Categorification, Kostant's problem and generalized Verma modules
Abstract: Let $\mathfrak{g}$ be a semi-simple complex finite-dimensional Lie algebra and M be a $\mathfrak{g}$-module. The classical Kostant's problem for M (as formulated by A. Joseph about 30 year ago) is whether the canonical inclusion of $U(\mathfrak{g})/\mathrm{Ann}(M)$ into the set of all $\mathbb{C}$-endomorphisms of M, locally finite with respect to the adjoint action of $\mathfrak{g}$, is surjective. Some first steps in understanding this problem were made by Joseph and Gabber-Joseph in the beginning of 80's. In particular, the answer is known to be positive for Verma modules and for simple highest weight modules of the form $L(w_0^S w_0\cdot\lambda)$, where $\lambda$ is dominant, $x_0$ is the longest element of the integral Weyl group and $w_0^S$ is the longest element of some parabolic subgroup. In type $B_2$ it is known (Joseph) that for $L(st\cdot 0)$ the answer to the above problem is negative.
Viewing category $\mathcal{O}$ from the point of view of categorification philosophy motivates deeper study of functorial actions on $\mathcal{O}$. In the present talk I will try to explain how this can be applied to
Because of the approach proposed by Milicic and Soergel, Kostant's problem is an important tool in understanding the structure of parabolically induced modules (generalized Verma modules). It turns out that the above type A result can be used to describe (in type A) the so-called rough structure of generalized Verma modules. In particular, the rough structure turns out to depend only on the annihilator of the simple module we induce from. Informally speaking, the rough structure means the description of simple subquotients parabolically induced from simple modules, whose annihilator is comparable with the annihilator of the simple module, which the original generalized Verma module is induced from. The fine structure of generalized Verma modules seems to be a very difficult question. First of all it depends on more than the annihilator of the starting module, moreover, generalized Verma modules can potentially have infinite length.
Braid group actions, symmetric algebras, centres and traces
Abstract: In this talk I will describe how Serre functors on derived categories are related to symmetric algebras and (if lucky) to functorial braid group actions. I will give several examples and explain the relevance of some of the appearing symmetric algebras (one of the easiest cases will be Khovanov's algebra). It will be used to give a categorical interpretation of the cohomology ring of the Springer fibres and the generalised Steinberg varieties. Finally I want to outline its connection to knot theory via a categorical Ocneanu trace.
Cyclotomic Hecke algebras and parabolic category O
Abstract: This talk will focus on some older results (joint with Kleshchev) giving a higher level analogue of the classical Schur-Weyl duality, in which the symmetric group gets replaced by generate cyclotomic Hecke algebras and the general linear group gets replaced by a parabolic version of the Bernstein-Gelfand-Gelfand category O. From this one can deduce all the known results about the representation theory of degenerate cyclotomic Hecke algebras over the complex numbers (analogous to results of Ariki about the cyclotomic Hecke algebras themselves but at a generic parameter not at a root of unity) directly from the Kazhdan-Lusztig conjecture. In the other direction, we explain how to reconstruct parabolic category O from these Hecke algebras.
Graded lift of tensoring with finite dimensional representations
Abstract: We explain how the functor of tensoring with a finite dimensional representation on the integral category O may be lifted to its graded version in such a way, that this operation will be weekly associative.
Cohomology for quantum groups via the geometry of the nullcone
Abstract: Given a semi-simple algebraic group G over an algebraically closed field k of characteristic p > 0, one of the central problems is to find a character formula for the finite-dimensional simple G-modules. For p > h-1, a character formula is given by a conjectural formula due to Lusztig.
In the analogous world of quantum groups over the complex numbers, this formula is verified for l > h where q is a primitive l th root of unity. The first proof used an equivalence of categories due to Kazhdan and Lusztig between quantum groups and affine Lie algebras. Kashiwara and Tanisaki subsequently verified the character formula for affine Lie algebras. A second proof was found recently by Arkhipov, Bezrukavnikov and Ginzburg. One of the key component of their proof involved employing the computation of the cohomology of quantum groups for l > h due to Ginzburg and Kumar in 1993.
The main purpose of this talk is to demonstrate how to compute cohomology for quantum groups when l < k+1. This computation entails many beautiful results:
Our results show that the cohomology ring is finitely generated. This allows us to define support varieties and compute the support varieties for quantum Weyl modules in the case when (l,p) = 1 where p is any prime for the underlying root system.
This talk represents joint work with Christopher Bendel, Brian Parshall, and Cornelius Pillen.
Calculating the cohomology of the line bundles on flag varieties in characteristic p
Abstract: Let G be a connected reductive group over an algebraiclly closed field of characteristic p and let B be a Borel subgroup. The character of the cohomology of the line bundle on the flag variety G/B is not well understood (by constrast with the situation in characteristic zero where this is given by Weyl's character formula, via the Borel-Weil-Bott Theorem). We describe some general methods of calculation and a complete solution for the case $\mathrm{SL}_3(k)$.
The center of the small quantum group: a cohomological approach
Abstract: The small quantum group $\mathfrak{u}$ is a finite dimensional Hopf algebra that appears as a subquotient of the quantized enveloping algebra $U_q(\mathfrak{g})$ when the quantization parameter is a root of unity. Being the kernel of the quantum Frobenius map, $\mathfrak{u}$ plays an important role in structure and representation theory of $U_q(\mathfrak{g})$. I will present a cohomological description of the center of the regular block of $\mathfrak{u}$, based on the functorial relation between the derived category of the principal block in modules over $\mathfrak{u}$ and a derived category of coherent sheaves on the Springer resolution $\widetilde{\mathcal{N}}$.
This is a joint work with R.Bezrukavnikov.
Two Constructions of the Fock Space and Symplectic Duality
Abstract: We discuss a duality involving pairs of symplectic resolutions related to the Springer Resolution in Type A, and the relationship between this duality and representations of Convolution Algebras.
This is joint work in progress with N. Proudfoot and T. Braden.
Bethe ansatz and Schubert calculus
Abstract: I will discuss recent developments in the Bethe ansatz for quantum integrable models associated with the Lie algebra $\mathrm{gl}_N$ which shows its intimate relation to the geometry of Schubert cells. This relation was used in the recent proof of B. and M.Shapiro conjecture in real enumerative algebraic geometry. The main conjecture is that the Bethe algebra of the quantum model (a commutative algebra generated by all integrals of motion) is isomorphic to the algebra of functions on a suitable intersection of Schubert cells. The conjecture has been proved recently for N=2.
Brauer algebras and Kazhdan-Lusztig polynomials
Abstract: We give a proof of the restriction rules from Gl(N) to O(N) using Brauer algebras. This suggests a description of the maximum semisimple quotient of the Brauer algebras via certain parabolic Kazhdan-Lusztig polynomials.
Bases, Braids, and Beyond
Abstract: Chevalley's basis for simple Lie algebras has been the preferred basis choice for over a century, and it has given much insight into the structure and representations of the Lie algebras. This talk will focus on different choices of basis for sl(2) that make transparent connections with the modular group, the braid group on 3 strands and its representations, hyperbolic Kac-Moody Lie algebras, and beyond. These bases arose in a natural way from combinatorial investigations of association schemes and tridiagonal pairs of linear transformations.
Another look at irreducible representations of quantum groups
Abstract: We use methods from Hopf algebra theory and some algebraic combinatorics to provide constructions of irreducible representations of quantized enveloping algebras (also at a root of 1), which lead to an other kind of character formulas.
Existence and combinatorial model for Kirillov-Reshetikhin crystals
Abstract: We provide the explicit combinatorial structure of the Kirillov-Reshetikhin crystals $B^{r,s}$of type $D_n(1)$, $B_n(1)$, and $A_{2n-1}(2)$. This is achieved by constructing the crystal analogue sigma of the automorphism of the $D_n(1)$ (resp. $B_n(1)$ or $A_{2n-1}(2)$) Dynkin diagram that interchanges the 0 and 1 node. The involution sigma is defined in terms of new plus-minus diagrams that govern the $D_n$ to $D_{n-1}$ (resp. $B_n$ to $B_{n-1}$, or $C_n$ to $C_{n-1}$) branching. It is also shown that the crystal $B^{r,s}$ is perfect. This work is based on the preprint math.QA/0704.2046.
In recent work in collaboration with Masato Okado, we also address the question of the existence of these crystals, which I will explain in my talk.
Vanishing integrals of Macdonald polynomials
Abstract: If one integrates a Schur function $s_\lambda$ over the orthogonal group, the integral is zero unless $\lambda$ has all parts even.
A similar statement is true for Macdonald polynomials, where one modifies the density appropriately. This modification is dictated by the representation theory of the affine Hecke algebra.
This is joint work with E. Rains.
De Concini-Procesi compactifications of semi-simple groups, Jacquet functors and semi-regular bimodules over semi-simple Lie algebras
Abstract: This is joint work with Dmitry Donin. We recall the geometry of the De Cocini-Procesi compactification of a simple Lie group G over complex numbers. We define the Jacquet functor on the category of Harish-Chandra bimodules over the corresponding Lie algebra g and its geometric realization due to Emmerton, Nadler and Vilonen. Finally we provide a localization for the semi-regular bimodule over g in terms of this geometric Jacquet functor.
Slices for biparabolics and centralizers in type A
Abstract: The celebrated Kostant slice theorem for the coadjoint action of a semisimple Lie algebra g has a remarkable potential generalization to certain non-reductive Lie subalgebras q of g. These include (truncated) biparabolics on the one hand and centalizers of nilpotent elements on the other. Indeed for type A, we show that such a generalization can be realized in all cases using the same general technique, though the biparabolic case is significantly more difficult to carry out. Moreover we show that shift of argument applied to Y(q) gives a Poisson commutative subalgebra T(q) of S(q) having the maximum possible GK dimension, namely 1/2(dim q + index q). In addition when index q = rank g, T(q) has a linearization similar to that of Y(q) as provided by the slice theorem. It is hence polynomial and maximal commutative. Outside type A, such theorems generally fail, though do hold in many cases.
Cluster algebras of finite type via principal minors
Abstract: This is a joint work in progress with Shih-Wei Yang. We give an explicit unified geometric realization for the cluster algebra of an arbitrary finite type having principal coefficients at some acyclic cluster (all this terminology will be explained). The cluster algebra in question is realized as the coordinate ring of a certain double Bruhat cell in the complex semisimple algebraic group of the same Cartan-Killing type. In this realization the set of cluster variables is some explicitly determined collection of (generalized) principal minors.
From link patterns to the intersections of the components of a Springer fiber of nilpotent order 2
Abstract: Consider the variety of strictly upper-triangular $n\times n$ matrices of nilpotent order 2. The (Borel) subgroup B of invertible upper-triangular $n\times n$ matrices acts on this variety by conjugation. The number of B-orbits is equal to the number of involutions in the symmetric group $S_n$.
We show that the link pattern associated to an involution is the "right" combinatorial object for the description of the geometry of the closures of such B-orbits. Link patterns play the same role for the description of these B-orbits as Young diagrams for the description of the of nilpotent orbits under the action of $\mathrm{GL}_n$. In particular, we give a simple formula for computing the dimension of a B-orbit in terms of a link pattern, we define a new partial order on link patterns giving us the description of the B-orbit closure.
We apply this order to give the full picture of the intersections of the irreducible components of a Springer fiber of nilpotent order 2. We use meanders to compute the number of the components in the intersection and their codimensions. We show that these intersections are reducible and not of pure dimension in general, however the intersections of codimension 1 are irreducible.
Families for complex reflection groups and generalised Cologero-Moser spaces
Abstract: This is joint work with Maurizio Martino (Bonn). I will discuss conjectural relationship between families of irreducible representations for complex reflection groups (with unequal parameters) and the singularities of generalised Calogero-Moser spaces. These spaces are defined to be the centre of rational Cherednik algebras and their structure is of interest in the representation theory of these algebras.
Revised 26.06.2007
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