  We give a embedding of the Lagrangian Grassmannian LG(n) inside an ordinary
Grassmannian that is well-behaved with respect to the Wronski map. As a
consequence, we obtain an analogue of the Mukhin-Tarasov-Varchenko theorem for
LG(n). The restriction of the Wronski map to LG(n) has degree equal to the
number of shifted or unshifted tableaux of staircase shape. For special fibres
one can define bijections, which, in turn, gives a bijection between these two
classes of tableaux. The properties of these bijections lead a geometric proof
of a branching rule for the cohomological map H*(Gr(n,2n)) x H*(LG(n)) ->
H*(LG(n)), induced by the diagonal inclusion LG(n) -> LG(n) x Gr(n,2n). We also
discuss applications to the orbit structure of jeu de taquin promotion on
staircase tableaux.
