I was studying Sheaves on Manifolds by Kashiwara and Schapira, and while the singular support seems like a complicated invariant I cannot seem to find a counterexample to the following:


Let $X$ be a smooth complex variety and $\mathcal{F}=IC(U,\mathcal{L})$ be an irreducible perverse sheaf, where $\mathcal{L}$ is a local system on $U\subset X$. Then $SS(\mathcal{F})=T\_{\overline{U}}^\*X$, where the latter means the conormal bundle at $\overline{U}$. 


This seems too easy of an answer to be true, but I still cannot find either a counterexample or a proof, and I cannot think of how to get an explicit answer using the Riemann-Hilbert correspondance either. Any help?


PS. I have asked the question already [in Stack Exchange](https://math.stackexchange.com/q/3676427) but it wasn't answered and I thought it may be more appropriate here after all?



Singular support of an irreducible perverse sheaf
Take $X=\mathbb C$, $U=\mathbb C^\times$, and $\mathcal L$ a nontrivial rank 1 local system (with monodromy $\mu \neq 1$, say).


Then the singular support of $IC(U,\mathcal L)$ is the union $T^\ast \_X X \cup T^\ast\_0 X$ of the zero section (which is what your conjecture would give in this case) with the cotangent fiber.


Note that here we have $ j\_\ast \mathcal L[1] \cong IC(U,\mathcal L) \cong j\_! \mathcal L[1]$. One can compute the singular support either using the sheaf definition from Kashiwwara-Schapira, or by considering the associated $D$-module $D\_{\mathbb C}/D\_{\mathbb C}(x\partial\_x - \log(\mu))$ under the Riemann-Hilbert correspondence.


