Relative Frobenius Formula

Abstract

For a finite group $G$, Frobenius found a formula for the values of the function ${\sum }_{\mathrm{Irr}G}(\dim \pi {)}^{-s}$ for even integers $s$, where $\mathrm{Irr}G$ is the set of irreducible representations of $G$. We generalize this formula to the relative case: for a subgroup $H$, we find a formula for the values of the function ${\sum }_{\mathrm{Irr}G}(\dim \pi {)}^{-s}(\dim {\pi }^H{)}^{-t}$. We apply our results to compute the E-polynomials of Fock--Goncharov spaces and to relate the Gelfand property to the geometry of generalized Fock--Goncharov spaces.