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Colloquium April 21

Date: Friday April 21

Time: 16:00 to 17:00

Place: 455 McBryde (Commons Room)

Speaker: Dan Nakano of Georgia Tech

Title: Quantum group cohomology via the geometry of the nullcone

Abstract

Given a semisimple 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 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 z is a primitive lth 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 verfied the character formula for affine Lie algebras. A second proof was found recently by Arkhipov, Bezrukavnikov and Ginzburg. One of the key components 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>h. This computation entails many beautiful results:

  1. Realization of the ``restricted nullcone'' due to Carlson, Lin, Nakano and Parshall
  2. Combinatorics involving the decomposition of the exterior algebra via the Steinberg representation. Our decomposition results makes use of MAGMA computations on root systems for exceptional Lie algebras.
  3. Powerful vanishing results on line bundle cohomology proved via complex algebraic geometry (i.e. Grauert-Riemenschnieder theorem).
  4. Normality results on the closures of nilpotent orbits due to Kraft-Procesi, Sommers, Broer, Kumar-Lauritzen-Thomsen.
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 bad prime for the underlying root system. This talk represents joint work with Christopher Bendel, Brian Parshall, and Cornelius Pillen.

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