  Let V be a vector bundle on a scheme X endowed with a nondegenerate
symplectic or orthogonal form. Let G be a Grassmannian bundle parametrizing
maximal isotropic subbundles of V. The main goal of the paper is to give
formulas for the classes of the loci in G parametrizing those subbundles which
intersect successive members of of a fixed flag of isotropic subbundles of V in
dimensions greater than 1,2,3,... . The formulas which we give are quadratic
expressions in Q-polynomials of the tautological bundle on G and flag
S-polynomials of the members of the flag. These formulas globalize to
Lagrangian and orthogonal degeneracy loci. They have especially nice form for
the loci of points in X where two maximal isotropic subbundles intersect in
dimension exceeding a given number. (A problem for finding formulas for these
loci was posed by J. Harris several years ago.) In this case, our formulas are
simple quadratic expressions of Q-polynomials applied to E and F. The work
generalizes formulas given in [P.Pragacz, Algebro- geometric applications of
Schur S- and Q-polynomials, SLN 1478 (1991), 130-191].
  One of its applications is computation of the classes of Brill-Noether loci
in Pryms in [De Concini, P. Pragacz, On the class of Brill-Noether loci for
Prym varieties, Math. Ann. 302 (1995), 687-697].
