A stable splitting for spaces of commuting elements in unitary groups

TL;DR

This paper extends Miller's stable splitting to spaces of commuting elements in unitary groups, revealing a stable wedge decomposition after inverting m!, and explores similar decompositions for symplectic and orthogonal groups.

Contribution

It introduces a stable splitting for spaces of commuting elements in unitary groups, including the failure of integral splitting and analogous results for other classical groups.

Findings

Stable splitting holds after inverting m!

Splitting does not hold integrally due to Steenrod operations

Provides decompositions for symplectic and orthogonal groups

Abstract

We prove an analogue of Miller's stable splitting of the unitary group $U(m)$ for spaces of commuting elements in $U(m)$. After inverting $m!$, the space $\text{Hom}(\mathbb{Z}^n,U(m))$ splits stably as a wedge of Thom-like spaces of bundles of commuting varieties over certain partial flag manifolds. Using Steenrod operations we prove that our splitting does not hold integrally. Analogous decompositions for symplectic and orthogonal groups as well as homological results for the one-point compactification of the commuting variety in a Lie algebra are also provided.