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. 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 characters of the rational irreducible $G$-modules stipulates that $p\ge h$. The main result in this paper provides a surprisingly uniform answer for the cohomology algebra $\opH^\bullet(u_ζ,{\mathbb C})$ 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 $\opH^\bullet(u_ζ,M)$ is a finitely generated $\opH^\bullet(u_ζ,\mathbb C)$-module, and we obtain new results on the theory of support varieties for $u_{\zeta }$.

References in corpus (1)

• Representations and cohomology for Frobenius-Lusztig kernels