Morse Theory and Tilting Sheaves

Tilting perverse sheaves arise in geometric representation theory as natural bases for many categories. (See [1] or Section 3 below for definitions and some background results.) For example, for category O of a complex reductive group G, tilting modules correspond to tilting sheaves on the Schubert stratification S of the flag variety B. In this setting, there are many characterizations of tilting objects involving their categorical structure. (See [13] and [14].) In this paper, we propose a new geometric construction of tilting sheaves. Namely, we show that they naturally arise via the Morse theory of sheaves on the opposite Schubert stratification S. In the context of the flag variety B, our construction takes the following form. Let F be a complex of sheaves on B with bounded constructible cohomology. We will refer to such an object as simply a sheaf. Given a point p ∈ B, and the germ F at p of a complex analytic function, we have the corresponding local Morse group (or vanishing cycles) M p,F (F) which measures how F changes at p as we move in the family defined by F . The local Morse groups play a central role in the theory of perverse sheaves. For an arbitrary sheaf on B, its local Morse groups are graded by cohomological degree. The perverse sheaves on B are characterized by the fact that their local Morse groups (with respect to sufficiently generic functions F ) are concentrated in degree 0. In other words, the local Morse groups are t-exact functors (with respect to the perverse t-structure), and any sheaf on which they are concentrated in degree 0 is in turn perverse. If we restrict to sheaves F on the Schubert stratification S, then for any function F at a point p, it is possible to find a sheaf Mp,F on a neighborhood of p which represents the functor M p,F (though Mp,F is not constructible with respect to S.) To be precise, for any F on S, we have a natural isomorphism