Isospectral deformations of the Lagrangian Grassmannians

Jacques Gasqui[1]; Hubert Goldschmidt[2]

  • [1] Université Joseph Fourier Institut Fourier 100 rue des Maths BP 74 38402 Saint-Martin d’Hères (France)
  • [2] Columbia University Department of Mathematics MC 4406 2990 Broadway New York, NY 10027 (USA)

Annales de l’institut Fourier (2007)

  • Volume: 57, Issue: 7, page 2143-2182
  • ISSN: 0373-0956

Abstract

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We study the special Lagrangian Grassmannian S U ( n ) / S O ( n ) , with n ≥ 3 , and its reduced space, the reduced Lagrangian Grassmannian X . The latter is an irreducible symmetric space of rank n - 1 and is the quotient of the Grassmannian S U ( n ) / S O ( n ) under the action of a cyclic group of isometries of order n . The main result of this paper asserts that the symmetric space X possesses non-trivial infinitesimal isospectral deformations. Thus we obtain the first example of an irreducible symmetric space of arbitrary rank ≥ 2 , which is both reduced and non-infinitesimally rigid. Our result may be viewed as a generalization of the construction which we had given previously for the reduced Grassmannian of 3 -planes in ℝ 6 ; in fact, this space is isometric to the reduced space of S U ( 4 ) / S O ( 4 ) .

How to cite

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Gasqui, Jacques, and Goldschmidt, Hubert. "Isospectral deformations of the Lagrangian Grassmannians." Annales de l’institut Fourier 57.7 (2007): 2143-2182. <http://eudml.org/doc/10293>.

@article{Gasqui2007,
abstract = {We study the special Lagrangian Grassmannian $SU(n)/SO(n)$, with $n\ge 3$, and its reduced space, the reduced Lagrangian Grassmannian $X$. The latter is an irreducible symmetric space of rank $n-1$ and is the quotient of the Grassmannian $SU(n)/SO(n)$ under the action of a cyclic group of isometries of order $n$. The main result of this paper asserts that the symmetric space $X$ possesses non-trivial infinitesimal isospectral deformations. Thus we obtain the first example of an irreducible symmetric space of arbitrary rank $\ge 2$, which is both reduced and non-infinitesimally rigid. Our result may be viewed as a generalization of the construction which we had given previously for the reduced Grassmannian of $3$-planes in $\mathbb\{R\}^6$; in fact, this space is isometric to the reduced space of $SU(4)/SO(4)$.},
affiliation = {Université Joseph Fourier Institut Fourier 100 rue des Maths BP 74 38402 Saint-Martin d’Hères (France); Columbia University Department of Mathematics MC 4406 2990 Broadway New York, NY 10027 (USA)},
author = {Gasqui, Jacques, Goldschmidt, Hubert},
journal = {Annales de l’institut Fourier},
keywords = {Symmetric space; special Lagrangian Grassmannian; reduced Lagrangian Grassmannian; Radon transform; infinitesimal isospectral deformation; symmetric form; Guillemin condition; symmetric space; isospectral deformation},
language = {eng},
number = {7},
pages = {2143-2182},
publisher = {Association des Annales de l’institut Fourier},
title = {Isospectral deformations of the Lagrangian Grassmannians},
url = {http://eudml.org/doc/10293},
volume = {57},
year = {2007},
}

TY - JOUR
AU - Gasqui, Jacques
AU - Goldschmidt, Hubert
TI - Isospectral deformations of the Lagrangian Grassmannians
JO - Annales de l’institut Fourier
PY - 2007
PB - Association des Annales de l’institut Fourier
VL - 57
IS - 7
SP - 2143
EP - 2182
AB - We study the special Lagrangian Grassmannian $SU(n)/SO(n)$, with $n\ge 3$, and its reduced space, the reduced Lagrangian Grassmannian $X$. The latter is an irreducible symmetric space of rank $n-1$ and is the quotient of the Grassmannian $SU(n)/SO(n)$ under the action of a cyclic group of isometries of order $n$. The main result of this paper asserts that the symmetric space $X$ possesses non-trivial infinitesimal isospectral deformations. Thus we obtain the first example of an irreducible symmetric space of arbitrary rank $\ge 2$, which is both reduced and non-infinitesimally rigid. Our result may be viewed as a generalization of the construction which we had given previously for the reduced Grassmannian of $3$-planes in $\mathbb{R}^6$; in fact, this space is isometric to the reduced space of $SU(4)/SO(4)$.
LA - eng
KW - Symmetric space; special Lagrangian Grassmannian; reduced Lagrangian Grassmannian; Radon transform; infinitesimal isospectral deformation; symmetric form; Guillemin condition; symmetric space; isospectral deformation
UR - http://eudml.org/doc/10293
ER -

References

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  1. J. Gasqui, H. Goldschmidt, Radon transforms and the rigidity of the Grassmannians, (2004), Princeton University Press, Princeton, NJ, Oxford Zbl1051.44003MR2034221
  2. J. Gasqui, H. Goldschmidt, Infinitesimal isospectral deformations of the Grassmannian of 3 -planes in  ℝ 6 , Mém. Soc. Math. Fr. (N.S.) (2007) Zbl1152.53040
  3. Victor Guillemin, Some microlocal aspects of analysis on compact symmetric spaces, Seminar on Microlocal Analysis 93 (1979), 79-111, Princeton Univ. Press, Princeton, N.J. Zbl0425.58020MR560313
  4. S. Helgason, Differential geometry, Lie groups, and symmetric spaces, (1978), Academic Press, Orlando, FL Zbl0451.53038MR514561

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