In the article titled Tilting Exercises (See http://arXiv.org/abs/math/0301098v3) the authors define a notion of tilting perverse sheaves on an algebraic vareity $X$ with respect to stratification $\lbrace X_{\nu} \rbrace$. It appears that §1.3 onwards they assume that all the morphisms $i_{\nu} : X_{\nu} \hookrightarrow X$ are affine. They further assume that all the strata $X_{\nu}$ have no higher nontrivial cohomology. They obtain an equivalence of bounded homotopy category of tilting sheaves with the full subcategory $D$ (of complexes of sheaves constant along the strata of $X$) of $D(X)$ (=the bounded derived category of constructible sheaves). This is §1.5, Proposition of loc. cit.

It is certainly clear that there are spaces of interest where the above assumptions on the strata are not satisfied.

Q1. Has the theory of perverse tilting sheaves been explored out of the context of the paper cited above? For example consider the algebraic variety given by compactification of the moduli of principally polarized abelian varieties or loosely speaking the minimal or some toroidal compactification of locally symmetric space associated to $\mathrm{Sp}(2g)$ and some congruence subgroup. Is it possible to understand how big or small the subcategory of tilting perverse sheaves will be?

Q2. If the notion of tilting perverse sheaves has not been explored in more general context then I would like to ask if there are certain philosophical reasons that it may not be so meaningful to pursue the theory of tilting sheaves in more generality?

Any comments or suggestions are welcome. Thank you.