Monday, 13 November 2023
In this talk, we introduce crystal bases for the quantum groups associated with general linear and orthosymplectic Lie superalgebras. We introduce a family of highest weight representations which have crystal bases, together with their combinatorial models.
The aim of the lectures is to give a gentle introduction to the theory of vertex algebras, and present the concept of associated variety. The associated varieties are certain Poisson varieties attached to vertex algebras whose geometry capture important information. They are related to other interesting objects of the theory: the arc spaces and Zhu’s algebras.
The lectures will focus on the classification, combinatorial construction, characters, and crystals of integrable representations of quantum affine algebras. Following the paper arXiv1907.11796 , the classification fall into three cases: positive level, negative level and level zero, of which the level 0 case is the most “interesting”.
Any symmetric closed subset of a finite crystallographic root system must be a closed subroot system. This is not true for real affine root systems. We shall discuss about this by describing the classification result. In the latter part of the talk, we explore the correspondence between symmetric closed subsets of real affine root systems and the regular subalgebras generated by them.
In this talk, we will discuss the branching problems for the pair $(\mathrm{Sp}(2m) \subset \mathrm{Sp}(2n))$. Specifically, we'll explore how an irreducible representation $\pi$ of $\mathrm{Sp}(2n)$ decomposes when restricted to $\mathrm{Sp}(2m)$. In our presentation, we will observe these branching multiplicities as a striking combinatorial matrix.
The cyclic characters of a group G are the characters induced from the cyclic subgroups of G. In the case of classical Coxeter groups, Kraskiewicz and Weyman gave the decomposition of cyclic characters of Coxeter groups obtained by inducing characters of the cyclic group generated by a Coxeter element. The cyclic characters of Symmetric Group are well studied and described in terms of a statistic on the Young tableaux called the multi major index. In this talk, we will see a description of the cyclic characters of the alternating group. This is joint work with Amritanshu Prasad and Velmurugan S.
Tuesday, 14 November 2023
In this talk, we introduce crystal bases for the quantum groups associated with general linear and orthosymplectic Lie superalgebras. We introduce a family of highest weight representations which have crystal bases, together with their combinatorial models.
The aim of the lectures is to give a gentle introduction to the theory of vertex algebras, and present the concept of associated variety. The associated varieties are certain Poisson varieties attached to vertex algebras whose geometry capture important information. They are related to other interesting objects of the theory: the arc spaces and Zhu’s algebras.
The lectures will focus on the classification, combinatorial construction, characters, and crystals of integrable representations of quantum affine algebras. Following the paper arXiv1907.11796 , the classification fall into three cases: positive level, negative level and level zero, of which the level 0 case is the most “interesting”.
Let g be a symmetrizable Kac-Moody Lie algebra with Cartan subalgebra h. We prove that unique factorization holds for tensor products of parabolic Verma modules. We prove more generally a unique factorization result for products of characters of parabolic Verma modules when restricted to certain subalgebras of h. These include fixed point subalgebras of h under subgroups of diagram automorphisms of g.
The type $SL_2$ Macdonald polynomials $P_m$ and $E_m$ are $q,t$-deformation of characters of $SL_2$ representations. We give compact formulas for the $E$-expansion of the product $E_\ell P_m$ and $P$-expansion of the product $P_\ell P_m$ by using techniques from double affine Hecke algebra. This is based on joint work with Arun Ram.
We show that the set of all flagged reverse plane partitions of a given skew shape and a flag is a disjoint union of Demazure crystals (up to isomorphism). As a result, the flagged dual stable Grothendieck polynomial is shown to be key positive.
Wednesday, 15 November 2023
In this talk, we introduce crystal bases for the quantum groups associated with general linear and orthosymplectic Lie superalgebras. We introduce a family of highest weight representations which have crystal bases, together with their combinatorial models.
The aim of the lectures is to give a gentle introduction to the theory of vertex algebras, and present the concept of associated variety. The associated varieties are certain Poisson varieties attached to vertex algebras whose geometry capture important information. They are related to other interesting objects of the theory: the arc spaces and Zhu’s algebras.
The lectures will focus on the classification, combinatorial construction, characters, and crystals of integrable representations of quantum affine algebras. Following the paper arXiv1907.11796 , the classification fall into three cases: positive level, negative level and level zero, of which the level 0 case is the most “interesting”.
For any n xn Hermitian matrix A, let e(A) = (e_A(1), ..., e_A(n)) be its set of eigenvalues written in descending order. We recall the following classical problem.
Problem 1. (The Hermitian eigenvalue problem) Given two n-tuples of non- increasing real numbers: e' = (e'(1), ..., e'(n)) and e" = (e"(1), ..., e"(n)), determine all possible f = (f(1), ..., f(n)) such that there exist Hermitian matrices A, B, C with e(A) = e'; e(B) = e"; e(C) = f and A + B + C = 0. Even though this problem goes back to the nineteenth century, the first significant result was obtained by H. Weyl in 1912. With contributions from several mathematicians over the century, this problem was finally solved by combining the works of Klyachko (1998), Knutson-Tao (1999), Belkale (2001) and Knutson-Tao-Woodward (2004).
This problem can be generalized for an arbitrary reductive algebraic group as follows: Let G be a connected algebraic group with a maximal compact subgroup K and let g and k be their Lie algebras. Consider the Cartan decomposition g = k+p. Choose a maximal subalgebra (which is necessarily abelian) a of p and let a+ be a dominant chamber in a. Then, any K-orbit in p intersects a+ in a unique point. Then the analogue of the above Hermitian eigenvalue problem is the determination of the following subset C_m (for any m > 2) of (a+)^m: C_m := {(a_1, ..., a_m) in (a+)^m such that there exists (x_1, ..., x_m) in (a+)^m with x_1+ ...+ x_m = 0} and x_i in AdK. a_i}. By works of several mathematicians including Berenstein-Sjamaar (2000), Kapovich-Leeb-Millson (2005), Belkale-Kumar (2006) and Ressayre (2008), C_m has been determined in terms of an irredundant set of inequalities. This series of talks will give a complete solution of the problem. The main tools used are: Geometric Invariant Theory and Topology. We will assume familiarity with basic algebraic geometry and topology. Otherwise, the lectures will be self-contained.
We will also discuss the parallel `saturated' tensor product decomposition problem.
Thursday, 16 November 2023
For any n xn Hermitian matrix A, let e(A) = (e_A(1), ..., e_A(n)) be its set of eigenvalues written in descending order. We recall the following classical problem.
Problem 1. (The Hermitian eigenvalue problem) Given two n-tuples of non- increasing real numbers: e' = (e'(1), ..., e'(n)) and e" = (e"(1), ..., e"(n)), determine all possible f = (f(1), ..., f(n)) such that there exist Hermitian matrices A, B, C with e(A) = e'; e(B) = e"; e(C) = f and A + B + C = 0. Even though this problem goes back to the nineteenth century, the first significant result was obtained by H. Weyl in 1912. With contributions from several mathematicians over the century, this problem was finally solved by combining the works of Klyachko (1998), Knutson-Tao (1999), Belkale (2001) and Knutson-Tao-Woodward (2004).
This problem can be generalized for an arbitrary reductive algebraic group as follows: Let G be a connected algebraic group with a maximal compact subgroup K and let g and k be their Lie algebras. Consider the Cartan decomposition g = k+p. Choose a maximal subalgebra (which is necessarily abelian) a of p and let a+ be a dominant chamber in a. Then, any K-orbit in p intersects a+ in a unique point. Then the analogue of the above Hermitian eigenvalue problem is the determination of the following subset C_m (for any m > 2) of (a+)^m: C_m := {(a_1, ..., a_m) in (a+)^m such that there exists (x_1, ..., x_m) in (a+)^m with x_1+ ...+ x_m = 0} and x_i in AdK. a_i}. By works of several mathematicians including Berenstein-Sjamaar (2000), Kapovich-Leeb-Millson (2005), Belkale-Kumar (2006) and Ressayre (2008), C_m has been determined in terms of an irredundant set of inequalities. This series of talks will give a complete solution of the problem. The main tools used are: Geometric Invariant Theory and Topology. We will assume familiarity with basic algebraic geometry and topology. Otherwise, the lectures will be self-contained.
We will also discuss the parallel `saturated' tensor product decomposition problem.
The aim of the lectures is to give a gentle introduction to the theory of vertex algebras, and present the concept of associated variety. The associated varieties are certain Poisson varieties attached to vertex algebras whose geometry capture important information. They are related to other interesting objects of the theory: the arc spaces and Zhu’s algebras.
The lectures will focus on the classification, combinatorial construction, characters, and crystals of integrable representations of quantum affine algebras. Following the paper arXiv1907.11796 , the classification fall into three cases: positive level, negative level and level zero, of which the level 0 case is the most “interesting”.
In this talk, we delve into the fascinating intersection of graph theory, Lie superalgebras, and chromatic polynomials. Specifically, we explore the intricate relationship between the Generalized $\bold k$- Chromatic polynomial of a supergraph associated with a BKM Lie superalgebra $\mathfrak L$ and its free roots, where a free root is a root unaffected by Serre relations. By investigating the multi-coloring possibilities of the graph corresponding to these free roots using $q$ colors, we uncover a polynomial expression in $q$ known as the Generalized $\bold k$-Chromatic polynomial.
We establish a connection between this polynomial and the multiplicity of the free root space by employing the denominator identity. This association provides a Lie theoretic interpretation of Chromatic polynomials.
In this talk, we shall introduce the notion of weakly integrable modules over affine Kac—Moody algebras after developing the necessary background and henceforth completely classify the irreducible weakly integrable representations for these Lie algebras. Finally, if time permits, we shall discuss the analogous classification problem for extended affine Lie algebras of nullify 2. Our results generalize the well-known work of Chari—Pressley concerning the classification of irreducible integrable modules over affine Kac—Moody algebras.
Quantum loop algebras are infinite-dimensional algebras which are naturally involved in the theory of integrable systems as well as in the study of cluster algebras, quiver varieties and Coulomb branches. The category $\mathcal{C}$ of finite-dimensional modules over such an algebra $U_q(\mathfrak{g})$ is an interesting example of a non-semisimple category with generic braidings; that is, isomorphisms of $V \otimes W$ onto $W \otimes V$ for "generic" simple objects $V$ and $W$. Recently, Hernandez constructed similar isomorphisms for a subcategory $\mathcal{O}^-$ of the category of representations of the Borel subalgebra $U_q(\mathfrak{b})$ of $U_q(\mathfrak{g})$. This subcategory $\mathcal{O}^-$, defined by Hernandez--Leclerc in the context of monoidal categorifications of cluster algebras, has the same Grothendieck ring as that of another subcategory $\mathcal{O}^+$ of modules over $U_q(\mathfrak{b})$. This last observation raises the following questions:
Is the category $\mathcal{O}^+$ generically braided? Can we lift the isomorphism between the Grothendieck rings of $\mathcal{O}^+$ and $\mathcal{O}^-$ into an isomorphism of categories?
In this talk, we will answer the above questions positively and give, if time permits, potential applications of our results to generalized quantum affine Schur--Weyl dualities (in the spirit of Kang--Kashiwara--Kim--Oh--Park) and to the representation theory of shifted quantum affine algebras.
Friday, 17 November 2023
For any n xn Hermitian matrix A, let e(A) = (e_A(1), ..., e_A(n)) be its set of eigenvalues written in descending order. We recall the following classical problem.
Problem 1. (The Hermitian eigenvalue problem) Given two n-tuples of non- increasing real numbers: e' = (e'(1), ..., e'(n)) and e" = (e"(1), ..., e"(n)), determine all possible f = (f(1), ..., f(n)) such that there exist Hermitian matrices A, B, C with e(A) = e'; e(B) = e"; e(C) = f and A + B + C = 0. Even though this problem goes back to the nineteenth century, the first significant result was obtained by H. Weyl in 1912. With contributions from several mathematicians over the century, this problem was finally solved by combining the works of Klyachko (1998), Knutson-Tao (1999), Belkale (2001) and Knutson-Tao-Woodward (2004).
This problem can be generalized for an arbitrary reductive algebraic group as follows: Let G be a connected algebraic group with a maximal compact subgroup K and let g and k be their Lie algebras. Consider the Cartan decomposition g = k+p. Choose a maximal subalgebra (which is necessarily abelian) a of p and let a+ be a dominant chamber in a. Then, any K-orbit in p intersects a+ in a unique point. Then the analogue of the above Hermitian eigenvalue problem is the determination of the following subset C_m (for any m > 2) of (a+)^m: C_m := {(a_1, ..., a_m) in (a+)^m such that there exists (x_1, ..., x_m) in (a+)^m with x_1+ ...+ x_m = 0} and x_i in AdK. a_i}. By works of several mathematicians including Berenstein-Sjamaar (2000), Kapovich-Leeb-Millson (2005), Belkale-Kumar (2006) and Ressayre (2008), C_m has been determined in terms of an irredundant set of inequalities. This series of talks will give a complete solution of the problem. The main tools used are: Geometric Invariant Theory and Topology. We will assume familiarity with basic algebraic geometry and topology. Otherwise, the lectures will be self-contained.
We will also discuss the parallel `saturated' tensor product decomposition problem.
The aim of the lectures is to give a gentle introduction to the theory of vertex algebras, and present the concept of associated variety. The associated varieties are certain Poisson varieties attached to vertex algebras whose geometry capture important information. They are related to other interesting objects of the theory: the arc spaces and Zhu’s algebras.
The lectures will focus on the classification, combinatorial construction, characters, and crystals of integrable representations of quantum affine algebras. Following the paper arXiv1907.11796 , the classification fall into three cases: positive level, negative level and level zero, of which the level 0 case is the most “interesting”.
Let g = g(A) be a general Borcherds–Kac–Moody (BKM) C-Lie algebra, for Borcherds–Cartan matrix A, and V a highest weight g-module with weight-set wt V . Over Kac–Moody g, we know wt V for:
1) integrable V classically;
2) simple (highest weight) V by Dhillon and Khare, via parabolic Vermas;
3) all highest weight modules V recently by the author and Khare, via holes & higher order Vermas. Over BKM g, even for integrable V , to our knowledge wt V is unknown.
In this talk, we will determine the weight-sets wt V for all highest weight modules V over general BKM g, extending and subsuming at once the formulas in settings 1)–3). Our formula is:
i) explicit, non-recursive and cancellation-free; and
ii) uniform across all types of g (semisimple to Kac– Moody to general BKM) and all V (and all highest weights λ).
We begin with (the combinatorial problem of) describing the comple- ments of the upper-sets in Zn(≥0), thereby determining wt V for all V over g = H ⊕ · · · ⊕ H for H the 3-dimensional Heisenberg Lie algebra and over g = g(A) with A(ii) = 0 ∀i (in connection to finite-dimensional Heisenberg Lie algebras). This reveals a refined notion of holes to study wt V . We construct a novel class (to our knowledge) of “higher order parabolic Vermas” M(λ, H) over BKM g (only singleton holes in H in real directions), subsuming and generalizing i) all integrable V and ii) parabolic Vermas over Kac–Moody g.
We determine wt M(λ, H) over all λ and H, for every BKM g. In particular, we determine wt V for all integrable V – perhaps interestingly in the flavour of Weyl orbit weight-formulas for integrable V over Kac–Moody g (see e.g. Kac’s book). For all V over all BKM g, we recover wt V as the union of wt M(λ, H) with H ⊇ HV . (This talk is based on a recent joint work with Souvik Pal.)
Let g be a finite-dimensional simple Lie algebra over the complex field C and g[t] be the Lie algebra of polynomial mappings from C to g, which is its associated current algebra. We study the structure of the finite-dimensional representations of
the current Lie algebra of type A1, sl2[t], which are obtained by taking tensor products of local Weyl modules with Demazure modules. We show that these representations admit a Demazure flag and obtain a closed formula for the graded multiplicities of the level 2 Demazure modules in the filtration of the tensor product of two local Weyl modules for sl2[t]. Using Pieri formulas, we have also expressed the product of two specialized Macdonald polynomials in terms of specialized Macdonald polyomials.
Furthermore, we show that the tensor product of a local Weyl module with an irre- ducible sl2[t] module admits a Demazure filtration and derive graded character of such tensor product modules. This helps us express the product of a specialized Macdon- ald polynomial with a Schur polynomial in terms of Schur polynomial. Our findings provide evidence for the conjecture that the tensor product of Demazure modules of levels m and n respectively has a filtration by Demazure modules of level m + n.
Quantum toroidal algebras Uq(g_tor) occur as the Drinfeld quantum affinizations of quantum affine algebras. In particular, they contain (and are generated by) a horizontal and vertical copy of the affine quantum group. In type A, Miki obtained an automorphism of Uq(g_tor) exchanging these subalgebras, which has since played a crucial role in the investigation of its structure and representation theory.
In this talk, we shall construct an action of the extended double affine braid group B on the quantum toroidal algebra in all untwisted types. In the simply laced cases, using this action and certain involutions of B we obtain automorphisms and anti-automorphisms of Uq(g_tor) which exchange the horizontal and vertical subalgebras, thus generalising the results of Miki.
Monday, 20 November 2023
Shifted quantum affine algebras and their truncations emerged from the study of quantized Coulomb branches. I will report on a joint work in progress with C. Geiss and B. Leclerc : we show that the Grothendieck ring of the category O for the shifted quantum affine algebras has the structure of a cluster algebra, with initial seeds parametrized by reduced expressions of the associated (finite) Weyl group W. The initial cluster variables are constructed from a new Weyl group action introduced in a joint work with Frenkel (Weyl group symmetry of q-characters, arXiv:2211.09779).
I will explain how to realize quantum loop groups and shifted quantum loop groups of arbitrary type, possibly non symmetric, using critical convolution algebras. This generalizes Nakajima’s famous construction of symmetric quantum loop groups via quiver varieties. It also provides the first geometrical description of certain families of simple modules such as Kirillov-Reshetikhin or the prefundamental modules (for negative-shift quantum loop groups).
The multispecies totally asymmetric long-range exclusion process is an interacting particle system with multiple species of particles on a finite ring where the hopping rates are site-dependent. (The homogeneous variant on Z is also known as the Hammersley–Aldous–Diaconis process.) In its simplest variant with a single species, a particle at a given site will hop to the first available site clockwise. We show that the partition function of this process is intimately related to the classical Macdonald polynomial. We also show that well-known families of symmetric polynomials appear as expectations in the stationary distribution of important observables. This is joint work with James Martin (Oxford), Omer Angel (UBC) and
Lauren Williams (Harvard).
Let \mathfrak{g} be a simple Lie algebra over \mathbb{C}. The KZ connection is a connection on the constant bundle associated with a set of n finite-dimensional irreducible representations of \mathfrak{g} and a
nonzero \kappa \in \mathbb{C}, over the configuration space of n-distinct points on the affine line. Via the work of Schechtman--Varchenko, Looijenga, Belkale--Brosnan--Mukhopadhyay when \kappa \in \mathbb{Q} the associated local systems can be seen to be realizations of naturally defined motivic local systems.
In this talk, we will discuss a basic factorization for the nearby cycles of these motivic local systems as some of the n points coalesce. This leads to the construction of a family (parametrized by \kappa) of deformations over \mathbb{Z}[t] of the representation ring of \mathfrak{g}. We will also discuss how similar constructions for conformal blocks in genus 0 lead to the construction of a family of deformations of the fusion rings. This is a joint work with Prakash Belkale and Najmuddin Fakhruddin.
Tuesday, 21 November 2023
I will review some stabilization phenomena in the representation theory/geometry of type GL, with focus on the more recent construction of the stable limit DAHA of type GL and the structure of its standard representation.
The semi-infinite flag manifold Q associated with a simple algebraic group G is a "level-zero" variant of the affine flag variety of G. The geometry of Q is quite different from that of the standard affine flag variety; in particular, its Schubert varieties are both infinite-dimensional and infinite-codimensional in Q. Despite this difficulty, a notion of equivariant K-group for Q has been introduced by Kato, Naito, and Sagaki and, through further work of Kato, this K-group has had important applications to the quantum K-theory of (partial) flag varieties G/P. In this talk, I will discuss a nil-DAHA action on the equivariant K-group of Q and explain how it can be used to find algebraic and combinatorial "inverse" Chevalley formulas in this setting (for G of ADE type). This is partly based on joint works with Kouno, Lenart, Naito, and Sagaki.
In this talk, we extend the notion of Weyl modules to toroidal Lie algebras with n variables. Further, using the work of Rao, we identify the level one global Weyl module of toroidal Lie algebra with the suitable submodule of its Fock space representation up to a twist. As an application, we compute the graded character of the level one local Weyl module of toroidal Lie algebra, which generalises the recent work of Kodera.
We introduce a new family of simple representations of quantum affine sl_n which generalizes both prime snake modules and the so called HL-modules. We show that such modules are prime, real and admit a determinantal formula in terms of the action of the Nil Hecke algebra on the monoid of multisegments. This generalizes the determinantal formula of Tadic-Lapid-Minguez for ladder representations of p-adic groups via affine Hecke algebras and a Schur--Weyl duality.
Wednesday, 22 November 2023
We consider reduced imaginary Verma modules for the untwisted quantum affine algebras and define a crystal-like base which we call imaginary crystal base using the Kashiwara algebra constructed by Ben Cox, Vyacheslav Futorny and Kailash Misra. We prove the existence of the imaginary crystal base for any object in a suitably defined category of modules containing the reduced imaginary Verma modules for the untwisted quantum affine algebras. This is a joint work with J. Arias and V. Futorny.
Given a Coxeter system, we can associate with it its braid monoid. The Coxeter group and the Hecke monoid are natural quotients of it, and both are known to have a lot of applications and interesting representation theory. A particularly important role for geometric applications is played by parabolic elements of the Coxeter group, which form a commutative submonoid of the Hecke monoid. The aim of the present work (joint with Arkady Berenstein and Jianrong. Li) is to classify those homomorphisms of braid monoids which factor through to homomorphisms of both Coxeter groups and Hecke monoids (we call them admissible) and those homomorphisms of Hecke monoids which preserve parabolic elements and their generalizations. As a byproduct, we also obtain a procedure for embedding a non-simply laced Coxeter system into a simply laced one in a way which is compatible with all three structures.
Classically, the Chevalley restriction theorem on the g-invariants in S(g*), for a Lie (super)algebra g, can be seen as a graded version of the Harish-Chandra theorem on the center of the universal enveloping algebra.
We shall discuss the corresponding version of the Chevalley restriction theorem for a super-symmetric spaces, namely for the associated graded space of the invariant differential operators on G/K, where G is a Lie supergroup and K is the subgroup of fixed points of an involution on G. Joint work with Siddhartha Sahi and Vera Serganova.
We study general highest weight modules V over an arbitrary complex Kac-Moody algebra \g (one may assume this to be \sl(n) throughout the talk, without sacrificing novelty). We present a "first order" invariant for every V, which yields the convex hull of its weights (for which we classify the faces), but also the Weyl group stabilizer of its character - and for simple (non-integrable) V, the weights of V as well. This connects to earlier results by Satake, Borel-Tits, Cellini-Marietti, Casselman, and Vinberg for finite-dimensional V over semisimple \g (aka Weyl polytopes); and to results of Chari and coauthors.
We then introduce the notions of holes in a module V (over Kac-Moody \g) and of higher order Verma modules. Using these, we present positive formulas (without cancellations) for the weights of arbitrary highest weight modules V. We will end with BGG resolutions and Weyl-Kac character formulas for classes of higher order Verma modules. (Partly joint with Gurbir Dhillon and with G.V.K. Teja.)
We introduce Toroidal algebras, Full Toroidal algebras and Extended Affine Lie Algebras. We study their representations. Among them we pay more attention to the Hamiltonian Extended affine Lie algebra.
Thursday, 23 November 2023
The Brauer category is a diagrammatic monoidal category describing the representation theory of the orthogonal and symplectic groups. Its endomorphism algebras are Brauer algebras, which replace the group algebra of the symmetric group in the orthogonal and symplectic analogues of Schur-Weyl duality. However, the Brauer category is missing one important piece of the picture---the spin representation. We will introduce a larger category, the spin Brauer category, that remedies this deficiency. This is joint work with Peter McNamara.
Quantum Grothendieck ring in this talk is a deformation of the Grothendieck ring of the monoidal category of finite-dimensional modules over the quantum loop algebra, endowed with a canonical basis consisting of the so-called simple (q,t)-characters. We discuss a collection of isomorphisms among the quantum Grothendieck rings of different Dynkin types respecting the canonical bases, via which the (q,t)-characters of non-simply-laced type inherit several good properties from those of the unfolded simply-laced type. We also discuss their cluster theoretical interpretation, which particularly yields non-trivial birational relations among the (q,t)-characters of different Dynkin types. This is a joint work with David Hernandez, Se-jin Oh, and Hironori Oya.
We propose a new geometric model for the center of the small quantum group using the cohomology of certain affine Springer fibers. More precisely, we establish an isomorphism between the equivariant cohomology of affine Springer fibers for a split element and the center of the deformed graded modules for the small quantum group, and an embedding from the invariant part of the nonequivariant cohomology under the action of the extended affine Weyl group to the G-invariant part of the center of the small quantum group, which we conjecture to be an isomorphism.
We propose a geometric approach to the Feigin-Loktev fusion product of modules over the current algebra and compute it in some new cases. Moreover, this establishes a connection between two categories with cluster structure in representation theory: the one of modules over the affine quantum group and the one of perverse coherent sheaves on the affine Grassmannian. Using this connection, we are able to establish the existence of analogs of Q-systems of perverse coherent sheaves in simply-laced types, which conjecturally stand for cluster relations. Based on arXiv:2308.05268.
We explore on abelianizations of a semi simple, complex Lie algebra and the induced degenerations on their simple, finite-dimensional modules and generalized flag varieties. One instance of such abelianization is the PBW degeneration, and within this framework, it has been shown that in type A and C the degenerate module is a Demazure module, the degenerate flag variety is in fact a Schubert variety. “How far should/could you degenerate the Lie structure to stay in a non-trivial Lie-theoretic setup?” could be also the title of this talk, as we explore the Dynkin diagram combinatorics to answers this question for all finite types.
Friday, 24 November 2023
I will describe such modules with finite dimensional weight spaces with respect to its Cartan subalgebra, where nonzero center acts nontrivially on the modules.