Reductive subgroup schemes of a parahoric group scheme

Let $K$ be the field of fractions of a complete discrete valuation ring $A$ with residue field $k$, and let $G$ be a connected and reductive linear algebraic group over $K$. Bruhat-Tits associate to $G$ various parahoric group schemes $P$. Such $P$ are smooth and affine group schemes over A, but in general they are not reductive.

Assume that $G$ splits over an unramified extension of $K$ and that $P$ is one of these parahoric group schemes. Under these assumptions, we prove that there is a reductive subgroup scheme $M$ of $P$ such that $M_k$ is a Levi factor of the special fiber $P_k$, and such that $M_K$ is a reductive subgroup of $G$ containing a maximal torus. In fact, $M_K$ is - at least geometrically - the centralizer of the image of a homomorphism $\mu_N \to G$ for some $N > 1$. The talk will describe the construction of M, and it will describe some application of the existence of $M$ to the study of $G(K)$-orbits on nilpotent elements of $\mathrm{Lie}(G)$.