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Let G a semsisimple connect'ed group over $k$, $B$ a Borel and $P$ a parabolic subgroup of $G$ with Weyl group W_{P}.

For $w\in W_{P}\backslash W/W_{P}$, how can we solve the singularities of $X_{w}=\overline{PwP}/P\subset G/P$?

By solving, the singularities, I want that the the resulting

$\pi:Y\rightarrow X_{w}$

is an isomorphism on $PwP/P$.

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    $\begingroup$ I remark that the OP has never voted up or down any question or answer. $\endgroup$
    – Mikhail Borovoi
    Commented May 10, 2013 at 12:09
  • $\begingroup$ $P$-orbit closures are $B$-orbit closures. en.wikipedia.org/wiki/Bott%E2%80%93Samelson_variety $\endgroup$
    – Allen Knutson
    Commented May 10, 2013 at 12:34
  • $\begingroup$ yes but a resolution of singularities of $\overline{BwP}$ is birational on $BwP$ and not on $PwP$ a priori $\endgroup$
    – prochet
    Commented May 10, 2013 at 12:42
  • $\begingroup$ Why that's true! This sounds like a rather hard question. $\endgroup$
    – Allen Knutson
    Commented May 11, 2013 at 3:54
  • $\begingroup$ Perhaps this paper may help you in some special cases : arxiv.org/abs/math/0601117 $\endgroup$
    – Libli
    Commented May 11, 2013 at 15:02

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