Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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.

share|improve this question
add comment

1 Answer 1

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.)

share|improve this answer
    
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 $G/B$. –  Jiuzu Hong May 10 '10 at 12:16
    
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. –  Allen Knutson May 10 '10 at 15:50
    
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 $G/B$. –  Jim Humphreys May 10 '10 at 16:26
    
Small edit to last line of comment: "This many orbit closures at any rate ..." –  Jim Humphreys May 10 '10 at 17:54
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.