Commuting Varieties of Upper Triangular Matrices and Representation Homology

We establish a connection between commuting varieties Cg(G) (potentially of higher genus), which are associated with a group scheme G consisting of upper triangular matrices, and the representation homology HR_*(Σg,G) of a Riemann surface Σg with coefficients in the group G. As an outcome, we provide a numerical criterion for determining whether commuting varieties C(Uₙ) form a complete intersection for groups consisting of upper triangular unipotent matrices.