Invariant rings of the special orthogonal group have nonunimodal h -vectors

For K an infinite field of characteristic other than two, consider the action of the special orthogonal group SOt(K) on a polynomial ring via copies of the regular representation. When K has characteristic zero, Boutot's theorem implies that the invariant ring has rational singularities; when K has positive characteristic, the invariant ring is F-regular, as proven by Hashimoto using good filtrations. We give a new proof of this, viewing the invariant ring for SOt(K) as a cyclic cover of the invariant ring for the corresponding orthogonal group; this point of view has a number of useful consequences, for example, it readily yields the a-invariant and information on the Hilbert series. Indeed, we use this to show that the h-vector of the invariant ring for SOt(K) need not be unimodal.