Let $G$ be a compact Lie group and $T$ a choice of maximal torus. Denote the corresponding Lie algebras by $\mathfrak{g}$ and $\mathfrak{t}$. Elements of $\mathfrak{t} \otimes \mathbb{R}^3$ are called Cartan triples. Note that given any root $\alpha$ of $G$, one may construct the linear map
$$ \alpha \otimes 1_{\mathbb{R}^3} \colon \mathfrak{t} \otimes \mathbb{R}^3 \to \mathbb{R}^3,$$
where $1_{\mathbb{R}^3}$ denotes the identity map on $\mathbb{R}^3$. A Cartan triple $x$ is said to be regular if it is not in the kernel of any linear map $\alpha \otimes 1_{\mathbb{R}^3}$, no matter what root $\alpha$ of $G$ one takes. We also denote by $\Delta$ the following union
$$ \Delta = \bigcup_{\alpha} \operatorname{ker}(\alpha \otimes 1_{\mathbb{R}^3}),$$
the union being over the roots $\alpha$ of $G$. Thus the space of regular Cartan triples is
$$ (\mathfrak{t} \otimes \mathbb{R}^3) \setminus \Delta.$$
The Berry-Robbins problem, as stated and generalized to a Lie theoretic setting by Atiyah and Bielawski, asks if there exists a smooth map
$$f \colon (\mathfrak{t} \otimes \mathbb{R}^3) \setminus \Delta \to G/T,$$
which is $SU(2)$ and $W$ equivariant. The Lie group $SU(2)$ acts on the domain by fixing the $\mathfrak{t}$ "factor" and acting on $\mathbb{R}^3$ via the usual adjoint representation (the double cover $SU(2) \to SO(3)$) and acts on the target space via a preferred Lie group homomorphism $SU(2) \to G$ (for the sake of this post, one may assume that this homomorphism is a regular homomorphism, in the sense of Kostant's work on $sl(2)$ triplets). The Weyl group $W$ of $(G,T)$ acts naturally on $\mathfrak{t}$ and thus on $\mathfrak{t} \otimes \mathbb{R}^3$ by fixing the $\mathbb{R}^3$ "factor". This action descends to an action on the space of regular Cartan triples of $G$. And $\sigma \in W$ acts naturally on $G/T$ by multiplication from the right by $\sigma^{-1}$, since $W = N(T)/T$.
Atiyah and Bielawski solved this problem using Nahm's equations. However, there is another approach, due to Atiyah and further expanded together with Paul Sutcliffe, for the case $G = U(n)$, but for this approach to work, a linear independence conjecture has to hold.
I have expanded the latter approach, to include also $G = Sp(m), \, SO(2m) \text{ and } SO(2m+1)$, also based on linear independence conjectures.
For the exceptional cases, I mostly focused on $G_2$, but I only got a smooth map
$$f: (\mathfrak{t} \otimes \mathbb{R}^3) \setminus \Delta \to SO(7)/SO(2)^3,$$
where $SO(2)^3$ is a choice of maximal torus in $SO(7)$ (while $\mathfrak{t}$ is the two-dimensional real Lie algebra of a maximal torus in $G_2$), which is $SU(2)$ and $W$ equivariant, also relying on a linear independence conjecture (for some action of the Weyl group $W$ of $G_2$ on $SO(7)/SO(2)^3$ which I did not define here).
I am thinking to write up what I have. However, I anticipate the dreaded questions "what makes these maps interesting?", "what is the motivation for studying such maps?" and so on.
The motivation for studying the Berry-Robbins problem comes from the work of Berry and Robbins problem on the spin-statistics theorem. They had a neat construction for the case of $n = 2$ identical particles and wanted to generalize their construction to any number of identical particles, which led to that problem. This is thus motivated essentially by the fact that the unitary groups are groups of symmetries of the various Hilbert spaces occurring in Quantum Mechanics.
I am basically trying to motivate my work. I find such maps interesting, but I would like to convince others (including reviewers) that they are indeed interesting!
Ideas and comments are greatly appreciated.
Edit: If someone is interested, I wrote up my constructions for $SO(2m+1)$ and $SO(2m)$. I also reviewed the Atiyah and Sutcliffe construction, which is the original construction, corresponding to $G = U(n)$ and I also reviewed my variant/construction corresponding to $G = Sp(m)$. Such constructions, old and new, can be found in my preprint "Constructions of $SU(2)$ and Weyl equivariant maps for all classical groups". Kindly note that the construction for $SO(2m)$ was more sophisticated, and it did take me a number of years actually to think about it (though I had a rather unsatisfactory "ancestor" of such a construction years ago). Anyway, I do thank the Mathoverflow Users in general. The platform and the Users make research go more smoothly.
