Mirabolic Robinson-Shensted-Knuth correspondence

Abstract

The set of orbits of GL(V) in Fl(V) × Fl(V) × V is finite, and is parametrized by the set of certain decorated permutations in a work of Magyar, Weyman, and Zelevinsky. 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) × Fl(V) × V. We also give conjectural applications to the classification of unipotent mirabolic character sheaves on GL(V) × V.