Cohomology for Quantum Groups via the Geometry of the Nullcone

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 characters of the rational irreducible $G$-modules stipulates that $p \geq h$. The main result in this paper provides a surprisingly uniform answer for the cohomology algebra $\operatorname{H}^\bullet(u_\zeta,{\mathbb C})$ of the small quantum group.