Mirabolic Robinson-Schensted-Knuth correspondence

Abstract

The set of orbits of $GL(V)$ in $Fl(V)\times Fl(V)\times V$ is finite, and is parametrized by the set of certain decorated permutations in a work of Solomon. We describe a Mirabolic RSK correspondence (bijective) between this set of decorated permutations and the set of triples: a pair of standard Young tableaux, and an extra partition. It gives rise to a partition of the set of orbits into combinatorial cells. We prove that the same partition is given by the type of a general conormal vector to an orbit. We conjecture that the same partition is given by the bimodule Kazhdan-Lusztig cells in the bimodule over the Iwahori-Hecke algebra of $GL(V)$ arising from $Fl(V)\times Fl(V)\times V$. We also give conjectural applications to the classification of unipotent mirabolic character sheaves on $GL(V)\times V$.

28 pages. The list of conditions for the formulas for the action of generators on the mirabolic bimodule over the Hecke algebra corrected; the mirabolic RSK example modified; other minor changes. Document style reverted to amsart

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