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On induced representations. (English) Zbl 0657.22020

The mathematical heritage of Hermann Weyl, Proc. Symp., Durham/NC 1987, Proc. Symp. Pure Math. 48, 1-13 (1988).
[For the entire collection see Zbl 0644.00001.]
In this paper, one of the Symposium dedicated to the works of H. Weyl, the author outlines some of the most outstanding results belonging to the progeny of the famous character formula of H. Weyl for the irreducible representations of a semisimple compact Lie group K. First comes the Borel-Weil-Bott theorem which describes the irreducible representations themselves, as modules of holomorphic sections of a line bundle over K/T, where T is a maximal torus. Then, starting with the expression \(\chi(g)\) of a character \(\chi\) of a permutation group as the number of fixed points of a permutation g, the author introduces by analogy the famous Atiyah-Bott-Lefschetz formula for the number of fixed points of a diffeomorphism transversal to the identity. Finally he sketches two more recent developments. One is linked to his own use of Morse theory to study the homology of a homogeneous space K/U; it is the “geometric quantization”, starting with the study of the coadjoint action by Kirillov. The other, pioneered first by physicists, is the study of the “loop group” \(\Lambda\) K of all smooth maps of \(S_ 1\) into K, which contains the group \(\Omega\) K of pointed maps as a subgroup; that group and its representations are described in detail in the recent book of A. Pressley and G. Segal [“Loop groups” (1986; Zbl 0618.22011; reprint 1988)].
Reviewer: J.Dieudonn√©

MSC:

22E65 Infinite-dimensional Lie groups and their Lie algebras: general properties
22E46 Semisimple Lie groups and their representations
58D15 Manifolds of mappings
14F40 de Rham cohomology and algebraic geometry
58A14 Hodge theory in global analysis
01A65 Development of contemporary mathematics