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 $(Y,\mathrm{\Delta })$ and show that $X$ is F-split (F-regular) if and only if the pair $(Y,\mathrm{\Delta })$ if F-split (F-regular). We relate splittings of $X$ compatible with $H$-invariant subvarieties to compatible splittings of $(Y,\mathrm{\Delta })$, 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.