  We define a class of amenable Weyl group elements in the Lie types B, C, and
D, which we propose as the analogues of vexillary permutations in these Lie
types. Our amenable signed permutations index flagged theta and eta
polynomials, which generalize the double theta and eta polynomials of Wilson
and the author. In geometry, we obtain corresponding formulas for the
cohomology classes of symplectic and orthogonal degeneracy loci.
