We present a nearby cycle sheaf construction in the context of symmetric spaces. This construction can be regarded as a replacement for the Grothendieck-Springer resolution in classical Springer theory.
We extend the results of [GVX] to the setting of a stable polar representation G|V (G connected, reductive over C), satisfying some mild additional hypotheses. Given a G-equivariant rank one local system L on the general fiber of the quotient map f :
V --> V/G, we compute the Fourier transform of the corresponding nearby cycle sheaf P on the zero-fiber of f. Our main intended application is to the theory of character sheaves for graded semisimple Lie algebras over C.
We establish a Springer correspondence for classical symmetric pairs making use of Fourier transform, a nearby cycle sheaf construction and parabolic induction. In particular, we give an explicit description of character sheaves for classical symmetric pairs.
We extend the scope of a former paper to vector bundle problems involving more than one vector bundle. As the main application, we obtain the solution of the well-known moduli problems of vector bundles associated with general quivers.
We prove stability of logarithmic tangent sheaves of singular hypersurfaces D of the projective space with constraints on the dimension and degree of the singularities of D. As main application, we prove that determinants and symmetric determinants h
ave stable logarithmic tangent sheaves and we describe an open dense piece of the associated moduli space.