Workshop on Representation Theory and Geometry 2007

Title. Bases and Crystal Bases for Representations of Reductive Groups.

In the fifties Gelfand posed the question of how to find a good basis for irreducible representations of reductive groups. In a paper with Zeitlin he defined special bases for irreducible representations of the general linear group and orthogonal groups. But the breakthrough came in the 80s with the geometrization of the theory of representations and in the early 90s when Lusztig defined the canonical basis for representations of arbitrary reductive groups and Kashiwara defined the crystal basis which a way to parameterize such bases. As was shown by Joseph the crystal basis is the unique way to parameterize bases in irreducible representations in a way compatible with tensor products. In my lectures I explain some of these constructions.

I will discuss work generalizing the classification by I. Dimitrov and I. Penkov of Borel subalgebras of \gl_\infty. Root-reductive Lie algebras are unions of finite-dimensional reductive Lie algebras under inclusions preserving the root spaces with respect to nested Cartan subalgebras. The main general result is that a Borel subalgebra of a root-reductive Lie algebra is the simultaneous stabilizer of a certain type of generalized flag in each of the standard representations. For the three infinite-dimensional simple root-reductive Lie algebras more precise results are obtained. The map sending a maximal closed (isotropic) generalized flag in the standard representation to its stabilizer hits Borel subalgebras, yielding a bijection in the cases of \sl_\infty and \sp_\infty; in the case of \so_\infty the fibers are of size one and two. Finally, a large class of Borel subalgebras of a general root-reductive Lie algebra are seen to correspond bijectively with Borel subalgebras of the commutator subalgeba, which are well understood in terms of the special cases.

Almost-commuting variety and Cherednik algebras

I will speak on the geometry of a scheme that is closely related to the set of pairs of n by n-matrices with rank 1 commutator, and describe applications to Cherednik algebras and symplectic reflection algebras. This is a joint work with V. Ginzburg.

Classical and quantum infinitesimal Hecke algebras of sl(2)

The simplest case of infinitesimal Hecke algebras (introduced by Etingof, Ginzburg and Gan), are a family of PBW deformations of the smash product algebra of sl(2) and its natural two-dimensional representation. This family can naturally be quantized; the original family of algebras is then a "classical limit" of the quantized family. I will characterize complete reducibility in the quantum case.

The moduli space of genus zero real stable curves, and commutors for quantum groups.

The moduli space of genus zero stable curves with n marked points over the complex numbers is well understood. In recent years, progress has been made in studying the same space over the reals. One important observation is that its fundamental group acts by isomorphisms on the set of tensor products of n representations for U_q(g). The action is constructed using a commutor, which is a natural isomorphism from V(x)W to W(x)V, satisfying certain properties. This leads to an analogy with the configuration space of n-tuples of complex numbers, since the fundamental group of configuration space, i.e. the braid group, has a well known action on tensor products of representations for U_q(g). In this talk I will discuss various results about the moduli space of real genus zero stable curves, keeping in mind the above analogy.

Gaudin models with irregular singularities

(this is joint work with B. Feigin and E. Frenkel.) I will explain how to diagonalise the quantum Hamiltonians arising from the Casimir connection by using affine Kac-Moody algebras at critical level. This mirrors the construction of Feigin, Frenkel and Reshetikhin who diagonalised the Gaudin Hamiltonians arising from the Knizhnik-Zamolodchikov connection, and leads to a new class of quantum integrable systems generalizing the Gaudin model. Two interesting new features appear in the contruction: the use of non-highest weight representations of affine Lie algebras and connections (more precisely opers) with irregular (as opposed to regular) singularities on the Riemann sphere which describe the spectrum of the algebras of quantum Hamiltonians.

The geometry of Soergel bimodules

We will discuss how Soergel bimodules appear geometrically as cohomology of perverse sheaves on Lie groups, and show how this geometry is helpful in understanding the structure of Soergel bimodules, their Hochschild homology, and hopefully a little bit about Khovanov-Rozansky's triply graded knot homology.

TBA

Quantum Symmetric Algebras and their Applications

I will introduce quantum (or braided) symmetric algebras, which were defined as q-analogs of the symmetric algebra of a Lie algebra module in a joint work with A. Berenstein. I will show that many important quantized coordinate rings naturally appear as quantum symmetric algebras, and then explain how these results apply to the study of canonical bases and classical invariant theory.