Maximal Levi subgroups acting on the Euclidean building of $\mathrm{GL}_n(F)$

Abstract

In this paper we give a complete invariant of the action of ${GL}_n\left(F\right)\times {GL}_m\left(F\right)$ on the Euclidean building $\mathcal B_e$${GL}_{n+m}\left(F\right)$, where $F$ is a discrete valuation field. We then use this invariant to give a natural metric on the resulting quotient space. In the special case of the torus acting on the tree $\mathcal B_e$${GL}_2\left(F\right)$, we obtain an algorithm for calculating the distance of any vertex in the tree to any fixed apartment.