MIT Infinite Dimensional Algebra Seminar (Fall 2011)

Frenkel and Ben-Zvi gave a method for attaching a space of conformal blocksto the data of a smooth complex algebraic curve, a quasi-conformal vertexalgebra, and modules placed at points. Furthermore, when the vertex algebrahas conformal structure, one obtains sheaves of conformal blocks withprojectively flat connection on moduli spaces of smooth curves with markedpoints. I'll describe how logarithmic geometry can be employed tocanonically extend these sheaves to the semistable locus, where theconnection acquires at most logarithmic singularities. When one has a finitegroup G acting by automorphisms of the conformal vertex algebra, one mayconstruct equivariant intertwining operators by varying ramified G-covers ofthe projective line.

I will discuss some numerical invariants of representations for algebras obtained by quantizing symplectic resolutions over a characteristic p field. I will first present an application to a proof of a generalization for Macdonald positivity conjecture by Haiman (this also gives a new proof of the original conjecture proved by Haiman). Time permitting I will describe Okounkov's conjectural description for classes of irreducible modules over rational DAHA over a characteristic p, and an approach to its proof.

We will present a new approach to the theorem of Beilinson and Drinfeld completely describing the derived global sections of spherical D-modules on the affine Grassmannian at critical level as modules over the corresponding affine Kac-Moody algebra. This approach is independent of the Feigin-Frenkel theorem relating the critical-level center of the affine Kac-Moody algebra attached to a semisimple Lie algebra to the space of opers for the Langlands dual group, and gives a new proof of this theorem via a construction of Beilinson-Drinfeld-Frenkel-Gaitsgory.

It is known that the quasi-triangular groups $QTr_n$, for $n \geq 4$ (given by generators $R_{ij}$, for $1 \leq i \ne j \leq n$, modulo the quantum Yang-Baxter equations) are not 1-formal: the Malcev Lie algebra $M_{QTr_n}$ of $QTr_n$ is not isomorphic to the (completed) rational holonomy Lie algebra of $QTr_n$. Nonetheless, one can ask whether the $QTr_n$ satisfy a 'graded 1-formality': i.e. is there such an isomorphism at the level of the respective associated gradeds. This corresponds to the associated graded of the universal enveloping algebra $U_{QTr_n}$ of $M_{QTr_n}$ (with respect to the filtration by powers of the augmentation ideal) being 'quadratic', that is, being generated in degree 1 and having only relations generated in degree 2.We give a general criterion for determining whether, if an augmented algebra $K$ over $\Q$ is filtered by powers of its augmentation ideal $I$, the associated graded algebra $gr_I K$ is quadratic. We apply this criterion to $U_{QTr_n}$, and show that it is quadratic, and hence that the $QTr_n$ are 'graded 1-formal'.

This will be a pair of essentially independent talks following up on my colloquium talk on October 14.

Topic 1: Classifying q-hypergeometric identities(this is joint work with van Diejen)

Abstract: One of the benefits of generalizing to the elliptic level is that there are not only fewer identities to consider, but those identities tend to have more interesting symmetries. In particular, one can often obtain rather different looking ordinary or $q$-hypergeometric identities via different limits of the same elliptic identity. I'll describe joint work with van Diejen in which we have systematically explored $q$-hypergeometric limits of elliptic hypergeometric integrals and associated biorthogonal functions. For instance, we find that a large class of $q$-hypergeometric functions are classified by a certain $E_7$-invariant polytope: there is a different $q$-hypergeometric function attached to each simplicial face of this polytope, and these functions satisfy transformations coming from the action of $E_7$.

Topic 2: Elliptic analogues of Painlev\'e theory

Abstract: Another class of special functions which gained prominence in the late 20th century (though they date from 1906) are the Painlev\'etranscendents, solutions of certain nonlinear second-order differential equations. In 2001, Sakai constructed an elliptic analogue of these equations (and classified their degenerations), but Sakai's construction (though very pretty) does not account for two important roles of the Painlev\'e transcendents: the fact that many random matrix integrals (and analogues) are Painlev\'e transcendents, and the fact that Painlev\'e transcendents describe monodromy-preserving deformations of linear differential equations. I'll describe an alternate path to Sakai's construction (and generalizations) based on extending these two facts.

This talk is based on a joint work with Iain Gordon, arXiv:1109.2315.Cyclotomic rational Cherednik algebras are certain associative algebrasconstructed from the complex reflection groups $G(l,1,n)$ and certaincomplex valued parameters. In my talk I am going to describe three results:1) Existence of equivalence between the Cherednik category O and anappropriate version of the parabolic category O for the general linear algebra. This equivalences exist under some strong restriction on parameters.2) Existence of auto-equivalences constituting an $S_l$-action of theCherednik cateogories O for different values of parameters. These equivalences exist under some mild restriction on the parameters.3) Existence of derived "translation" equivalences between the Cherednikcategories O. These equivalences exist for all parameters.All necessary information about the Cherednik algebras and theircategories O will be recalled.

The theory of MV cycles associated to a complex reductive group $G$ has proven to be a rich source of structures related to representation theory. We investigate double MV cycles, which are analogues of MV cycles in the case of an affine Kac-Moody group. We prove an explicit formula for the Braverman-Finkelberg-Gaitsgory crystal structure on double MV cycles, generalizing a finite-dimensional result of Baumann and Gaussent. As an application, we give a geometric construction of the Naito-Sagaki-Saito crystal via the action of $\hat{SL}_n$ on Fermionic Fock space. In particular, this construction gives rise to an isomorphism of crystals between the set of double MV cycles and the Naito-Sagaki-Saito crystal. As a result, we can independently prove that the Naito-Sagaki-Saito crystal is the $B(\infty)$ crystal. In particular, our geometric proof works in the previously unknown case of $\hat{\mathfrak{sl}}_2$.I will spend the first half of my talk reviewing some aspects of the theory of MV cycles for finite-dimensional groups. The story gives rise to MV polytopes and a surprising connection with Lusztig's canonical basis. In the second half, I will introduce double MV cycles. Here the finite-dimensional story does not naively generalize. Nonetheless, in type A, I will present a method to parameterize double MV cycles. This method gives rise to exactly the combinatorics of the Naito-Sagaki-Saito crystal. If I have time, I will discuss some related work and some open problems.

This talk will be divided into two parts. In the first partwe are going to review some well-knownresults about spherical and Iwahori Hecke algebras of a reductivegroup G over a local non-archimedianfield and then explain how to generalize those results to the casewhen G is an affine Kac-Moody group.In the second part we are going to talk about Eisenstein series foraffine Kac-Moody groups (as in the first part,we'll start by reviewing the corresponding results for finitedimensional G and then proceed to the case when G isaffine).Based on joint works with H.Garland, M.Finkelberg, D.Kazhdan and M.Patnaik.

The Poisson vertex algebra cohomology can be used to establish thevalidity of the so called Lenard-Magri scheme of integrability for abi-Hamiltonian PDE. I will discuss some recent work (in collaboration withV. Kac) where we computed the PVA cohomology for any quasiconstantcoefficient Hamiltonian structure K of arbitrary order, with invertibleleading coefficient.

Kashiwara developed combinatorial objects called crystals to study the representation theory of complex simple Lie groups and Lie algebras. The construction is quite involved, but one can often realize the same combinatorics by more elementary means. One useful realization is based on the Mirkovic-Vilonen polytopes of the title. I will describe what these polytopes are, and why they are interesting. I will then explain current work giving an analogous construction for symmetric affine Kac-Moody algebras. For affine sl(2) the construction is purely combinatorial. For other symmetric affine types the definitions are combinatorial, but we need some geometry (quiver varieties) to prove that everything works. This is joint work with Pierre Baumann, Thomas Dunlap and Joel Kamnitzer.

The nonsymmetric q-Whittaker function was constructedin the author's paper "A new take" together with the Dunkloperator in the q-Toda theory. The approach was based ona special (spinor) variant of the limiting procedure from theq,t-theory due to Ruijsenaars and Etingof. An algebraicoutput was a new polynomial-type representation of nil-daha,which appeared not an induced one in any standard way.The algebraic meaning of this representation was recentlyclarified in full in terms of ``nil-nil daha", which is expected toplay the role of the canonical (crystal) limit $t\to 0$ in the dahatheory. The talk will be devoted to $A_1$, which is actually theonly case when this new theory is completed.