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Note: writing my thoughts in this post made me realize I now have a research project.

Motivation: the generalized AS functions are natural and they allow one to generalize the Atiyah problem on configurations.

Questions to try to answer in the research project:

1. Can one recover $V_\lambda$, or equivalently $\lambda$, by just knowing the corresponding generalized AS function?
2. How is information about $V_\lambda$ encoded by such generalized AS functions?

Thank you MO!

What kind of applications may such concepts have? I don't know. I just feel these are natural objects to consider.

What kind of applications may such concepts have? I don't know. I just feel these are natural objects to consider. These generalizations of the Atiyah-Sutcliffe (AS) determinant remind me a bit of characters, except they are more like determinants rather than traces, or actually, they can be defined essentially as successive contractions of a multi-qubit using a complex symplectic form on $\mathbb{C}^2$, until we get a scalar.

From this point of view it makes sense to ask the following questions. Can one recover $V_\lambda$, or equivalently $\lambda$, by just knowing the corresponding generalized AS function? How is information about $V_\lambda$ encoded by such generalized AS functions?

I guess it is hard to get help this way since this work is unpublished and I haven't given all the details (in particular, the precise definition of the generalized AS functions), yet (almost) all of the building blocks were already there in my article in the Journal of Exp. Math. ("Weights, Weyl-Equivariant Maps and a Rank Conjecture").

Edit: I had swept this under the rug, but it is possible to take care of the phase ambiguity of $\psi(\mathbf{x})$ by dividing it with a non-zero scalar-valued quantity with the same kind of homogeneity. So $\psi$ is really a smooth function on $\mathcal{C}$ with values in the vector space $$ \bigotimes_{\alpha \in \Phi} \mathbb{C}^2. $$

Edit 1: I had swept this under the rug, but it is possible to take care of the phase ambiguity of $\psi(\mathbf{x})$ by dividing it with a non-zero scalar-valued quantity with the same kind of homogeneity. So $\psi$ is really a smooth function on $\mathcal{C}$ with values in the vector space $$ \bigotimes_{\alpha \in \Phi} \mathbb{C}^2. $$

Edit 2: given an irreducible representation $V$ of $\mathfrak{g}$ corresponding to a dominant integral weight $\lambda$, we can choose a vector $0 \neq v \in V$ corresponding to the weight $\lambda$. If we then consider the principal $\mathfrak{su}(2)$ in $\mathfrak{g}$ with special element $H$ given by $2 \delta^\vee$, where $\delta^\vee$ is half the sum of the positive coroots (I forgot the corresponding formulas for the $E$ and $F$ elements), then under that representation, the $\mathfrak{su}(2)$ orbit of $v$ defines a representation of $\mathfrak{su}(2)$ which I think is irreducible (is this always the case, please?). Let us denote it by $W_\lambda$.

We could have started with an element say $\lambda'$ in the Weyl orbit $W.\lambda$ of $\lambda$. A construction of mine, generalizing the Atiyah problem on configurations, assigns, to each choice of $\lambda' \in W.\lambda$ a smooth function from $\mathcal{C}$ to $\mathbb{P}(W_\lambda)$, where $\mathbb{P}(.)$ denotes the projectivization operation.

Moreover, using the previously mentioned smooth functions defined on $\mathcal{C}$, one can construct a smooth complex-valued function defined on $\mathcal{C}$, which generalizes the Atiyah-Sutcliffe determinant, starting from a choice of $V$. Such a function is Weyl-invariant and is also invariant under an action of $SO(3)$ on $\mathcal{C}$. I feel like such a function is naturally associated to each choice of irreducible representation of $\mathfrak{g}$ (this vaguely reminds me of the geometric Langlands program, which I know very little about, but I am getting too speculative now, so I will stop).

What kind of applications may such concepts have? I don't know. I just feel these are natural objects to consider.