Explicit inducing data for Gross-Reeder-Yu supercuspidals of split reductive groups

Let K be a non-archimedean local field and G a split connected reductive affine algebraic K-group. The simple supercuspidals of the 2010 Duke paper of Gross-Reeder, whose definition was extended by the 2014 JAMS paper of Reeder-Yu beyond split simply connected groups, are very special representations r(x,L) of G(K) attached to the barycenter x of an alcove A and a certain linear functional L. It is known from Reeder-Yu that each irreducible subrepresentation r of r(x,L) is compactly induced from the subgroup H(x,L) of g in G(K) which fix x and L. To fully understand r, it is necessary to understand H(x,L) explicitly. In this paper, I give a factorization of H(x,L) into a product of three natural and explicit subgroups. If G is semisimple and simply connected, then H(x,L) is already known from Gross-Reeder, but in all other cases there are additional elements, coming from the stabilizer in G(K) of A, whose precise form is very delicate and highly dependent on L.