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.
Peter Fiebig Sheaves on the alcoves and modular representations
Abstract: The main topic is the problem of determining the characters of reductive algebraic groups over a field of positive characteristic. Since Williamson gave countless (unexpected) examples of characters that differ from the character formula given by George Lusztig in 1980 (which holds for large enough characteristics), the community is hoping for a revised formula, or an effective algorithm, that provides a solution in the general case. For this, one surely needs a much better understanding on the torsion phenomena occurring for small primes. In my talk I first want to give an understandable introduction to the problem, and then I want to introduce a new category related to modular representation theory that hopefully will help us understand the problem in greater depth.
Alejandro Ginory The Verlinde Formula and Twisted Affine Lie Algebras
Abstract: In the category of integrable highest weight modules for affine Lie algebras, the usual tensor product fails to preserve the so-called level (i.e., the scalar action of the canonical central element). For untwisted affine Lie algebras, a product structure called the fusion product makes the subcategory of modules at a fixed level into a braided monoidal category (in particular, with non-negative integral structure constants). In this talk, I will discuss a fusion product with integral structure constants on the space of characters of twisted affine Lie algebras. Surprisingly, in the A^(2)_{2r} case and for certain natural quotients for the other twisted cases, these algebras have negative structure constants “half” the time, (depending on the parity of the level). We will discuss these and other new features in the twisted cases, and their representation-theoretic meaning.
Changjian Su Motivic Chern classes, K-theoretic stable basis and Iwahori invariants of principal series
Abstract: Let G be a split reductive p-adic group. In the Iwahori-invariants of an unramified principal series representation of G, there are two bases, one of which is the so-called Casselman basis. In this talk, we will prove a conjecture of Bump–Nakasuji–Naruse about certain transition matrix between these two bases. We will first relate the Iwahori invariants to Maulik–Okounkov’s stable envelopes and Brasselet–Schurmann–Yokura’s motivic Chern classes for the Langlands dual groups. Then the conjecture follows from a K-theoretic generalization of Kumar’s smoothness criterion for the Schubert varieties. This is based on joint work with P. Aluffi, L. Mihalcea and J. Schurmann.
Michael Strayer Minuscule Kac-Moody settings unified by new poset coloring properties
Abstract: R.M. Green axiomatically defined full heap posets and used them to build elegant minuscule-like doubly infinite representations of many affine Kac-Moody algebras. Two other classes of colored posets, namely minuscule and d-complete posets, have also been used in many applications to Lie theory. For instance, the d-complete posets correspond to “dominant lambda-minuscule” elements of Kac-Moody Weyl groups. We present new poset coloring properties that unify the above three classes of colored posets. They can be used to characterize various Kac-Moody (sub)algebra representations. These include the full heap representations of Green, the minuscule representations of semisimple Lie algebras, and the Demazure modules of dominant minuscule Weyl group elements. We give Dynkin diagram-indexed classifications of the colored posets that satisfy the two most important sets of these properties.
Yiqiang Li Quiver varieties and symmetric pairs
Abstract: To a simply-laced Dynkin diagram, one can attach a complex simple Lie algebra, say g, and a class of Nakajima (quiver) varieties. The latter provides a natural home for a geometric representation theory of the former. If the algebra g is further equipped with an involution, it leads to a so-called symmetric pair (g,k), where k is the fixed-point subalgebra under involution. In this talk, I’ll present a recent study of fixed-point loci of Nakajima varieties under certain involutions and provide bridges at several levels between symmetric pairs and Nakajima varieties.
Sami Assaf Nonsymmetric Macdonald polynomials and Demazure characters
Abstract: Macdonald introduced symmetric functions in two parameters that simultaneously generalize Hall—Littlewood symmetric functions and Jack symmetric functions. Opdam and Macdonald independently introduced nonsymmetric polynomial versions of these that Cherednik then generalized to any root system. Ion showed that these nonsymmetric Macdonald polynomials with one parameter specialized to 0 arise as characters for affine Demazure modules. Recently, I used the Haglund—Haiman—Loehr combinatorial formula for nonsymmetric Macdonald polynomials in type A to show that, in fact, the specialized nonsymmetric Macdonald polynomials are graded sums of finite Demazure characters in type A. In this talk, I’ll present joint work with Nicolle Gonzalez where we construct an explicit Demazure crystal for specialized nonsymmetric Macdonald polynomials, giving rise to an explicit formula for the Demazure expansion in terms of certain lowest weight elements. Connecting back with the symmetric case, this gives a refinement of the Kostka—Foulkes polynomials defined by the Schur expansion of Hall—Littlewood symmetric functions.
This talk assumes no prior knowledge of Macdonald polynomials, Demazure characters, or crystals.
Richárd Rimányi Motivic characteristic classes, Hall algebras, and DT type identities
Abstract: The equivariant Chern-Schwartz-MacPherson (CSM) class and the equivariant Motivic Chern (MC) class are fine characteristic classes of singular varieties in cohomology and K theory—and their theory overlaps with the theory of Okounkov’s stable envelopes. We study CSM and MC classes for the orbits of Dynkin quiver representations. We show that the problem of computing the CSM and MC classes of all these orbits can be reduced to some basic classes $c^o_\beta$, $C^o_\beta$ parameterized by positive roots $\beta$. We prove an identity in a deformed version of Kontsevich-Soibelman’s Cohomological (and K-theoretic) Hall Algebra (CoHA, KHA), namely, that a product of exponentials of $c^o_\beta$, $C^o_\beta$ classes formally depending on a stability function Z, does not depend on Z. This identity—which encodes infinitely many identities among rational functions in infinitely many variables—has the structure of Donaldson-Thomas type quantum dilogarithm identities. Using a wall-crossing argument we present the $c^o_\beta$, $C^o_\beta$ classes as certain commutators in the CoHA, KHA.
Natalia Kolokolnikova K-theoretic Thom polynomial and the rationality of the singularities of the A2 loci
Abstract: I will discuss the definitions of two K-theoretic invariants of the singularity loci, prove that they are not always equal and tell how this problem is connected to the study of the rationality of the singularities of the singularity loci. I will prove that the singularities of the A2 loci are rational in some very specific cases, but are not rational in general.
Daping Weng Cyclic Sieving Phenomenon of Plane Partitions and Cluster Duality of Grassmannian
Abstract: Fix two positive integers $a$ and $b$. Scott showed that a homogeneous coordinate ring of the Grassmannian $Gr_{a, a+b}$ has the structure of a cluster algebra. This homogeneous coordinate ring can be decomposed into a direct sum of irreducible representations of $GL_{a+b}$ which correspond to integer multiples of the fundamental weight $w_a$. By proving the Fock-Goncharov cluster duality conjecture for the Grassmannian using a sufficient condition found by Gross, Hacking, Keel, and Kontsevich, we obtain bases parametrized by plane partitions for these irreducible representations. As an application, we use these bases to show a cyclic sieving phenomenon of plane partitions under a certain sequence of toggling operations. This is joint work with Linhui Shen.
Nicolas Ressayre On the tensor semigroup of affine Kac-Moody Lie algebras.
Abstract: In this talk, we are interested in the decomposition of the tensor product of two representations of a symmetrizable Kac-Moody Lie algebra g. Let P+ be the set of dominant integral weights. For λ ∈ P+, L(λ) denotes the irreducible, integrable, highest weight representation of g with highest weight λ. Consider the tensor cone
Γ(g):={(λ1,λ2,μ)∈P+3 |∃N >1 L(Nμ)⊂L(Nλ1)⊗L(Nλ2)}.
If g is finite dimensional, Γ(g) is a polyhedral convex cone described by Belkale-Kumar by an explicit finite list of inequalities. In general, Γ(g) is nor polyhedral, nor closed. We will describe the closure of Γ(g) by an explicit countable family of linear inequalities, when g is untwisted affine. This solves a Brown-Kumar’s conjecture in this case.
Simon Salamon Wolf spaces and Fano contact manifolds
Abstract: The correspondence between Riemannian symmetric spaces with holonomy a subgroup of Sp(n)Sp(1) and complex homogeneous spaces with a holomorphic contact structure was discovered by Joseph Wolf in 1965, yet the possibility of non-homogeneous manifolds with positive curvature subscribing to this model remains open. I shall explain what is known about positive quaternion-Kaehler manifolds and their Fano twistor spaces, in the light of recent results of Buczynski-Wisniewski-Weber on torus actions on contact Fano manifolds.
Chiara Damiolini Twisted conformal blocks
Abstract: Let G be a simple and simply connected algebraic group over a field. We can attach to a n-tuple of representations of G the sheaf of conformal blocks: a vector bundle on M_{g,n} whose fibres are identified with global sections of a certain line bundle on the stack of G-torsors. We generalize the construction of conformal blocks to the case in which G is replaced by a group H defined over curves in terms of covering data. In this case the associated conformal blocks define a sheaf on a Hurwitz space and have properties analogous to the classical case.
Baiying Liu On the local converse theorem for GL_n
Abstract: In this talk, I will introduce a complete proof of a standard conjecture on the local converse theorem for generic representations of GLn(F), where F is a non-archimedean local field. This is a joint work with Prof. Herve Jacquet. I will also briefly talk about extensions of the local converse theorem to the setting of l-adic families, which is a joint work with Gilbert Moss.
Johannes Flake Dirac cohomology, Hopf-Hecke algebras, and infinitesimal Cherednik algebras
Abstract: Dirac operators have played an important role in the representation theory of reductive Lie groups, but also of various Hecke algebras. We show how these cases can be studied uniformly, using smash products of Hopf algebras, PBW deformations, and superalgebras. We prove a generalized version of a result known as Vogan’s conjecture in certain special cases, which relates Dirac cohomology with central characters. We discuss infinitesimal Cherednik algebras as a novel special case, and we obtain partial results on the classification of the class of algebras to which our theory applies.