Abstract :
Extending material from the theory of -modules to Lie algebroids (see also [S. Chemla, A duality property for complex Lie algebroids, Math. Z. (1999) 367–388]), we introduce an inverse image functor and show that it preserves coherence under appropriate circumstances. As in the case of -modules, a suitable non characteristicity notion enables us to give a sufficient condition in order for the duality functor and the inverse image functor to commute. This generalizes a result for -modules due to Kashiwara, Kawai, and Sato but, even in the case of -modules, our proof is different from theirs. In particular, we obtain a new duality formula for complexes of modules over an ordinary Lie algebra and, as a special case, we get a new adjunction formula for modules over Lie algebras. Moreover, our result will shed some light on the behaviour of Poisson cohomology (a notion introduced by Lichnerowitz) under a Poisson map (in the analytic case).
