Let p:X \to Y be a map of smooth algebraic varieties. Let $C \subset T^\* X$ be a (locally closed) submanifold. Denote by $p_\*(C) \subset T^\* Y$ the following set:

$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \ \ \ \ \ \ $ {$(y,v) \in T^\*(Y)|\exists x \in p^{-1}(y) \text{ with } (x,(d_x(p))^\*(v)) \in C$ } .

This opreration can describe (to some extent) what happens to the singular support of a D-module when one takes its direct image.

My question is: when can one claim that $p_*$ of a (conic) Lagrangian manifold is Lagrangian? I heard it is not true in general. Is it true when p is proper? What are the counter examples?

Thank you very much, Rami