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Representation Theory

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On the exotic Grassmannian and its nilpotent variety

Authors: Lucas Fresse and Kyo Nishiyama
Journal: Represent. Theory 20 (2016), 451-481
MSC (2010): Primary 14L30; Secondary 14L35, 14M15, 17B08
Published electronically: November 28, 2016
MathSciNet review: 3576071
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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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Additional Information

Lucas Fresse
Affiliation: Université de Lorraine, CNRS, Institut Élie Cartan de Lorraine, UMR 7502, Vandoeuvre-lès-Nancy, F-54506, France

Kyo Nishiyama
Affiliation: Department of Physics and Mathematics, Aoyama Gakuin University, Fuchinobe 5-10-1, Sagamihara 252-5258, Japan

Received by editor(s): April 6, 2016
Received by editor(s) in revised form: October 9, 2016
Published electronically: November 28, 2016
Additional Notes: The first author was supported by the ISF Grant Nr. 797/14 and by the ANR project NilpOrbRT (ANR-12-PDOC-0031).
The second author was supported by JSPS KAKENHI Grant Numbers #25610008 and #16K05070.
Article copyright: © Copyright 2016 American Mathematical Society

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