Cohomology of the minimal nilpotent orbit

Abstract

We compute the integral cohomology of the minimal non-trivial 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 sub-root system generated by the long simple roots. The modulo $\ell$ reduction of the Springer correspondent representation involves the sign representation exactly when $\ell$ divides the order of this cohomology group. The primes dividing the torsion of the rest of the cohomology are bad primes.

29 pages, v2 : Leray-Serre spectral sequence replaced by Gysin sequence only, corrected typos

References in corpus (1)

• Decomposition numbers for perverse sheaves

Cited by in corpus (10)

• Parity Sheaves
• Decomposition numbers for perverse sheaves
• Perverse sheaves and modular representation theory
• Modular representations of reductive groups and geometry of affine Grassmannians
• Correspondance de Springer modulaire et matrices de décomposition
• Modular Springer correspondence, decomposition matrices and basic sets
• Kumar's criterion modulo p
• Singularities of closures of spherical $B$-conjugacy classes of nilpotent orbits
• The second cohomology groups of nilpotent orbits in classical Lie algebras
• Generating varieties for affine Grassmannians