On the exotic Grassmannian and its nilpotent variety

Abstract

Given a decomposition of a vector space V = V 1 ⊕ V 2 V=V_1\oplus V_2 , the direct product X \mathfrak {X} of the projective space P ( V 1 ) \mathbb {P}(V_1) with a Grassmann variety G r k ( V ) \mathrm {Gr}_k(V) can be viewed as a double flag variety for the symmetric pair ( G , K ) = ( G L ( V ) , G L ( V 1 ) × G L ( V 2 ) ) (G,K)=(\mathrm {GL}(V),\mathrm {GL}(V_1)\times \mathrm {GL}(V_2)) . Relying on the conormal variety for the action of K K on X \mathfrak {X} , we show a geometric correspondence between the K K -orbits of X \mathfrak {X} and the K 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 ( G L ( V ) , S p ( V ) ) (\mathrm {GL}(V),\mathrm {Sp}(V)) .