# Irreducibility of Gelfand-Serganova strata

To keep the notations simple I'll restrict my attention to the complete flag variety although the question should be equally valid for partial flag varieties. Consider $$G=SL_n(\mathbb C)$$ with Borel $$B\subset G$$, Cartan $$T\subset B$$ and flag variety $$F=G/B$$. The Plücker embedding $$F\subset\mathbb P(\wedge^1\mathbb C^n)\times\ldots\times\mathbb P(\wedge^{n-1}\mathbb C^n)$$ equips $$F$$ with Plücker coordinates $$\{X_{i_1,\ldots,i_k}\}$$.

The Gelfand-Serganova strata in $$F$$ are the equivalence classes with respect to any of the following three equivalence relations (which are shown to coincide).

1. $$x\sim y$$ iff the set of Plücker coordinates vanishing in $$x$$ is the same as that for $$y$$.
2. $$x\sim y$$ iff the set of $$T$$-fixed points contained in the orbit closure $$\overline{Tx}$$ is the same as that in $$\overline{Ty}$$.
3. $$x\sim y$$ iff for any Borel $$B'\supset T$$ the orbits $$B'x$$ and $$B'y$$ coincide, i.e. $$x$$ and $$y$$ lie in the same Schubert cell with respect to $$B'$$. In other words, the strata are the nonempty intersections of $$n!$$ Schubert cells, one for every Borel containing $$T$$.

Unfortunately, as shown in the above paper, the closure of a stratum is not necessarily a union of strata. This rules out many nice properties one could hope for in such a setting. However, I still have a feeling that the strata (and their closures) might always be irreducible. Is that so and has this been discussed in the literature?