Generalization of Kolyvagin's reduction relations.

Let $T_p$ be the simplest $p$-Hecke correspondence on a Siegel variety of genus
$g$.
There is the following formula for reduction of $T_p$ at $p$:

$$ ilde T_p = Phi_0 + Phi_1 + dots + Phi_g$$
where $Phi_0$ is the Frobenius map, $Phi_g$ is the Verschibung correspondence
and other $Phi_i$ are some intermediate correspondences. This representation of
$ ilde T_p$ as a finite sum of other correspondences comes from other
partition of the above Grassmann variety.
We consider the same $V subset X$ as earlier. Our purpose is to describe the reduction at $p$ of irreducible components of $T_p(V)$ in terms of $Phi_i( ilde V)$.
The key object of the answer is the intersection of two above partitions of the Grassmann variety.