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).


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.

  • $\begingroup$ 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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