Cohomology of the Minimal Nilpotent Orbit

Abstract

We compute the integral cohomology of the minimal nontrivial nilpotent orbit in a complex simple (or quasi-simple) Lie algebra. We find by a uniform approach that the middle cohomology group is isomorphic to the fundamental group of the parabolic subsystem generated by the long simple roots. The modulo ℓ reduction of the Springer correspondent representation (in the parametrization of the original paper by Springer) involves the trivial representation exactly when ℓ divides the order of this cohomology group. The primes dividing the torsion of the rest of the cohomology are bad primes.