We classify all products of flag varieties with finitely many orbits under the diagonal action of the general linear group. We also classify the orbits in each case and construct explicit representatives. Academic Press

DEDICATED TO PROFESSOR DAVID BUCHSBAUM ON HIS RETIREMENTWe classify products of partial flag varieties of the symplectic group for which the diagonal action has finitely many orbits. © 2000 Academic PressIn another paper (P. Magyar, J. Weyman, and A. Zelevinsky, Adv. Math., 141, 97-118), we examined the following question in the case of the group G = GL m :Problem. Given a reductive algebraic group G, find all k-tuples of parabolic subgroups P 1 P k such that the product of flag varieties G/P 1 × · · · × G/P k has finitely many orbits under the diagonal action of G. In this case we call G/P 1 × · · · × G/P k a multiple flag variety of finite type.In this paper, we solve this problem for the symplectic group G = S P 2n . We also give a complete enumeration of the orbits and explicit representatives for them. The cases in our classification where one of the parabolics is a Borel subgroup, P 1 = B, are exactly those for which G/P 2 × · · · × G/P k is a spherical variety under the diagonal action of G = S P 2n , and our results specialize to classify the B-orbits on these spherical varieties.Our main tool is, as in our other paper (P. Magyar, J. Weyman, and A. Zelevinsky, Adv. Math., to appear), the algebraic theory of quiver representations. Rather unexpectedly, it turns out that two multiple symplectic flags

Abstract. Schubert polynomials generalize Schur polynomials, but it is not clear how to generalize several classical formulas: the Weyl character formula, the Demazure character formula, and the generating series of semistandard tableaux. We produce these missing formulas and obtain several surprising expressions for Schubert polynomials.The above results arise naturally from a new geometric model of Schubert polynomials in terms of Bott-Samelson varieties. Our analysis includes a new, explicit construction for a BottSamelson variety Z as the closure of a B-orbit in a product of flag varieties. This construction works for an arbitrary reductive group G, and for G = GL(n) it realizes Z as the representations of a certain partially ordered set.This poset unifies several well-known combinatorial structures: generalized Young diagrams with their associated Schur modules; reduced decompositions of permutations; and the chamber sets of Berenstein-Fomin-Zelevinsky, which are crucial in the combinatorics of canonical bases and matrix factorizations. On the other hand, our embedding of Z gives an elementary construction of its coordinate ring, and allows us to specify a basis indexed by tableaux.Mathematics Subject Classification (1991). 14M15, 16G20.

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