A smooth projective variety X is called D-affine if every left D-module is globally generated over D (the sheaf of differential operators) and has vanishing higher cohomology. A well known result of Beilinson-Bernstein asserts that over the field of complex numbers flag varieties of semi-simple algebraic groups are D-affine. So far, Flag varieties are the only known examples of smooth projective D-affine varieties. Moreover, Kashiwara-Lauritzen showed that over fields of positive characteristic the grassmannian Gr(2,5) is not D-affine and the classification of D-affine flag varieties remains an open problem.

In the first part of the talk, I will briefly recall the basics of D-affine varieties. Then, I will survey both the problem of classification of D-affine varieties over fields of arbitrary characteristic and the problem of classification of D-affine flag varieties in positive characteristic. Finally, I will present my recent result: Over a field of positive characteristic the smooth even-dimensional quadric hypersurface of dimension at least four is not D-affine. This result implies that various other flag varieties fail to be D-affine and refines a previous result of A. Langer who showed that odd-dimensional quadrics are D-affine if the characteristic of the base field is sufficiently large.