On the exotic Grassmannian and its nilpotent variety

Fresse, Lucas; Nishiyama, Kyo

doi:10.1090/ert/489

On the exotic Grassmannian and its nilpotent variety
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by Lucas Fresse and Kyo Nishiyama

Represent. Theory 20 (2016), 451-481

DOI: https://doi.org/10.1090/ert/489

Published electronically: November 28, 2016

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Abstract:

Given a decomposition of a vector space $V=V_1\oplus V_2$, the direct product $\mathfrak {X}$ of the projective space $\mathbb {P}(V_1)$ with a Grassmann variety $\mathrm {Gr}_k(V)$ can be viewed as a double flag variety for the symmetric pair $(G,K)=(\mathrm {GL}(V),\mathrm {GL}(V_1)\times \mathrm {GL}(V_2))$. Relying on the conormal variety for the action of $K$ on $\mathfrak {X}$, we show a geometric correspondence between the $K$-orbits of $\mathfrak {X}$ and the $K$-orbits of some appropriate exotic nilpotent cone. We also give a combinatorial interpretation of this correspondence in some special cases. Our construction is inspired by a classical result of Steinberg (1976) and by the recent work of Henderson and Trapa (2012) for the symmetric pair $(\mathrm {GL}(V),\mathrm {Sp}(V))$.

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