F-Split and F-Regular Varieties with a Diagonalizable Group Action

Let H be a diagonalizable group over an algebraically closed field
k of positive characteristic, and X a normal k-variety with an H-action. Under
a mild hypothesis, e.g. H a torus or X quasiprojective, we construct a certain
quotient log pair (Y,Δ) and show that X is F-split (F-regular) if and only
if the pair (Y,Δ) if F-split (F-regular). We relate splittings of X compatible
with H-invariant subvarieties to compatible splittings of (Y,Δ), as well as
discussing diagonal splittings of X. We apply this machinery to analyze the
F-splitting and F-regularity of complexity-one T-varieties and toric vector
bundles, among other examples