We study the special Lagrangian Grassmannian , with , and its reduced space, the reduced Lagrangian Grassmannian . The latter is an irreducible symmetric space of rank and is the quotient of the Grassmannian under the action of a cyclic group of isometries of order . The main result of this paper asserts that the symmetric space possesses non-trivial infinitesimal isospectral deformations. Thus we obtain the first example of an irreducible symmetric space of arbitrary rank , 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 -planes in ; in fact, this space is isometric to the reduced space of .
Nous étudions la grassmannienne lagrangienne spéciale , avec , et son espace réduit , qui est l’espace symétrique irréductible de rang quotient de par l’action d’un groupe cyclique d’isometries d’ordre . Notre résultat principal est la construction de déformations infinitésimales isospectrales non triviales de . Nous obtenons ainsi les premiers exemples en rang quelconque d’espaces symétriques irréductibles réduits et non infinitésimalement rigides. Notre résultat peut être vu comme une généralisation de la construction que nous avions donnée dans un précédent papier pour la grassmannienne réduite des -plans de , espace qui est en fait isométrique à l’espace réduit de .
Keywords: Symmetric space, special Lagrangian Grassmannian, reduced Lagrangian Grassmannian, Radon transform, infinitesimal isospectral deformation, symmetric form, Guillemin condition
Mot clés : espace symétrique, grassmannienne lagrangienne speciale, grassmannienne lagrangienne réduite, transformée de Radon, déformation isospectrale, forme symétrique, condition de Guillemin
Gasqui, Jacques 1; Goldschmidt, Hubert 2
@article{AIF_2007__57_7_2143_0, author = {Gasqui, Jacques and Goldschmidt, Hubert}, title = {Isospectral deformations of the {Lagrangian} {Grassmannians}}, journal = {Annales de l'Institut Fourier}, pages = {2143--2182}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {57}, number = {7}, year = {2007}, doi = {10.5802/aif.2329}, mrnumber = {2394538}, zbl = {1140.44001}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2329/} }
TY - JOUR AU - Gasqui, Jacques AU - Goldschmidt, Hubert TI - Isospectral deformations of the Lagrangian Grassmannians JO - Annales de l'Institut Fourier PY - 2007 SP - 2143 EP - 2182 VL - 57 IS - 7 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.2329/ DO - 10.5802/aif.2329 LA - en ID - AIF_2007__57_7_2143_0 ER -
%0 Journal Article %A Gasqui, Jacques %A Goldschmidt, Hubert %T Isospectral deformations of the Lagrangian Grassmannians %J Annales de l'Institut Fourier %D 2007 %P 2143-2182 %V 57 %N 7 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.2329/ %R 10.5802/aif.2329 %G en %F AIF_2007__57_7_2143_0
Gasqui, Jacques; Goldschmidt, Hubert. Isospectral deformations of the Lagrangian Grassmannians. Annales de l'Institut Fourier, Volume 57 (2007) no. 7, pp. 2143-2182. doi : 10.5802/aif.2329. https://aif.centre-mersenne.org/articles/10.5802/aif.2329/
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