# 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 30, 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 $Y^{tw}(\mathfrak{sp}_2)$.

## Recent talks

### 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.

### 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.