Functional analysis
Homogeneous Hermitian holomorphic vector bundles and the Cowen–Douglas class over bounded symmetric domains
Comptes Rendus. Mathématique, Volume 354 (2016) no. 3, pp. 291-295.

It is known that all the vector bundles of the title can be obtained by holomorphic induction from representations of a certain parabolic group on finite-dimensional inner product spaces. The representations, and the induced bundles, have composition series with irreducible factors. We write down an equivariant constant coefficient differential operator that intertwines the bundle with the direct sum of its irreducible factors. As an application, we show that in the case of the closed unit ball in $Cn$ all homogeneous n-tuples of Cowen–Douglas operators are similar to direct sums of certain basic n-tuples.

Il est bien connu que les fibrés vectoriels homogènes holomorphes hermitiens peuvent être obtenus par induction holomorphe à partir des representations de dimension finie d'un certain groupe parabolique. Les représentations, ainsi que les fibrés induits, ont des séries de composition à quotients irréductibles. On montre qu'il existe un opérateur différentiel invariant à coefficients constants qui entrelace le fibré et la somme directe de ses quotients irréductibles. Comme application, on montre que tous les n-tuples d'opérateurs homogènes de la classe de Cowen–Douglas associés à la boule dans $Cn$ sont similaires à des sommes directes de certains n-tuples fondamentaux.

Accepted:
Published online:
DOI: 10.1016/j.crma.2015.11.002

1 Lehman College, Bronx, NY 10468, USA
2 Department of Mathematics, Indian Institute of Science, Bangalore 560012, India
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Adam Korányi; Gadadhar Misra. Homogeneous Hermitian holomorphic vector bundles and the Cowen–Douglas class over bounded symmetric domains. Comptes Rendus. Mathématique, Volume 354 (2016) no. 3, pp. 291-295. doi : 10.1016/j.crma.2015.11.002. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.11.002/

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[2] H.P. Jakobsen The last possible place of unitarity for certain highest weight modules, Math. Ann., Volume 256 (1981), pp. 439-447

[3] A. Korányi; G. Misra A classification of homogeneous operators in the Cowen–Douglas class, Adv. Math., Volume 226 (2011), pp. 5338-5360

[4] I. Satake Algebraic Structures of Symmetric Domains, Princeton University Press, Princeton, NJ, USA, 1980

Cited by Sources:

This research was supported, in part, by a DST–NSF S&T Cooperation Program. The second author also acknowledges the support he has received through the J.C. Bose National Fellowship of the Department of Science and Technology and the University Grants Commission Centre for Advanced Studies.