Work with complex varieties and constructible sheaves on the complex analytic site. All functors will be tacitly derived.

Let $X$ be a variety acted upon by a connected linear algebraic group. Let $X = \bigsqcup X_i$ be the orbit decomposition (assumed to be finite). Let $f_i\colon X_i \hookrightarrow X$ be the inclusion. If you wish, assume each $f_i$ is affine.

Let $\underline{\mathbb{C}}$ denote the constant (rank $1$) sheaf. Consider the cohomology sheaves $H^k(f_j^*f_{i*}\underline{\mathbb{C}})$. These are certainly semi simple local systems on $X_j$ (by equivariance). Without imposing any further conditions, is there any reason these will be the trivial local system (of some rank)? In particular, I don't want to impose any conditions on the isotropy groups of points.

Generalizing, one could replace $\underline{\mathbb{C}}$ by an arbitrary equivariant local system and ask for constraints on the monodromy of the analogous restriction. Apart from the fact that one gets equivariant local systems this way, are there any other constraints?


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