Let $G = SL_n$; then for any tuple $\lambda$ such that $\sum \lambda_i = n$, define $f_\lambda(g)$ as the product of the determinants of successive minors of lengths $\lambda_i$ of $g$ (e.g. for $\lambda = (1, 1, ...), f_\lambda = \prod a_{ii}$ while for $\lambda = (n), f_\lambda = det(g)$). Then for any point $a = \sum_{\lambda} c_\lambda \lambda$ in the lattice generated by these tuples (that is, the lattice such that the basis consists of the set of tuples), define $f_a = \prod_\lambda f_\lambda^{c_\lambda}$.
This is a function on $G$; by considering $G$ as a space with a left $G$-action by translation, we can take the $G$-invariant projection of this function. This gives us a constant function; call it $h(a)$.
Is anything known about $h(a)$? I know how to calculate it when $c_\lambda = 0$ for all $\lambda$ of length greater than 2, but not in general.
Edit: I have a conjecture after checking a few more cases, but no idea how to prove it. The conjecture is:
For each $\lambda$, let $\mu_j = \sum_{i = 1}^j \lambda_i$ be an increasing tuple whose final coordinate is $n$. For each positive root $\alpha$ (the choice of Borel and Cartan don't really matter), let $\chi_\alpha(\lambda)$ be equal to 1 if there is some fundamental weight $\omega_k$ such that $k$ appears in $\mu$ and $\omega_k < \alpha$, and 0 otherwise. Finally, let $\rho$ be the sum of the fundamental weights. Then my conjecture is:
$h(a) = \prod_{\alpha \in R^+} \frac{\rho(\alpha)}{\rho(\alpha) + \sum_\lambda c_\lambda \chi_\alpha(\lambda)}$
This conjecture works when $\sum c_\lambda = 1$ or in the case where all $\lambda$ are of length at most 2.
