  For $K$ an infinite field of characteristic other than two, consider the
action of the special orthogonal group $\operatorname{SO}_t(K)$ 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 $\operatorname{SO}_t(K)$ 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 $\operatorname{SO}_t(K)$
need not be unimodal.
