Torsion primes for commuting varieties of complex reductive Lie algebras

Determine the primes at which the compactly supported integral cohomology of the commuting variety of a complex reductive Lie algebra has torsion, thereby characterizing the torsion primes for commuting varieties in complex reductive Lie algebras.

Background

The paper completely characterizes the torsion primes in the integral cohomology of identity components of spaces of commuting tuples in complex reductive groups and compact Lie groups: precisely the primes dividing the order of the Weyl group. It also proves the analogous result for compactly supported cohomology of commuting tuples in compact real Lie algebras.

The authors explain that their group-theoretic methods do not extend directly to complex reductive Lie algebras. In particular, unlike the compact real Lie algebra case, the total rank of compactly supported rational cohomology for commuting varieties in complex reductive Lie algebras is not determined solely by the rank of the Lie algebra; the comparison of the rank-two examples $\mathfrak{sp}_4^\mathbb{C}$ and $\mathfrak{gl}_2^\mathbb{C}$ demonstrates this obstruction. Consequently, the torsion-prime characterization for complex reductive Lie algebras remains unresolved.