a graphical calculus of $B$-decorated string diagrams. From this calculus we produce algebras consisting of closed planar diagrams and of closed annular diagrams. The action of annular diagrams on planar diagrams can be used to make clockwise (or counterclockwise) annular diagrams into an inner product space. I will explain how this gives a graphical realization of the space of symmetric functions equipped with the Jack inner product. This is joint work with Tony Licata and Alistair Savage. June 8, 2017 1-2 p.m. Surge 284 Yilan Tan The irreducibility of the Verma Modules for Yangians and twisted Yangians Abstract. In this talk, we first introduce the definition of Yangians for the reductive complex Lie algebra $\mathfrak{gl}(N)$, describe their finite-dimensional representations and give necessary and sufficient conditions for the irreducibility of the Verma modules. Next, the definition of twisted Yangians is introduced. In the end, we give necessary and sufficient conditions for the irreducibility of the Verma modules for twisted Yangian $Y^{tw}(\mathfrak{sp}_2)$.

### Winter 2017

 January 17, 2017 1-2 p.m. Surge 284 Ting Xue (Melbourne) The Springer correspondence for symmetric spaces and Hessenberg varieties Abstract. The Springer correspondence relates nilpotent orbits in the Lie group of a reductive algebraic group to irreducible representations of the Weyl group. We develop a Springer theory in the case of symmetric spaces using Fourier transform, which relates nilpotent orbits in this setting to irreducible representations of Hecke al gebras at $q=-1$. We discuss applications in computing cohomology of Hessenberg varieties. Examples of such varieties include classical objects in algebraic geometry: Jacobians, Fano varieties of $k$-planes in the intersection of two quadrics, etc. This is based on joint work with Tsao-hsien Chen and Kari Vilonen. January 24, 2017 1-2 p.m. Surge 284 Xinli Xiao Nakajima's double of representations of COHA Abstract. Given a quiver $Q$ with/without potential, one can construct an algebra structure on the cohomology of the moduli stacks of representations of $Q$. The algebra is called Cohomological Hall algebra (COHA for short). One can also add a framed structure to quiver $Q$, and discuss the moduli space of the stable framed representations of $Q$. Through these geometric constructions, one can construct two representations of Cohomological Hall algebra of Q over the cohomology of moduli spaces of stable framed representations. One would get the double of the representations of Cohomological Hall algebras by putting these two representations together. This double construction implies that there are some relations between Cohomological Hall algebras and some other algebras. In the talk we will focus on two specific examples: $A_1$ quiver and the Jordan quiver. February 9, 2017 1-2 p.m. Surge 284 Daniele Rosso Symmetry, Combinatorics and Geometry Abstract. Symmetry is a concept that most people grasp intuitively, and it has important applications in several branches of mathematics, as well as other sciences. We will focus on a surprising connection between the geometry of subspaces of a vector space (like lines and planes) and an algorithm that was originally defined for purposes of combinatorics (arranging and counting things in various ways). February 28, 2017 1-2 p.m. Surge 284 Deniz Kus (Universität Bonn, Germany) Graded tensor products for Lie (super)algebras Abstract. In this talk I will discuss the construction of graded tensor products for the current algebra associated to a Lie (super)algebra. For the ortho--symplectic Lie superalgebra we will show that these representations can be filtered by the corresponding graded tensor products for the underlying reductive Lie algebra. In the second part of my talk, I will discuss the appearence of graded tensor products in PBW theory and categorification. One of the future goals is to understand which 2-representation of the categorified quantum group corresponds to graded tensor products. March 2, 2017 4-5 p.m. Surge 284 Georgia Benkart (University of Wisconsin Madison) Richard Block lecture: Tracing a Path -- From Walks on Graphs to Invariant Theory Abstract. Molien's 1897 formula for the Poincaré series of the polynomial invariants of a finite group has given rise to many results in combinatorics, coding theory, mathematical physics, algebraic geometry, and representation theory. This talk will focus on analogues of Molien's formula for tensor invariants and will discuss various connections with representation theory, and the McKay Correspondence. The approach is via walking on graphs. March 7, 2017 1-2 p.m. Surge 284 Ryo Fujita (Kyoto University, Japan) Tilting modules in affine higest weight categories Abstract. Affine highest weight category, introduced by Kleshchev, is a generalization of the notion of highest weight category. For example, some module categories over central completions of (degenerate) affine Hecke algebras (more generally KLR algebras of finite type) and polynomial current Lie algebras are known to be affine highest weight. In this talk, we consider tilting modules in affine highest weight categories and explain that a complete collection of indecomposable tilting modules exists if our category has a large center. As an application, we gives a simple criterion for an exact functor between two affine highest weight categories to give an equivalence. We can apply this criterion to the Arakawa-Suzuki functor on the deformed category $\mathcal{O}$ for $\mathfrak{gl}_{m}$.

### Fall 2016

 October 25, 2016 1-2 p.m. Surge 284 Sarah Kitchen (UMich) Harish-Chandra and Generalized Harish-Chandra Modules Abstract. The representation theory of real reductive Lie groups can be studied by complex algebraic and geometric methods using infinitesimal approximations called Harish-Chandra modules. Harish-Chandra modules are representations of the corresponding complex reductive Lie algebra which are locally finite with respect to the complexification of the maximal compact subgroup of the original real Lie group. Generalized Harish-Chandra modules are a generalization in which the maximal compact group is replaced by an arbitrary reductive subalgebra of the complex Lie algebra. In this talk we will review the geometric classification of simple Harish-Chandra modules, and compare this case to the setting of generalized Harish-Chandra modules. This work is part of a program initiated by Ivan Penkov and Gregg Zuckerman.

### Spring 2016

 (WEDNESDAY!) June 8, 2016 1-2 p.m. Surge 284 Peter McNamara (Queensland) Consequences of a categorified braid group action Abstract. It is well known that the braid group acts on a quantised enveloping algebra by algebra automorphisms. We discuss the categorification of this braid group action and some of its consequences. Applications include constructing reflection functors for KLR algebras, and a theory of restricting a categorical representation along a face of a Weyl polytope.