Title: Modular perverse sheaves on the affine Grassmannian

Abstract: Perverse sheaves on the affine Grassmannian of a reductive group G encode a great deal of representation-theoretic information. In characteristic 0, these sheaves have been studied in depth since the 1990s. In this talk, I will discuss recent advances in the positive-characteristic case, including the proof of the Mirkovic-Vilonen conjecture and the relationship with the Springer resolution of the Langlands dual group. This is joint work with L. Rider. I will also explain the connection to closely related independent work of Mautner-Riche.

Title: Joseph ideals and lisse minimal W-algebras

Abstract: Motivated by a recent work of Kawasetu, we consider a lifting of Joseph ideals for minimal nilpotent orbit closures to the setting of affine Kac-Moody algebras, and find new examples of affine vertex algebras whose associated varieties are minimal nilpotent orbit closure. As an application we obtain a new family of lisse (C2-cofinite) W- algebras that are not coming from admissible representation of affine Kac-Moody algebras.

Title: Finite-dimensional representations of the queer Lie superalgebra of half-integer weights

Abstract: We give an interpretation of the representation theory of the finite-dimensional modules of the queer Lie superalgebra of half-integer weights in terms of Brundan's work of finite-dimensional modules of integer weights by means of Lusztig's canonical basis. Using this new viewpoint we compute the characters of the finite-dimensional irreducible modules of half-integer weights. In the special cases of irreducible modules whose highest weights are either totally connected and totally disconnected we derive close for formulas for them that are reminiscent of Kac-Wakimoto character formula for classical Lie superalgebras. This is a joint work with Jae-Hoon Kwon.

Title: Affine cellularity of some infinite-dimensional algebras

Abstract: Koenig and Changchang Xi recently defined the notion of affine cellularity, which generalizes the notion of cellular algebras and provides a unified framework for studying the classification of the irreducible representations of algebras that do not need to be finite dimensional over a noetherian domain k. In this talk, we will show that Beck-Lusztig-Nakajima (BLN) algebras and affine quantum Schur algebras are affine cellular. Applying Koenig and Xi's approach, we also study the homological properties of these algebras.

Title: On orbifold theory

Abstract: Let V be a simple vertex operator algebra and G a finite automorphism group of V such that is regular. It is proved that every irreducible V^G-module occurs in an irreducible g-twisted V-module for some g ∈ G. Moreover, the quantum dimensions of each irreducible V^G-module is determined and a global dimension formula for V in terms of twisted modules is obtained.

Title: Quantum linear supergroups and their canonical bases

Abstract: It is well known that the most fundamental structure of a universal enveloping algebra associated with a symmetrizable Kac-Moody algebra is stored in a matrix---the Cartan matrix. It is also known that, if the associated Lie algebra is a matrix algebra, then a PBW basis is indexed by certain matrices. In this talk, we will show that, in the type A family---the family of quantum linear groups/supergroups and affine quantum linear groups---there is a new basis for every member of the family, which contains the set of generators, such that the structure constants associated with multiplying a basis element by a generator are completely determinedby the labelling matrices. This indicates that further structures of such an object are also stored in matrices.

                   The first type of such bases was constructed for quantum linear groups two decades ago by Beilinson-Lusztig-MacPherson, using a geometric setting (i.e., the partial flag varieties) for q-Schur algebras. However, for the super and affine cases, purely algebraic and combinatorial approaches have been developed. We will mainly focus on the construction of the quantum linear supergroups in the talk. By this new construction, we will also give a combinatorial construction for the canonical bases of the positive and negative parts and discuss their relationship with the Kazhdan-Lusztig bases for the q-Schur superalgebras and the induced bases for simple polynomial representations.

Title: Equivalence of representation categories of various quantum and super quantum groups

Abstract: Corresponding to a Cartan datum, there are several versions of quantum enveloping algebras, including original quantum group in the form of Lusztig, and quantum groups with many parameters, as well as supervision. We establish equivalences of several representation theories of these quantum groups under certain assumption by introducing a new multi-parameter quantum algebra and its modified form. This is a joint work with Yiqiang Li and Zongzhu Lin.

Title: Degrees of modules and varieties of elementary abelian Lie algebras

Abstract: （PDF Version）

Title: Coset vertex operator algebras from tensor decomposition of affine vertex operator algebras

Abstract: We will talk about some recent results on coset vertex operator algebras associated to affine vertex operator algebras.

Title: Kac-Wakimoto character formula for ortho-symplectic Lie superalgebras

Abstract: We give the classification of finite-dimensional tame modules over the ortho-symplectic Lie superalgebras, and show that their characters are given by the Kac-Wakimoto character formula, thus establishing the Kac-Wakimoto conjecture for the ortho-symplectic Lie superalgebras. This is a joint work with Shun-Jen Cheng.

Title: An elementary approach to monomial bases of quantum affine $\mathfrak{gl}_n$

Abstract: In 1990 Beilinson, Lusztig and MacPherson provided a geometric realization of modified quantum $\mathfrak{gl}_n$ and its canonical basis. An essential step of their work is a construction of a monomial basis. Recently, Du and Fu provided an algebraic construction of canonical basis for modified quantum affine $\mathfrak{gl}_n$, which among other results used an earlier difficult construction of a monomial basis using Ringel-Hall algebra of the cyclic quiver. In this talk, I will give an elementary algebraic construction of a monomial basis for affine Schur algebras and modified quantum affine $\mathfrak{gl}_n$. This is a joint work with Li Luo (Shanghai).

Title: Quantizations of nilpotent orbits

Abstract: I will explain some recent existence and uniqueness results for algebras of regular functions on nilpotent orbits and describe applications to computing the Goldie ranks of primitive ideals in universal envelopin algebras. The talk is based on http://arxiv.org/abs/1505.08048

Title: Specializations of symmetric Macdonald polynomials and pseudo QLS paths

Abstract: A symmetric Macdonald polynomial$P_{\lambda}(x; q, t)$ can be thought of as a certain graded character of the crystal (for an affine Lie algebra) of pseudoQLS paths of shape $\lambda$; however, the representation-theoretic meaning of this (rather large) crystal is not yet known. In this talk, I explain that the specialization $P_{\lambda}(x; q, 0)$ at $t = 0$ of the symmetric Macdonald polynomial $P_{\lambda}(x; q, t)$ is identical to a graded character of a canonical quotient$W(\lambda)$ of a special Demazure submodule (corresponding to the longest element $w_{0}$ of the finite Weyl group) of the level- zero extremal weight module of extremal weight $\lambda$ over a quantum affine algebra.Also, I would like to explain that the specialization $E_{w_{0} \lambda}(x; q, \infty)$ at $t = \infty$ of the nonsymmetric Macdonald polynomial $E_{w_{0} \lambda}(x; q, t)$ can be described as another graded character of the (same) canonical quotient $W(\lambda) $ above, but with a curious grading different from the one in the $t = 0$ case.

Title: On the center of quiver Hecke algebras

Abstract: I will explain how to relate the center of a cyclotomic quiver Hecke algebras to the cohomology of Nakajima quiver varieties using a current algebra action. This is a joint work with M. Varagnolo and E.Vasserot.

Title: Springer correspondence for complex reflection groups and related Kostka functions

Abstract: Let $V$ be a $2n$-dimensional vector space over an algebraically closed field of odd characteristic. Put $G = GL(V)$ and $H Sp(V)$, and consider the symmetric space $G/H$. A variety $X = G/H \times V^{r-1}$ is called an exotic symmetric space of level $r$. Similarly, for an $n$-dimensional vector space $V$, we consider the variety $X GL(V) \times V^{r-1}$, which is called an enhanced variety of level $r$. For either case, one can define the unipotent subvariety $X_0$ of $X$. Let $W = W_{n,r}$ be the complex reflection group defined as a semi direct product of the symmetric group $S_n$ and the cyclic group $(Z/rZ)^n$. In this talk, we show that there exists a natural bijective correspondence between the set of irreducible representations of $W$ and a certain set of intersection cohomologies arising from $X_0$ of exotic type. A similar correspondence also holds in the case of the enhanced variety, in a reduced form. Kostka functions associated to complex reflection groups are functions indexed by pairs of $r$-tuple of partitions, which is an analogue of the original Kostka polynomials indexed by pairs of partitions. It is expected, as in the classical case, that those Kostka functions are closely related to the geometry of $X_0$ (for both type). We will explain some results, in the case of enhanced type, supporting this conjecture.

Title: Frobenius map for the centers of Hecke algebras

Abstract: We introduce a commutative associative graded algebra structure on the direct sum Z of the centers of the Hecke algebras associated to the symmetric groups in n letters for all n. As a natural deformation of the classical construction of Frobenius, we establish an algebra isomorphism from the algebra Z to the ring of symmetric functions. This isomorphism provides an identification between several distinguished bases for the centers (introduced by Geck-Rouquier, Jones, Lascoux) and explicit bases of symmetric functions. This is a joint work with Weiqiang Wang.

Title: Canonical bases for tensor product modules

Abstract: We will explain the construction of canonical bases in tensor products of several lowest and highest weight integrable modules(and a theory of based modules for general quantum groups), generalizing Lusztig's work. This is joint work with Huanchen Bao .

Title: Representations in rational fuctions

Abstract: Representations in polynomials have been investigated for long time and are pretty well understood, although there are still many problems to be settled. It seems that representations in rational functions are not much studied. In this talk we will give some discussions to representations in rational functions.

Title: Finte W-superalgebras for basic Lie superalgebras and their applications

Abstract: In this talk we mainly consider the related topics on finite W-superalgebra $U(\frak{g}_\bbf},e)$ for a basic Lie superalgebra $\frak{g}_\bbf=(\frak{g}_\bbf)_\bz+(\frak{g}_\bbf)_\bo$ associated with a nilpotent element $e\in (\frak{g}_\bbf)_{\bo}$ both over the field of complex numbers $\bbf=\bbc$ and over $\bbf=\bbk$ an algebraically closed field of positive characteristic.

                  In the first part, we present the PBW theorem for $U(\frak{g}_\bbf,e)$. In contrast with finite W-algebras, one can find that the construction of $U(\frak{g}_\bbf,e)$ is divided into two cases in virtue of the parity of $\dim(\frak{g}_\bbf}(-1)_{\bo}$. A module of $\frak{g}_\bbk$ is said to be of Kac-Weisfeiler type if its dimension coincides with the one in the super Kac-Weisfeiler property presented by Wang-Zhao, which is the dimensional lower bound for the modular representations of a basic Lie superalgebra $\frak{g}_\bbk$ over an algebraically closed field $\bbk$ of positive characteristic $p$.

                 In the second part, we verify the existence of the Kac-Weisfeiler modules for $\frak{gl}_{m|n}$ over an algebraically closed field $\overline{\bbf}_p$ of characteristic $p>2$. We also establish the corresponding consequence for $\frak{sl}_{m|n}$ with restrictions $p>2$ and $p\nmid(m-n)$.

                 In the third part, we formulate a conjecture about the minimal dimensional representations of the finite $W$-superalgebra $U(\frak{g}_\bbc,e)$ over the field of complex numbers and demonstrate it with examples including all the cases of type $A$. Under the assumption of this conjecture, we verify the existence of the modules of Kac-Weisfeiler type for any basic Lie superalgebras in characteristic $p\gg0$.

                 This talk is joint work with Bin Shu.