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On the exotic Grassmannian and its nilpotent variety

Lucas Fresse, Kyo Nishiyama

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 and by the recent work of Henderson and Trapa for the symmetric pair $(\mathrm{GL}(V),\mathrm{Sp}(V))$.

On the exotic Grassmannian and its nilpotent variety

Abstract

Given a decomposition of a vector space , the direct product of the projective space with a Grassmann variety can be viewed as a double flag variety for the symmetric pair . Relying on the conormal variety for the action of on , we show a geometric correspondence between the -orbits of and the -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 and by the recent work of Henderson and Trapa for the symmetric pair .

Paper Structure

This paper contains 26 sections, 19 theorems, 136 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Multiple flag varieties
  3. Double flag variety of a symmetric pair
  4. Conormal variety
  5. Steinberg correspondence
  6. Exotic Grassmannian and exotic nilpotent variety
  7. Precise statement of main result
  8. Exotic nilpotent cone and its $K$-orbits
  9. Main results
  10. Parametrization of $K$-orbits in $\mathfrak{X}$
  11. Review on Spaltenstein varieties
  12. Nilpotent orbits
  13. Parabolic subgroups
  14. Spaltenstein varieties
  15. Definition of the map $\pi$
  16. ...and 11 more sections

Key Result

Proposition 1.3

Let $(G,K)=(\mathrm{GL}_{p+q}(\mathbb{C}),\mathrm{GL}_p(\mathbb{C})\times \mathrm{GL}_q(\mathbb{C}))$. In each of the following cases, the variety $\mathfrak{X}=G/P\times K/Q$ is of finite type.

Theorems & Definitions (48)

  • Example 1.1
  • Example 1.2
  • Proposition 1.3: see NO.2011
  • Remark 1.4
  • Theorem 1.7: cf., Theorem \ref{['theorem-2']}
  • Corollary 1.8: cf., Corollary \ref{['corollary-2-11']}
  • Proposition 2.1: see Johnson.2010.MR
  • Definition 2.2
  • Example 2.4
  • Remark 2.5
  • ...and 38 more