Infinitesimal isospectral deformations of the Grassmannian of 3-planes in 6
Mémoires de la Société Mathématique de France, no. 108 (2007), 98 p.

We study the real Grassmannian G n,n of n-planes in 2n , with n3, and its reduced space. The latter is the irreducible symmetric space G ¯ n,n , which is the quotient of the space G n,n under the action of its isometry which sends a n-plane into its orthogonal complement. One of the main results of this monograph asserts that the irreducible symmetric space G ¯ 3,3 possesses non-trivial infinitesimal isospectral deformations; it provides us with the first example of an irreducible reduced symmetric space which admits such deformations. We also give a criterion for the exactness of a form of degree one on G ¯ n,n in terms of a Radon transform.

Ce mémoire a pour cadre la grassmannienne G n,n des n-plans de 2n , avec n3, et son espace réduit G ¯ n,n , qui est l’espace symétrique irréductible, quotient de G n,n par l’involution envoyant un n-plan sur son orthogonal. Un de nos principaux résultats est la construction de déformations infinitésimales isospectrales non triviales sur G ¯ 3,3 , obtenant ainsi le premier exemple d’espace symétrique irréductible réduit et non infinitésimalement rigide. Nous donnons aussi un critère d’exactitude pour les formes différentielles de degré 1 sur G ¯ n,n , mettant en jeu la nullité d’une transformée de Radon.

DOI : https://doi.org/10.24033/msmf.420
Classification:  44A12,  53C35,  58A10,  58J53
Keywords: Symmetric space, Grassmannian, Radon transform, infinitesimal isospectral deformation, symmetric form, Guillemin condition
@book{MSMF_2007_2_108__1_0,
     author = {Gasqui, Jacques and Goldschmidt, Hubert},
     title = {Infinitesimal isospectral deformations of the Grassmannian of 3-planes in ${\mathbb{R}}^6$},
     series = {M\'emoires de la Soci\'et\'e Math\'ematique de France},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {108},
     year = {2007},
     doi = {10.24033/msmf.420},
     zbl = {1152.53040},
     mrnumber = {2447005},
     language = {en},
     url = {http://www.numdam.org/item/MSMF_2007_2_108__1_0}
}
Gasqui, Jacques; Goldschmidt, Hubert. Infinitesimal isospectral deformations of the Grassmannian of 3-planes in ${\mathbb{R}}^6$. Mémoires de la Société Mathématique de France, Serie 2, , no. 108 (2007), 98 p. doi : 10.24033/msmf.420. http://www.numdam.org/item/MSMF_2007_2_108__1_0/

[1] S. ArakiOn root systems and an infinitesimal classification of irreducible symmetric spaces, J. Math. Osaka City Univ. 13 (1964), p. 1–34. | MR 153782

[2] N. BourbakiEléments de mathématique, Groupes et algèbres de Lie, Chapitres 4, 5 et 6, Hermann, Paris, 1968. | MR 132805 | Zbl 0165.56403

[3] W. Fulton & J. HarrisRepresentation theory: a first course, Graduate Texts in Math., vol. 129, Springer-Verlag, New York, Berlin, Heidelberg, 1991. | MR 1153249 | Zbl 0744.22001

[4] J. Gasqui & H. GoldschmidtOn the geometry of the complex quadric, Hokkaido Math. J. 20 (1991), p. 279–312. | MR 1114408 | Zbl 0764.53048

[5] —, Radon transforms and spectral rigidity on the complex quadrics and the real Grassmannians of rank two, J. Reine Angew. Math. 480 (1996), p. 1–69. | MR 1420557 | Zbl 0861.53054

[6] —, Radon transforms and the rigidity of the Grassmannians, Ann. of Math. Studies, no. 156, Princeton University Press, Princeton, NJ, Oxford, 2004. | Zbl 1051.44003

[7] —, Infinitesimal isospectral deformations of the Lagrangian Grassmannians, Ann. Inst. Fourier (Grenoble) (to appear). | Numdam | Zbl 1140.44001

[8] J. Gasqui, H. Goldschmidt & H. WilfSome summation identities and their computer proofs, 2004, Available online at http://www.cis.upenn.edu/wilf/GoldschmidtSummationQuestion.pdf.

[9] F. Gonzales & S. HelgasonInvariant differential operators on Grassmann manifolds, Adv. in Math. 60 (1986), p. 81–91. | MR 839483 | Zbl 0613.58038

[10] R. Goodman & N. WallachRepresentations and invariants of the classical groups, Cambridge University Press, Cambridge, 1998. | MR 1606831 | Zbl 0901.22001

[11] E. GrinbergOn images of Radon transforms, Duke. Math. J. 52 (1985), p. 939–972. | MR 816395 | Zbl 0623.44005

[12] —, Aspects of flat Radon transforms, Contemp. Math. 140 (1992), p. 73–85. | MR 1197589 | Zbl 0785.44002

[13] —, Flat Radon transforms on compact symmetric spaces with application to isospectral deformations, Preprint.

[14] V. GuilleminOn micro-local aspects of analysis on compact symmetric spaces, in Seminar on micro-local analysis, by V. Guillemin, M. Kashiwara and T. Kawai, Ann. of Math. Studies, no. 93, Princeton University Press, University of Tokyo Press, Princeton, NJ, 1979, p. 79–111. | MR 560315

[15] S. HelgasonFundamental solutions of invariant differential operators on symmetric spaces, Amer. Math. J. 86 (1964), p. 565–601. | MR 165032 | Zbl 0178.17001

[16] —, Differential geometry, Lie groups, and symmetric spaces, Academic Press, Orlando, FL, 1978.

[17] —, Some results on invariant differential operators on symmetric spaces, Amer. Math. J. 114 (1992), p. 769–811.

[18] —, Geometric analysis on symmetric spaces, Math. Surveys Monogr., vol. 39, American Mathematical Society, Providence, RI, 1994.

[19] R. MichelProblèmes d’analyse géométrique liés à la conjecture de Blaschke, Bull. Soc. Math. France 101 (1973), p. 17–69. | Numdam | MR 317231 | Zbl 0265.53041

[20] M. Petkovšek, H. Wilf & D. ZeilbergerA=B, A K Peters, Ltd., Wellesley, MA, 1996.

[21] R. StrichartzThe explicit Fourier decomposition of L 2 (SO(n)/SO(n-m)), Canad. J. Math. 27 (1975), p. 294–310. | MR 380277 | Zbl 0275.43009

[22] C. TsukamotoInfinitesimal Blaschke conjectures on projective spaces, Ann. Sci. École Norm. Sup. (4) 14 (1981), p. 339–356. | Numdam | MR 644522 | Zbl 0481.53041

[23] T. VustOpération de groupes réductifs dans un type de cônes homogènes, Bull. Soc. Math. France 102 (1974), p. 317–333. | Numdam | MR 366941 | Zbl 0332.22018

[24] N. WallachReal reductive groups I, Academic Press, Boston, San Diego, 1988. | MR 929683 | Zbl 0666.22002