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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.

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    $\begingroup$ Good question! a) As [BBM] explain, the notion of tilting sheaf is closely tied to Beilinson's notion of "maximal extension" (used in Beilinson's "How to glue perverse sheaves"). There are situations where no maximal extension exists (e.g. $\mathbb{P}^2$ stratified by $\mathbb{P}^1$ and its complement -- see MacPherson-Vilonen "Elementary construction..."). b) Another simple obstruction is that tilting sheaves are meant to have no self-extensions. If the constant sheaves on strata have self-extensions (i.e. cohomology), then one is possibly already in trouble. ctd. $\endgroup$ Nov 27, 2022 at 19:42
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    $\begingroup$ c) One example which doesn't fit [BBM]'s assumptions, but where tilting sheaves play a key role is in the stratification of the affine Grassmannian that shows up in geometric Satake. Here it is not obvious (from a geometric point of view) that tilting sheaves exist. However they do (most of the time) and are intimate relatives of parity sheaves. This is a long and interesting story, possibly far from your interests; one should see the works of Juteau-Mautner-Williamson, Mautner-Riche and most recently Bezrukavnikov-Gaitsgory-Mirkovic-Rider-Riche. $\endgroup$ Nov 27, 2022 at 19:47
  • $\begingroup$ @GeordieWilliamson Thank you. I see this does answer my question. So the assumptions that the authors make are somehow minimal assumptions to ensure certain expected properties of tilting sheaves continue to hold for perverse tilting sheaves as well. May I request you to please expand your comments into an answer or post the comments itself as the answer? $\endgroup$
    – random123
    Nov 28, 2022 at 16:18

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At @random123's request, I'm will try to argue that looking for tilting perverse sheaves takes one very close to a setting in which [BBM] work.

Suppose that we have a suitably stratified variety \begin{equation} X = \bigsqcup_{\lambda \in \Lambda} X_\lambda \end{equation} and we wish to understand $D = D^b_\Lambda(X,k)$, the derived category of $\Lambda$-constructible sheaves of $k$-vector spaces.

Often we try to understand derived categories by finding a tilting generator $T$, i.e. an object $T$ which generates $D$ and has no higher extensions. In this case $D$ is equivalent to a suitable derived category of modules over $End(T)$ ("tilting theory").

In general, describing all tilting complexes in $T$ is hopeless. (In examples I am familiar with, there are more than I can possibly comprehend, and classifying them is hopeless -- it is somewhat analogous to classifying all $t$-structures on $D$.)

However, the theory of highest weight categories (aka quasi-hereditary algebras) leads one to look for tilting complexes $T$ which are perverse and may be written as a direct sum $T = \bigoplus T_\lambda$ in a way that is "compatible with the stratification", more precisely:

  1. $T_\lambda$ is supported on $\overline{X}_\lambda$.
  2. The set $\{ T_\mu \; | \; \mu \le \lambda \}$ generates the full subcategory of $D$ generated consisting of complexes supported on $\overline{X}_\lambda$.

Firstly, note that "one $T_\lambda$ per strata" implies that there are no local systems on strata, in other words that $\pi_1(X_\lambda) = \{ 1 \}$. (Or at least that fundamental groups have no finite-dimensional representations.)

Next, it is a nice exercise to show that these assumptions imply that (if one denotes by $j : X_\lambda \hookrightarrow \overline{X}_\lambda$ the inclusion, one has a surjection $Hom^i(T_\lambda, T_\lambda) \to Hom^i(j^*T_\lambda, j^*T_\lambda)$. The latter group is equal to $H^i(X_\lambda, k)$, because I hope I convinced you in the previous paragraph that $j^*T_\lambda$ is a shift of a constant sheaf on $X_\lambda$.

In other words, we are forced to take a stratification with strata which look a lot like they are contractible. Thus we are very close to the setting in which [BBM] work.

(Please see comments following question for more specialized comments.)

Aside: I understand that the title "tilting exercises" refers to the medieval practice of riding towards each other carrying long jousting poles. When reading this beautiful paper, I was mystified as to why they also cite Oliver Twist. Only recently did I understand, but leave this as a tilting exercise for the interested reader.

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