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The Iwahori orbits in the affine flag variety of an adjoint group $G$ are numbered by the extended affine Weyl group $W_e=W\ltimes X$ (where $X$ is the coweight lattice, and $W$ is the finite Weyl group). Let $\rho\in X$ be the halfsum of the positive coroots. Let $w\in W_e$ be the minimal representative of the double coset $W\rho W$. Let $S_w$ be the corresponding Schubert variety in the affine flags (say, $S_w$ is a point for $G=PGL(2)$, and a projective line for $G=PGL(3)$). I wonder if anything special is known about the singularities of $S_w$. Say, is it Gorenstein by any chance?

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