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"Any $L^+G$-orbit in Gr is known to be an orbit of the form Oλ for some λ ∈ $X_*(T)$, and Oλ = Oμ iff λ and μ are conjugate by W and the orbits form a finite stratification of each of the $Gr_i$." who can prove this sentence? please tell me.

I also want to know the information $L^+T$-orbit in Gr and LN-orbit in Gr ,where T is max torus and N=[B,B],(B is a borel group )

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  • $\begingroup$ Can you give a little more context, please? Where are these statements from? What exactly is the part that you are having difficulty with? $\endgroup$
    – Andrew Stacey
    Commented Sep 12, 2011 at 10:13
  • $\begingroup$ this sentence comes from“PERVERSE SHEAVES ON A LOOP GROUP AND LANGLANDS’ DUALITY” difficulty is the provement。 thank you！ $\endgroup$
    – yingjin bi
    Commented Sep 12, 2011 at 10:32
  • $\begingroup$ Could you edit your question a little and include a link to that article. Also, could you say whether or not you have read Loop Groups by Pressley and Segal as I suspect that the answer is in there. $\endgroup$
    – Andrew Stacey
    Commented Sep 12, 2011 at 10:38
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    $\begingroup$ This is the Bruhat decomposition for the affine Grassmannian, related to Birkhoff's theorem giving a BWB decomposition for loops. It can, indeed, be found in Pressley-Segal, though I believe it is proved there only for GL(n). Look also in Kumar's book. $\endgroup$
    – Michael Thaddeus
    Commented Sep 12, 2011 at 15:18
  • $\begingroup$ thank you ， Andrew Stacey and Michael Thaddeus 。I will read loop group $\endgroup$
    – yingjin bi
    Commented Sep 13, 2011 at 3:21

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