The aim of this seminar is to bring speakers from this area and outside to speak on topics related to Representation Theory (specially geometric and topological methods employed in Representation Theory). The speakers are expected to give their talks at a level suitable for graduate students. The seminar is organized by Shrawan Kumar.

Elijah Bodish Abstract: Given a semisimple Lie algebra and a positive integer L, one obtains the monoidal category of tilting modules for the associated quantum group at a primitive L-th root of unity. The category of tilting modules has a unique semisimple monoidal quotient. The images of indecomposable tilting modules with nonzero dimension exhaust the isomorphism classes of simple objects in the quotient category.

If L is larger than the Coxeter number of the Lie algebra, then the indecomposable tilting modules with non-zero dimension are exactly those with highest weight in the fundamental alcove (with respect to the L dilated affine Weyl group). When L is less than the Coxeter number much less is known. The state of the art is that in early 2020 Brundan-Entova-Etingof-Ostrik determined which tilting modules have nonzero dimension for gl_n when L < n+1 (note that the Coxeter number for gl_n is n). The Coxeter number for sp_{2n} is 2n. We will propose a method to determine which indecomposable tilting modules for sp_{2n} have non-zero dimension, when n< L< 2n+1. The answer appears to be related to Kazhdan-Lusztig cells in the anti-spherical Hecke module, which we hope will indicate what to expect in the much more complicated case when L< n+1.

David Rose Abstract: In his seminal 1996 paper, Kuperberg gives presentations for the categories of finite-dimensional representations of quantum groups associated to rank 2 simple complex Lie algebras (as braided pivotal categories). Such presentations underly various invariants in low-dimensional topology; in particular, they serve as a “foundation” for link homology theories. Kuperberg also poses the following problem: to find analogous descriptions of these categories for quantum groups associated with higher rank Lie algebras. In 2012, Cautis-Kamnitzer-Morrison solved this problem in type A using skew Howe duality, a technique that does not immediately extend to give a solution in other types.

In this talk, we will present a solution to Kuperberg’s problem in type C. Our proof combines results on pivotal categories and quantum group representations with diagrammatic/topological analogues of theorems concerning reduced expressions in the symmetric group. Time permitting, we’ll discuss some future directions. This work is joint with Elijah Bodish, Ben Elias, and Logan Tatham.

Dinakar Muthiah Abstract: Affine Grassmannians are objects of central interest in geometric representation theory. For example, the geometric Satake correspondence tells us that their singularities carry representation theoretic information. In fact, it suffices to work with affine Grassmannian slices, which retain all of this information.

Recently, Braverman, Finkelberg, and Nakajima showed that affine Grassmannian slices arise as Coulomb branches of certain quiver gauge theories. Remarkably, their construction works in Kac-Moody type as well. Their work opens the door to studying affine Grassmannians for Kac-Moody groups. Unfortunately, it is difficult at present to do any explicit geometry with the Coulomb branch definition. For example, a basic feature is that affine Grassmannian slices embed into one another. However, this is not apparent from the Coulomb branch definition. In this talk, I will explain why these embeddings are necessarily subtle. Nonetheless, I will show a way to construct the embeddings using fundamental monopole operators. This is joint work in progress with Alex Weekes.

Tommaso Botta Abstract: The notion of stable envelopes of a symplectic resolution, developed by Okounkov and his coauthors in the last decade, lies at the heart of the geometric approach to the representation theory of quantum groups and q-difference equations. Nakajima quiver varieties form a rich family of symplectic resolutions, whose geometry governs the representation theory of Kac-Moody Lie algebras and, via stable envelopes, their q-deformations. In this talk, I will introduce an inductive formula that produces the stable envelopes of an arbitrary Nakajima variety, taking as input the stable envelopes of two other Nakajima varieties with smaller dimension and framing vectors. Some explicit examples will be also discussed. This formula is a wide generalisation earlier results inherited form the theory of weight functions. Time permitting, I will also discuss some connections with cohomological Hall algebras (CoHa) and Cherkis bow varieties, which are object of ongoing research.

Ivan Loseu Abstract: Let O be a nilpotent orbit in a semisimple Lie algebra over the complex numbers. Then it makes sense to talk about filtered quantizations of O, these are certain associative algebras that necessarily come with a preferred homomorphism from the universal enveloping algebra. Assume that the codimension of the boundary of O is at least 4, this is the case for all birationally rigid orbits (but six in the exceptional type), for example. In my talk I will explain a geometric classification of faithful irreducible Harish-Chandra modules over quantizations of O, concentrating on the case of canonical quantizations — this gives rise to modules that could be called unipotent. The talk is based on a joint paper with Shilin Yu (in preparation).

Josh Kiers Abstract: Demazure modules are certain B-submodules of irreducible G-modules, where B is a Borel subgroup of a semisimple Lie group G. They arise as cohomology groups of line bundles on Schubert varieties, and their characters are described by the Demazure character formula. We will consider the weight polytopes associated to these characters, presenting basic formulas and results on them. We will show how the polytopes shed light on the supports of the characters, at least in classical types. This gives an alternate proof of a conjecture of Monical, Tokcan, and Yong, first proven by Fink, Meszaros, and St. Dizier.

Shiliang Gao Abstract: The Newell-Littlewood numbers are defined in terms of the Littlewood-Richardson coefficients. Both arise as tensor product multiplicities for a classical Lie group. A. Klyachko connected eigenvalues of sums of Hermitian matrices to the saturated LR-cone and established defining linear inequalities. We prove analogues for the saturated NL-cone. This is based on work with Gidon Orelowitz, Nicolas Ressayre and Alexander Yong; see arxiv.org/abs/2005.09012, arxiv.org/abs/2009.09904, and arxiv.org/abs/2107.03152.

Cristian Lenart Abstract: I present a combinatorial Chevalley formula for an arbitrary weight in the equivariant K-theory of semi-infinite flag manifolds, which are certain affine versions of finite flag manifolds G/B. The formula is expressed in terms of the so-called quantum alcove model. One application is a Chevalley formula in the equivariant quantum K-theory of G/B. Another application is that the so-called quantum Grothendieck polynomials represent Schubert classes in the (non-equivariant) quantum K-theory of the type A flag manifold. Both applications solve longstanding conjectures. Other results include the Chevalley formula for partial flag manifolds G/P and related combinatorics of the quantum alcove model. This is joint work with Takafumi Kouno, Satoshi Naito, and Daisuke Sagaki. The talk will be largely self-contained.

Prakash Belkale Abstract: Local systems are sheaves which describe the behavior of solutions of differential equations. A local system is rigid if local monodromy determines global monodromy. We give a construction which produces irreducible complex rigid local systems on a punctured Riemann sphere via quantum Schubert calculus and strange duality. These local systems are unitary and arise from a study of vertices in the polytopes controlling the multiplicative eigenvalue problem for the special unitary groups SU(n) (i.e., determination of the possible eigenvalues of a product of unitary matrices given the eigenvalues of the matrices). Roughly speaking, we show that the strange duals of the simplest vertices (which can be inductively determined) of these polytopes give (all) possible unitary irreducible rigid local systems.

We note that these polytopes are generalizations of the classical Littlewood-Richardson cones of algebraic combinatorics. Answering a question of Nicholas Katz, we show that there are no irreducible rigid local systems on a punctured Riemann sphere of rank greater than one, with finite global monodromy, all of whose local monodromies have orders dividing n, when n is a prime number.

Hang Huang Abstract: Tensors are just multi-dimensional arrays. Notions of ranks and border rank abound in the literature. Tensor decompositions also have a lot of application in data analysis, physics, and other areas of science. I will try to give a colloquium-style talk surveying my recent two results about tensor ranks and their application to matrix multiplication complexity. The first result relates different notions of tensor ranks to polynomials of vanishing Hessian. The second one computes the border rank of 3 X 3 permanent. I will also briefly discuss the newest technique we used to achieve our results: border apolarity.

Xuqiang Qin Abstract: Instanton bundles were first introduced on P^3 as stable rank 2 bundles E with c1(E)=0 and H^1(E(-2))=0. Torsion free generalizations and properties of moduli spaces of instanton bundles have been widely studied. Faenzi and Kuznetsov generalized the notion of instanton bundles to other Fano threefolds. In this talk, we look at semistable sheaves of rank 2 with Chern classes c1 = 0, c2 = 2 and c3 = 0 on some Fano threefolds of Picard number 1 and index 2. We show that the moduli space of such sheaves provides a smooth compactification of the moduli space of minimal instanton bundles.

David Nadler Abstract: I’ll introduce Betti Geometric Langlands through some key objects, conjectures and results. Then I’ll discuss recent and ongoing work with Zhiwei Yun devoted to constructing some of its expected topological field theory structures.

Chiara Damiolini Abstract: In this talk I will discuss geometric properties of certain sheaves on the moduli space of stable n-pointed curves which arise from modules over vertex operator algebras. These sheaves can be seen as a generalization of the notion of conformal blocks attached to representations of Lie algebras. Keeping this in mind, I will focus on exploring which properties of the classical conformal blocks still hold in this more exotic context, and which new open questions arise in this setting. This is based on a joint work with A. Gibney and N. Tarasca and ongoing work with A. Gibney.

Iva Halacheva Abstract: One approach to studying the representation theory of Lie algebras and their associated quantum groups is through combinatorial shadows known as crystals. While the original representations carry an action of the braid group, their crystals carry an action of a closely related group known as the cactus group. I will describe how we can realize this combinatorial action both geometrically, as a monodromy action coming from a family of ‘’shift of argument’’ algebras, as well as categorically through the structure of certain equivalences on triangulated categories known as Rickard complexes. Parts of this talk are based on joint work with Joel Kamnitzer, Leonid Rybnikov, and Alex Weekes, as well as Tony Licata, Ivan Losev, and Oded Yacobi.

Olivia Dumitrescu Abstract: We will review an axiomatic formulation of a 2D TQFT whose formalism is based on the edge-contraction operations on graphs drawn on a Riemann surface (cellular graphs). We will describe a new result, that ribbon graphs provide both cohomological field theory and a visual explanation of Frobenius-Hopf duality. This is based on work in progress with Motohico Mulase.

Calvin McPhail-Snyder Abstract: Quantum groups are a central part of the construction of quantum invariants of knots, links, and 3-manifolds. Existing work focuses mainly on the case where the quantization parameter q is generic, or on the semisimplified theory at q a root of unity. In this talk, I will discuss how to construct invariants of knots (and links) using the non-semisimple part of unrestricted quantum sl_2 at a root of unity. These “holonomy invariants” turn out to be very geometric: they depend on the extra data of a map from π_1 of the knot complement to SL_2(C), which is essentially a choice of hyperbolic structure.

Alex Weeks Abstract: Braverman-Finkelberg-Nakajima have recently given a mathematical construction of the Coulomb branches for 3d N=4 theories. From a representation-theoretic perspective, one reason that their work is especially appealing is that affine Grassmannian slices of ADE types arise this way, associated to quiver gauge theories. By allowing general quivers, Coulomb branches also provide a candidate definition for affine Grassmannian slices in all symmetric Kac-Moody types. In this talk I will discuss joint work with Nakajima, where we generalize the BFN construction of the Coulomb branch to incorporate “symmetrizers”. In this way we recover affine Grassmannian slices in BCFG type, and a candidate definition for symmetrizable Kac-Moody types.

Tsao-Hsien Chen Abstract: In an ongoing project of D. Ben-Zvi, Y. Sakellaridis and A. Venkatesh, the authors propose a conjectural generalization of the derived Satake equivalence for complex reductive groups to spherical varieties. I will describe a program aimed at establishing their conjecture in the case of symmetric varieties (an important class of spherical varieties). A key ingredient is the relation between the Satake category for symmetric varieties and the geometric Langlands for real reductive groups.

Anne Dranowski In their recent paper on the MV basis and DH measures, Baumann, Kamnitzer and Knutson showed that Mirkovic and Vilonen’s geometric Satake basis of singular algebraic cycles yields a biperfect basis of the coordinate ring of a unipotent subgroup. Moreover, they showed that the structure constants of multiplication of basis vectors in this ring are given by intersection forms. We present joint work in progress generalizing an isomorphism of Mirkovic and Vybornov towards effectively computing these intersections when G=GL_m. Time permitting we discuss alternate methods in the literature as well as applications.

Charlotte Chan Flag varieties and representations of p-adic groups
Abstract: Deligne–Lusztig varieties are subvarieties of flag varieties whose cohomology encodes the representations of reductive groups over finite fields. These give rise to so-called “depth-zero” supercuspidal representations of p-adic groups. In this talk, we discuss geometric constructions of positive depth supercuspidal representations and the implications of such realizations towards the Langlands program. This is partially based on joint work with Alexander Ivanov and joint work with Masao Oi.

Yau Wing Li Endoscopy for affine Hecke categories
Abstract: Affine Hecke categories are categorifications of Iwahori-Hecke algebras, which are essential in the classification of irreducible representations of loop group LG with Iwahori-fixed vectors. The affine Hecke category has a monodromic counterpart, which contains sheaves with prescribed monodromy under the left and right actions of the maximal torus. We show that the neutral block of this monoidal category is equivalent to the neutral block of the affine Hecke category (with trivial torus monodromy) for the endoscopic group H. It is consistent with the Langlands functoriality conjecture.

Rekha Biswal Macdonald polynomials and level two Demazure modules for affine sl_{n+1}
Abstract: An important result due to Sanderson and Ion says that characters of level one Demazure modules are specialized Macdonald polynomials. In this talk, I will introduce a new class of symmetric polynomialsindexed by a pair of dominant weights of sl_{n+1} which is expressed as linear combination of specialized symmetric Macdonald polynomials with coefficients defined recursively. These polynomials arose in my own work while investigating the characters of higher level Demazure modules. Using representation theory, we will see that these new family of polynomials interpolate between characters of level one and level two Demazure modules for affine sl_{n+1} and give rise to new results in the representation theory of current algebras as a corollary. This is based on joint work with Vyjayanthi Chari, Peri Shereen and Jeffrey Wand.

Dennis Tseng Equivariant degenerations of plane curve orbits
Abstract: In a series of papers, Aluffi and Faber computed the degree of the GL3 orbit closure of an arbitrary plane curve. We attempt to generalize this to the equivariant setting by studying how these orbits degenerate, yielding a fairly complete picture in the case of plane quartics. As an enumerative consequence, we will see that a general genus 3 curve appears 510720 times as a 2-plane section of a general quartic threefold. We also hope to survey the relevant literature and will only assume the basics of intersection theory. This is joint work with M. Lee and A. Patel.