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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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I remark that the OP has never voted up or down any question or answer. –  Mikhail Borovoi May 10 '13 at 12:09
$P$-orbit closures are $B$-orbit closures. en.wikipedia.org/wiki/Bott%E2%80%93Samelson_variety –  Allen Knutson May 10 '13 at 12:34
yes but a resolution of singularities of $\overline{BwP}$ is birational on $BwP$ and not on $PwP$ a priori –  prochet May 10 '13 at 12:42
Why that's true! This sounds like a rather hard question. –  Allen Knutson May 11 '13 at 3:54
Perhaps this paper may help you in some special cases : arxiv.org/abs/math/0601117 –  Franz May 11 '13 at 15:02

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