TITLE: How does one interpret the naive t-structure on constructible sheaves as a t-structure on D-modules?
QUESTION [10 upvotes]: 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 
      $\mathcal{D}{\operatorname{-mod}}$
  ?

REPLY [13 votes]: The t-structure is described in 
this paper of Kashiwara.
  It looks essentially like Donu and t3suji suggest in their comments: defined by conditions that look like middle-perversity support conditions.