Let $G$ be a connected, simply connected, semisimple, complex linear algebraic group with maximal torus $T$ and affine Grassmannian $\mathcal Gr$. It is well known that $\mathcal Gr$ admits a Bruhat decomposition $$ \mathcal Gr = \bigsqcup_{\lambda\in X_*(T)} \mathcal B \lambda $$ where $\mathcal B$ is the Iwahori subgroup, and the union is taken over coweights of $T$. Let $X_\lambda = \overline{\mathcal B\lambda}$ be the corresponding Schubert variety.

What is the singular/smooth locus of $X_w$?

It is known that under the Bruhat order, $$ X_\lambda = \overline{\mathcal B\lambda} = \bigsqcup_{\mu\leq \lambda} B\mu,$$ and I have some sneaking suspicion that the smooth locus should just be $\mathcal B\lambda$, but I cannot think of how to conclude that this is the case.

  • $\begingroup$ I'm pretty sure that's true for $\lambda$ dominant, but it's definitely not always true, e.g. if $\dim X_\lambda=1$. $\endgroup$ – Allen Knutson Mar 30 '15 at 19:39
  • $\begingroup$ @AllenKnutson Perhaps this is the only exception in the infinite dimensional case? (were we in finite dimensions, the large cell would also prove a counter example) $\endgroup$ – Tyler Holden Mar 30 '15 at 20:39

The smooth locus of a spherical orbit (i.e. $G(\mathcal O)$-orbit) on the affine Grassmannian is just the spherical orbit itself (but this orbit consists of several Iwahori-orbits). See

Malkin-Ostrik-Vybornov, The minimal degeneration singularities in the affine Grassmannians, Duke Math. J. 126 (2005), no. 2, 233–249

for a more precise result: they describe the singularity of each orbit closure along each spherical orbit open in the boundary. They quote

Evens-Mirković Characteristic cycles for the loop Grassmannian and nilpotent orbits, Duke Math. J. 97 (1999), no. 1, 109–126

for the result that the smooth locus of a spherical orbit is just the orbit itself. I have an alternative modular representation-theoretic fun proof using the geometric Satake correspondence (see a preprint of mine from 2008).

So the answer to your question was clearly no as stated on general grounds: for a general Kac-Moody Schubert variety, if w is maximal modulo some finite parabolic subgroup $W_I$, then the smooth locus of $\overline{BwB}/B$ contains $P_IwB$, which is the union of the Schubert cells corresponding to the elements of the coset $W_I w$, so more than one if $W_I$ is nontrivial (for $W$ finite, the case of the full flag variety is $W_I = W$). If one considers $G/P$ instead of $G/B$, I guess one has to consider double cosets...

But the result above tells you that for $\lambda$ dominant, the smooth locus does not get larger than that (considering the finite Weyl group as a maximal parabolic subgroup of the affine Weyl group). I don't know the answer for general $\lambda$, but there is an extensive literature about smooth (resp. rationally smooth, even p-smooth) loci of Schubert varieties.

A similar result holds for nilpotent cones: the smooth locus of a nilpotent orbit is just the orbit itself (this is a consequence of results by Namikawa and Kaledin; again, one can describe all minimal degeneration singularities: this was done by Kraft-Procesi in classical types, and by Fu-J.-Levy-Sommers in exceptional types).

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