Constructions of $SU(2)$ and Weyl equivariant maps for all classical groups

TL;DR

This paper constructs explicit $SU(2)$ and Weyl equivariant maps for all classical Lie groups, extending previous work and providing new constructions for orthogonal groups.

Contribution

The paper develops explicit constructions of equivariant maps for all classical groups, completing the set of such constructions beyond unitary and symplectic groups.

Findings

Constructed explicit equivariant maps for $SO(2m+1)$ and $SO(2m)$.

Extended the class of groups for which explicit maps are known.

Provided new tools for understanding the Berry-Robbins problem in classical groups.

Abstract

If $G$ is a compact Lie group, $T$ a maximal torus in $G$ (with Lie algebras $\mathfrak{g}$ and $\mathfrak{t}$ respectively) and $W$ the corresponding Weyl group, then the Berry-Robbins problem for $G$, as formulated by Sir Michael Atiyah and Roger Bielawski, asks whether there exists a continuous $SU(2) \times W$ equivariant map from the space of regular Cartan triples (an open subset of $\mathfrak{t} \otimes \mathbb{R}^3$) to $G/T$, where $SU(2)$ acts via a regular Lie group homomorphism $SU(2) \to G$. This was settled positively by Atiyah and Bielawski, and their maps are even smooth, but they are not explicit. For $G=U(n)$, there exists another construction due to Sir Michael Atiyah and developed further with Paul Sutcliffe, which is explicit, but relies on a linear independence conjecture. The author had previously found a similar type of construction for $G=Sp(m)$, also relying on a linear independence conjecture. In this paper, similar constructions are done for $SO(2m+1)$ and $SO(2m)$, thus exhausting the list of classical groups.