Authors:
Christopher P. Bendel, Daniel K. Nakano, Brian J. Parshall, and Cornelius Pillen

Title:
Cohomology for quantum groups via the geometry of the nullcone

Abstract:
Let $\zeta$ be a complex $\ell$th root of unity for an odd integer
$\ell>1$. For any complex simple Lie algebra $\mathfrak g$, let
$u_\zeta=u_\zeta({\mathfrak g})$ be the associated ``small" quantum
enveloping algebra. This algebra is a finite dimensional Hopf
algebra which can be realized as a subalgebra of the Lusztig
(divided power) quantum enveloping algebra $U_\zeta$ and as a
quotient algebra of the De Concini--Kac quantum enveloping algebra
${\mathcal U}_\zeta$. It plays an important role in the
representation theories of both $U_\zeta$ and ${\mathcal U}_\zeta$
in a way analogous to that played by the restricted enveloping
algebra $u$ of a reductive group $G$ in positive characteristic $p$
with respect to its distribution and enveloping algebras. In
general, little is known about the representation theory of quantum groups
(resp., algebraic groups) when $l$ (resp., $p$) is smaller than the
Coxeter number $h$ of the underlying root system. For example,
Lusztig's Conjecture concerning the irreducible modules can only be
formulated when $p \geq h$. The main result in this paper provides a
surprisingly uniform answer  for the  cohomology algebra
$\opH^\bullet(u_\zeta,{\mathbb C})$ of the small quantum group.
When $\ell>h$, this cohomology algebra has been calculated by
Ginzburg and Kumar \cite{GK}. Our result requires powerful tools
from complex geometry and a detailed knowledge of the geometry of
the nullcone of $\mathfrak g$. In this way, the methods point out
difficulties present in obtaining similar results for the restricted
enveloping algebra $u$ in small characteristics, though they do
provide some clarification of known results there also. Finally, we
establish that if $M$ is a finite dimensional $u_\zeta$-module, then
$\opH^\bullet(u_\zeta,M)$ is a finitely generated
$\opH^\bullet(u_\zeta,\mathbb C)$-module, and we obtain new results
on the theory of support varieties for $u_\zeta$.

Status: preprint/submitted for publication