I am trying to see when one can extend "partial" distances to bi-$\text{SO}_n$-invariant metrics on $\text{GL}_n^+$. On the way to there, I came to the following question:
Suppose we have a symmetric function $g:(\mathbb R^{>0})^n \to \mathbb R^{\ge 0}$-i.e. $g(\sigma_1,\dots,\sigma_n)$ does not depend on the order of the $\sigma_i$. (In my case I also know that $g(1,\dots,1)=0$ but I am not sure it matters here).
We can identify $(\mathbb R^{>0})^n$ with the space $\text{PDiag}$ of real positive-definite diagonal $n \times n$ matrices.
Question: Does there exist a function $f:\text{PDiag} \times \text{GL}_n^+ \to\mathbb R^{\ge 0}$ satisfying $$ \,\,\,\,f(\tilde \Sigma,U\Sigma V)= f(\Sigma,U^{-1}\tilde \Sigma V^{-1}) \, \, \text{ for every } \, U,V \in \text{SO}_n \, \, \text{and } \Sigma, \tilde \Sigma \in \text{PDiag}, \text{ and }\,\, \, \, f(\Sigma,\text{Id})=g(\Sigma)$$
Also, If we assume that $g$ is continuous, can we create a continuous extension $f$?
(Here $\text{GL}_n^+ $ refers to the group of real invertible $n \times n$ matrices with positive determinant).
If that matters, I am ready to replace $\text{SO}_n $ with $\text{O}_n $ and assume "bi-$\text{O}_n$-invariance" in the first requirement instead of "bi-$\text{SO}_n$-invariance".