# Mirabolic Robinson-Schensted-Knuth correspondence

Roman Travkin

Arxiv ID: 0802.1651•Last updated: 11/9/2021

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 $GL(V)\times V$.

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