Let $G$ be a reductive group, let $B$ be one of its Borel subgroups, and $T$ be a torus in $B$. $G/B$ is its flag variety. Let $y,w$ be two T-fixed points in $G/B$. Let $\mathcal{O}_{y,w}$ be the $B$-orbit of $(y,w)$ in $G/B \times G/B$.

I want to understand the geometry of the closure of $\mathcal{O}_{y,w}$ in $G/B \times G/B$.

Where can I find the reference?

Thanks in advance.

EDIT: $T$ is a maximal torus.


If $T$ is taken to be a maximal torus of $G$ lying in $B$, the question may be reformulated using the set-up in 1.1-1.2 of the fundamental paper: Representations of Reductive Groups Over Finite Fields, P. Deligne and G. Lusztig, The Annals of Mathematics, Second Series, Vol. 103, No. 1 (Jan., 1976), pp. 103-161 (available in JSTOR). Relative to any such fixed choice of the pair $(T,B)$, they identify the $G$-orbits in $G/B \times G/B$ with the elements of a canonical Weyl group (independent of choices). On the other hand, an old result of Chevalley identifies the $T$-fixed points of $G/B$ with the set of Borels containing $T$, thus with the Weyl group $N(T)/T$ of $G$ relative to $T$. So you are starting with a pair of such elements, which in the Deligne-Lusztig set-up determine a single element of the "absolute" Weyl group. I'm not immediately sure how your question about the closure of a $B$-orbit will translate into this framework, but looking at it this way may be helpful. (Special cases suggest that you may just get a copy of a Schubert variety, but this is probably oversimplified.)

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  • $\begingroup$ In my question, $T$ is maximal. I don't even know whether all B-orbits in the closure of $\mathcal{O}_{y,w}$ are of the form $\mathcal{O}_{y',w'}$, where $y',w'$ are T-fixed points in $G/B$. $\endgroup$ – JJH May 10 '10 at 12:16
  • $\begingroup$ The question is more about certain $G$-orbits in $(G/B)^3$ than $(G/B)^2$, with the important difference that there are infinitely many $G$-orbits there but only finitely many of the form inquired about. $\endgroup$ – Allen Knutson May 10 '10 at 15:50
  • $\begingroup$ I'm still not sure what the question is actually about (or its motivation), but the $B$-orbits in question will be few in number and presumably easier to characterize than general $B$-orbits. To be precise, there are $|W|^2$ pairs of the given type; maybe there will be just $|W|$ distinct "types" of orbits? This many orbits at any rate look just like Schubert varieties in $G/B$. $\endgroup$ – Jim Humphreys May 10 '10 at 16:26
  • $\begingroup$ Small edit to last line of comment: "This many orbit closures at any rate ..." $\endgroup$ – Jim Humphreys May 10 '10 at 17:54

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