Richardson varieties have Kawamata log terminal singularities

Let Xvw be a Richardson variety in the full flag variety X associated to a symmetrizable Kac–Moody group G. Recall that Xvw is the intersection of the finite-dimensional Schu-bert variety Xw with the finite-codimensional opposite Schubert variety Xv. We give an explicit Q-divisor Δ on Xvw and prove that the pair (X v w,Δ) has Kawamata log terminal singularities. In fact, −KXvw − Δ is ample, which additionally proves that (Xvw,Δ) is log Fano. We first give a proof of our result in the finite case (i.e., in the case when G is a finite-dimensional semisimple group) by a careful analysis of an explicit resolution of singularities of Xvw (similar to the Bott–Samelson–Demazure–Hansen resolution of the Schubert varieties). In the general Kac–Moody case, in the absence of an explicit reso-lution of Xvw as above, we give a proof that relies on the Frobenius splitting methods. In particular, we use Mathieu’s result asserting that the Richardson varieties are Frobe-nius split, and combine it with a result of Hara and Watanabe [14] relating Frobenius splittings with log canonical singularities