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All varieties are over $\mathbb{C}$.

Let $G$ be a connected reductive group, $B\subseteq G$ a Borel subgroup. Let $O_w$ be a $B$-orbit in $G/B$. I.e., $O_w$ is a Bruhat cell. In particular, it is simply connected. Write $IC(O_w, \underline{\mathbb{C}})$ for the minimal perverse extension of the (shifted) constant rank 1 local system on $O_w$ to all of $G/B$.

Let $i_v\colon O_v \hookrightarrow G/B$ be the inclusion of some (other) $B$-orbit.

Fact: if $O_v \subseteq \overline{O_w}$ (Zariski closure), then $i_v^*IC(O_w, \underline{\mathbb{C}})\neq 0$.

Is there an elementary proof of this? The only way I know how to prove this is to use Kazhdan-Lusztig theory.

Motivation: If $B$-orbits are replaced by $K$-orbits (for some other closed subgroup $K\subseteq G$ acting on $G/B$ with finitely many orbits), then, in general, the `Fact' doesn't hold. Namely, if one takes the $IC$-extension $IC(Y_w,\tau)$ corresponding to a $K$-orbit $Y_w$ and an irreducible $K$-equivariant local system $\tau$ on $Y_w$, then it is generally NOT true that 'if $Y_v\subseteq \overline{Y_w}$, then $i_v^*IC(\tau)\neq 0$' (I hope the notation is self explanatory). For instance, take $\mathbb{C}^*$ acting on $\mathbb{P}^1 = \mathbb{C} \sqcup \{\infty\}$ by $t\cdot z = t^{-2}z$, and take $\tau$ to be the (only) non-trivial irreducible $\mathbb{C}^*$-equivariant local system on the open orbit.

Consequently, one is led to asking for a classification of equivariant local systems $\tau$ on $K$-orbits whose $IC$-extension is simply the $!$-extension of $\tau$. I only know such a characterization (for $K$ a symmetric subgroup) in terms of combinatorics of a module for the Hecke algebra. It seems a bit too high powered to have to use the Hecke algebra (and secretly weights) for this (I am not asking for detailed multiplicity or $Ext$ information, just an irreducibility criterion for standard objects).

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The stalk of the IC sheaf you're interested in looking at is the intersection cohomology of an actual space, given by the intersection of the Schubert variety $\bar{O}_w$ with an orbit of an opposite Borel (through the unique fixed point of the torus given by the intersection of the Borels in $O_v$). This intersection cohomology is non-zero in degree 0. This fails for a non-trivial local system, so I think it's the triviality of the local system that's the important thing.

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  • $\begingroup$ Ah yes! For $K$-orbits, $K$-symmetric (maybe even spherical), the same sort of thing works at least if one works with $G/K$ instead of $B\backslash G$ and equivariant local systems. Same sort of contracting transverse slices as the Schubert case exist. Thanks! $\endgroup$ – Reladenine Vakalwe Nov 3 '13 at 0:45

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