Suppose $f:X\to Y$ is a map of smooth complex algebraic varieties. There is a pushforward functor

$f_\ast : D(X) \to D(Y)$

on the derived category of $D$-modules. This certainly does not preserve coherence of $D$-modules in general. However, if $f$ is proper then the pushforward preserves coherence, ~~$t$-structure~~, and purity. There is also natural bound on the singular support of the pushforward.

The situation for the pullback functor seems to be entirely analogous, but where the word "proper" is replaced by "smooth". However the pullback, the results for smooth pullbacks can be generalized to the case of $f^{+} M$, when $M$ is *non-characteristic* for $f$. This is a certain condition on the singular support of $M$ (see for the definition).

**Question:** Is there a notion for pushforwards analogous to non-characteristic for pullbacks?

It looks like the words "elliptic pair" might be relevant here, but I don't see exactly how this fits the analogy (this may be because I don't fully understand the definitions...).