Cohomology of the minimal nilpotent orbit

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 $\mathrm{\ell }$ reduction of the Springer correspondent representation involves the sign representation exactly when $\mathrm{\ell }$ divides the order of this cohomology group. The primes dividing the torsion of the rest of the cohomology are bad primes.