Cohomology for Quantum Groups via the Geometry of the Nullcone

Let be a complex th root of unity for an odd integer >1 . For any complex simple Lie algebra g , let u =u (g) be the associated "small" quantum enveloping algebra. This algebra is a finite dimensional Hopf algebra which can be realised as a subalgebra of the Lusztig (divided power) quantum enveloping algebra U and as a quotient algebra of the De Concini-Kac quantum enveloping algebra U . It plays an important role in the representation theories of both U and U 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 ph . The main result in this paper provides a surprisingly uniform answer for the cohomology algebra H (u ,C) of the small quantum group.