Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Module structure of certain induced representations of compact Lie groups


Author: E. James Funderburk
Journal: Trans. Amer. Math. Soc. 228 (1977), 269-285
MSC: Primary 22E45
DOI: https://doi.org/10.1090/S0002-9947-1977-0439992-5
MathSciNet review: 0439992
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let G be a compact connected Lie group and assume a choice of maximal torus and positive roots has been made. Given a dominant weight $ \lambda $, the Borel-Weil Theorem shows how to construct a holomorphic line bundle on whose sections G acts so that the holomorphic sections provide a realization of the irreducible representation of G with highest weight $ \lambda $. This paper studies the G-module structure of the space $ \Gamma $ of square integrable sections of the Borel-Weil line bundle. It is found that $ \Gamma = {\lim _{n \to \infty }}\Gamma (n)$, where $ \Gamma (n) \subset \Gamma (n + 1) \subset \Gamma $ and $ \Gamma (n)$ is isomorphic, as G-module, to

$\displaystyle V(\lambda + n\lambda ) \otimes V(n{\lambda ^\ast}),$

where $ V(\mu )$ denotes the irreducible representation of highest weight $ \mu $, '+' is the Cartan semigroup operation, and '$ ^\ast$' is the contragredient operation. Similar formulas hold for powers of the Borel-Weil line bundle.

References [Enhancements On Off] (What's this?)

  • [1] J. F. Adams, Lectures on Lie groups, Benjamin, New York, 1969. MR 40 #5780. MR 0252560 (40:5780)
  • [2] L. Auslander and B. Kostant, Polarization and unitary representations of solvable Lie groups, Invent. Math. 14 (1971), 255-354. MR 45 #2092. MR 0293012 (45:2092)
  • [3] J. G. T. Belinfante and B. Kolman, A survey of Lie groups and Lie algebras with applications and computational methods, SIAM, Philadelphia, 1973.
  • [4] A. Borel and F. Hirzbruch, Characteristic classes and homogeneous spaces. I, Amer. J. Math. 80 (1958), 458-538. MR 21 #1586. MR 0102800 (21:1586)
  • [5] C. Chevalley, Theory of Lie groups. I, Princeton Univ. Press, Princeton, N.J., 1946. MR 7, 412. MR 0082628 (18:583c)
  • [6] L. Fonda and G. C. Ghirardi, Symmetry principles in quantum physics, Dekker, New York, 1970.
  • [7] J. Funderburk, Module structure of certain induced representations of compact Lie groups, Ph.D. Thesis, Univ. of Wisconsin, Madison, May, 1975.
  • [8] J. E. Humphreys, Introduction to Lie algebras and representation theory, Springer-Verlag, New York, 1972. MR 48 #2197. MR 0323842 (48:2197)
  • [9] S. Kobayashi and K. Nomizu, Foundations of differential geometry, Vol. II, Interscience, New York, 1969. MR 38 #6501. MR 0238225 (38:6501)
  • [10] B. Kostant, Lie algebra cohomology and the generalized Borel-Weil theorem, Ann. of Math. (2) 74 (1961), 329-387. MR 26 #265. MR 0142696 (26:265)
  • [11] -, Quantization and unitary representations, I. Prequantization, Lectures in Modern Analysis and Applications, III, Lecture Notes in Math., vol. 170, Springer-Verlag, Berlin, 1970, pp. 87-208. MR 45 #3638. MR 0294568 (45:3638)
  • [12] H. Samelson, Notes on Lie algebras, Mathematical Studies, no. 23, Van Nostrand Reinhold, New York, 1969. MR 40 #7322. MR 0254112 (40:7322)
  • [13] J.-P. Serre, Represésentations linéaires et espaces homogènes kälériens des groupes de Lie compacts, Séminaire Bourbaki: 1953/54, Exposé 100, 2nd corr. ed., Secrétariat mathématique, Paris, 1959. MR 28 #1087.
  • [14] J.-M. Souriau, Structure des systèmes dynamiques, Maîtrises de Mathématiques, Dunod, Paris, 1970. MR 41 #4866. MR 0260238 (41:4866)
  • [15] S. Sternberg, Lectures on differential geometry, Prentice-Hall, Englewood Cliffs, N.J., 1964. MR 33 #1797. MR 0193578 (33:1797)
  • [16] J. Tarski, Partition function for certain simple Lie algebras, J. Mathematical Phys. 4 (1963), 569-574. MR 26 #5099. MR 0147584 (26:5099)
  • [17] Nolan R. Wallach, Harmonic analysis on homogeneous spaces, Dekker, New York, 1973. MR 0498996 (58:16978)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 22E45

Retrieve articles in all journals with MSC: 22E45


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1977-0439992-5
Keywords: Compact connected Lie group, highest weight, lowest weight, opposition involution, Cartan semigroup, tensor product, cyclic representation, Borel-Weil realization
Article copyright: © Copyright 1977 American Mathematical Society

American Mathematical Society