Existence of a Weyl–equivariant stable splitting of the maximal torus in SU(m)

Establish whether there exists a stable splitting of a maximal torus T ⊂ SU(m) that is equivariant under the Weyl group W = N_{SU(m)}(T)/T, analogous to the Weyl–equivariant torus splitting used in the U(m) and Sp(m) cases, in order to enable an analogous stable splitting for Hom(Z^n, SU(m)).

Background

The paper proves a stable splitting for spaces of commuting elements in U(m) after inverting m!, relying on a stable splitting of a maximal torus that is equivariant for the action of the Weyl group. This equivariant splitting underpins the construction of stable summands indexed by partitions and associated bundles of commuting varieties.

For Sp(m) and certain orthogonal groups, the authors’ methods apply because an appropriate Weyl–equivariant splitting is available. For SU(m), however, the approach would require an analogous Weyl–equivariant stable splitting of a maximal torus, and the authors explicitly state that they do not know whether such a splitting exists.