By the Riemann-Hilbert correspondence, there is an equivalence between 


(1) 
 $\mathcal{D}\operatorname{-mod}(X)$
, the (derived) category of holonomic D-modules on a complex variety X, and 


(2) 
 $D^b\_c(X)$
, the (derived) category of constructible sheaves on X. 


There is a "naive" t-structure we can put on both categories. In 
 $\mathcal{D}\operatorname{-mod}(X)$
, we can look at a t-structure whose heart 
 $\mathcal{D}\operatorname{-mod}^\heartsuit$
 is a complex (of D-modules) concentrated in degree 0. In 
 $D^b\_c(X)$
, we can look at the naive t-structure whose heart 
 $D^{b \heartsuit}\_c$
 is a complex (of constructible sheaves) concentrated in degree 0. 


It's known that if we transfer the naive t-structure on 
 $\mathcal{D}\operatorname{-mod}(X)$
 to 
 $D^b\_c(X)$
 (using the equivalence above), 
 $\mathcal{D}\operatorname{-mod}^\heartsuit$
 is identified with "perverse sheaves" on X.


My question is: 



> 
> If we map 
>  $D^{b\heartsuit}\_c$
>  to the category of D-modules using the Riemann-Hilbert correspondence, what subcategory of 
>  $\mathcal{D}\operatorname{-mod}$
>  do we get? Does this have a well-known name?
> 
> 
> More generally, is there some geometric/nice description of what the naive t-structure on 
>  $D^b\_c$
>  becomes on 
>  $\mathcal{D}{\operatorname{-mod}}$
>  ?
> 
> 
> 



How does one interpret the naive t-structure on constructible sheaves as a t-structure on D-modules?
The t-structure is described in 
[this paper of Kashiwara](http://arxiv.org/abs/math/0302086).
 It looks essentially like Donu and t3suji suggest in their comments: defined by conditions that look like middle-perversity support conditions.


