TY - JOUR

T1 - Homogeneous Hermitian holomorphic vector bundles and operators in the Cowen-Douglas class over the poly-disc

AU - Deb, Prahllad

AU - Hazra, Somnath

N1 - Funding Information:
The research of the first named author was supported through IISER Kolkata Ph.D. Fellowship ( 14RS009 ), J.C. Bose National Fellowship of G. Misra followed by a post-doctoral fellowship at Ben-Gurion University of the Negev , Israel ( 850053497 ). The research of the second named author was supported through research Fellowships of CSIR , IISc , NBHM post-doctoral Fellowship ( 0204/16/2018/R&D-II/6439 ) followed by GA CR grant no. 21-27941S . Some of the results in this paper are from the PhD thesis of the first named author submitted to the Indian Institute of Science Education and Research Kolkata and from the PhD thesis of the second named author submitted to the Indian Institute of Science.
Funding Information:
The research of the first named author was supported through IISER Kolkata Ph.D. Fellowship (14RS009), J.C. Bose National Fellowship of G. Misra followed by a post-doctoral fellowship at Ben-Gurion University of the Negev, Israel (850053497). The research of the second named author was supported through research Fellowships of CSIR, IISc, NBHM post-doctoral Fellowship (0204/16/2018/R&D-II/6439) followed by GA CR grant no. 21-27941S. Some of the results in this paper are from the PhD thesis of the first named author submitted to the Indian Institute of Science Education and Research Kolkata and from the PhD thesis of the second named author submitted to the Indian Institute of Science.
Publisher Copyright:
© 2022 Elsevier Inc.

PY - 2022/6/15

Y1 - 2022/6/15

N2 - In this article, we obtain two sets of results. The first set of results are for the case of the bi-disc while the second set of results describe in part, which of these carry over to the general case of the poly-disc.A classification of irreducible hermitian holomorphic vector bundles over D2, homogeneous with respect to Möb×Möb, is obtained assuming that the associated representations are multiplicity-free. Among these the ones that give rise to an operator in the Cowen-Douglas class of D2 of rank 1,2 or 3 are determined.Any hermitian holomorphic vector bundle of rank 2 over Dn, homogeneous with respect to the n-fold direct product of the group Möb is shown to be a tensor product of n hermitian holomorphic vector bundles over D. Among them, n−1 are shown to be the line bundles and one is shown to be a rank 2 bundle. Also, each of the bundles are homogeneous with respect to Möb.The classification of irreducible homogeneous hermitian holomorphic vector bundles over D2 of rank 3 (as well as the corresponding Cowen-Douglas class of operators) is extended to the case of Dn, n>2.It is shown that there is no irreducible n - tuple of operators in the Cowen-Douglas class B2(Dn) that is homogeneous with respect to Aut(Dn), n>1. Also, pairs of operators in B3(D2) homogeneous with respect to Aut(D2) are produced, while it is shown that no n - tuple of operators in B3(Dn) is homogeneous with respect to Aut(Dn), n>2.

AB - In this article, we obtain two sets of results. The first set of results are for the case of the bi-disc while the second set of results describe in part, which of these carry over to the general case of the poly-disc.A classification of irreducible hermitian holomorphic vector bundles over D2, homogeneous with respect to Möb×Möb, is obtained assuming that the associated representations are multiplicity-free. Among these the ones that give rise to an operator in the Cowen-Douglas class of D2 of rank 1,2 or 3 are determined.Any hermitian holomorphic vector bundle of rank 2 over Dn, homogeneous with respect to the n-fold direct product of the group Möb is shown to be a tensor product of n hermitian holomorphic vector bundles over D. Among them, n−1 are shown to be the line bundles and one is shown to be a rank 2 bundle. Also, each of the bundles are homogeneous with respect to Möb.The classification of irreducible homogeneous hermitian holomorphic vector bundles over D2 of rank 3 (as well as the corresponding Cowen-Douglas class of operators) is extended to the case of Dn, n>2.It is shown that there is no irreducible n - tuple of operators in the Cowen-Douglas class B2(Dn) that is homogeneous with respect to Aut(Dn), n>1. Also, pairs of operators in B3(D2) homogeneous with respect to Aut(D2) are produced, while it is shown that no n - tuple of operators in B3(Dn) is homogeneous with respect to Aut(Dn), n>2.

KW - Cowen-Douglas class

KW - Curvature

KW - Hermitian holomorphic homogeneous vector bundles

KW - Homogeneous operators

KW - Lie algebra

KW - Representation

UR - http://www.scopus.com/inward/record.url?scp=85123632658&partnerID=8YFLogxK

U2 - 10.1016/j.jmaa.2022.126031

DO - 10.1016/j.jmaa.2022.126031

M3 - Article

AN - SCOPUS:85123632658

SN - 0022-247X

VL - 510

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

IS - 2

M1 - 126031

ER -