Given an algebraic variety \(X\) over an algebraically closed field, and a closed embedding \(i : Y \hookrightarrow X\), a \(Y\)-tilting perverse sheaf \(M\) on \(X\) is defined to a be a perverse sheaf of \(X\) such that \(i^{!}\,M\) and \(i^*\,M\) are perverse sheaves on \(Y\). The authors first discuss this notion in terms of tilting extensions of a perverse sheaf defined on the open complement \(X \backslash Y\). Then, in the case of a stratified variety with smooth connected strata, they exhibit a bijection between the set of strata and the set of isomorphism classes of indecomposable tilting perverse sheaves.
The theory is applied to the case of the Schubert stratification of a flag variety \(G/B\), where \(G\) is a semisimple algebraic group. Among other results, W. Soergel’s Struktursatz [J. Am. Math. Soc. 3, No. 2, 421–445 (1990; Zbl 0747.17008)] is obtained as a corollary. A conjecture by Kapranov is also proved, namely that the Serre functor on the category of perverse sheaves on \(X\) is the square of the Radon transform.