Schubert calculus on Grassmannians is governed by Schur S-functions,
the one on Lagrangian Grassmannians by Schur Q-functions. There were
several attempts to give a unifying approach to both situations. We
propose to use Hall-Littlewood symmetric polynomials (invented by
Ph. Hall in the 1950s in his study of the combinatorial lattice
structure of finite abelian p-groups). With the projection in a
Grassmann bundle, there is associated its Gysin map, induced by
pushing forward cycles (topologists call it ``integration along
fibers''). We state and prove a Gysin formula for HL-polynomials in
these bundles. We discuss its two specializations, giving better
insights to previously known formulas.
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