The characters of semisimple Lie groups

@article{HarishChandra1956TheCO,
  title={The characters of semisimple Lie groups},
  author={Harish-Chandra},
  journal={Transactions of the American Mathematical Society},
  year={1956},
  volume={83},
  pages={98-163}
}
  • Harish-Chandra
  • Published 1956
  • Mathematics
  • Transactions of the American Mathematical Society
is of the trace class and the mapping Tr:f—*sp(ir(f)) is a distribution on G which is called the character of ir (see [6(d)]). In this paper we shall obtain some results on these characters. Let / be the rank of G. We say that an element xEG is regular if / is exactly the multiplicity of the eigenvalue 1 of the matrix which corresponds to x in the adjoint representation of G. The regular elements form an open and dense subset G' of G. We shall prove (Theorem 6) that TT coincides on G' (in the… 

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  • Harish-Chandra
  • Mathematics
    Proceedings of the National Academy of Sciences of the United States of America
  • 1957
TLDR
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References

SHOWING 1-10 OF 20 REFERENCES

On the characters of a semisimple Lie group

where dx is the Haar measure of G and C?(G) is the set of all (complexvalued) functions on G which are everywhere indefinitely differentia t e and which vanish outside a compact set. V is called the

On Some Types of Topological Groups

As is well-known the so-called fifth problem of Hilbert on continuous groups was solved by J. v. Neumann [14]2 for compact groups and by L. Pontrjagin [15] for abelian groups. More recently, it is

A theory of spherical functions. I

The fact that there exists a close connection between the classical theory of spherical harmonics and that of group representations was first established by E. Cartan and H. Weyl in well known

The Classical Groups

: We consider the generalized doubling integrals of Cai, Friedberg, Ginzburg and Kaplan. These generalize the doubling method of Piatetski-Shapiro and Rallis and represent the standard L-function for

Theory of Lie Groups

This famous book was the first treatise on Lie groups in which a modern point of view was adopted systematically, namely, that a continuous group can be regarded as a global object. To develop this

Theory of Lie Groups (PMS-8)

This famous book was the first treatise on Lie groups in which a modern point of view was adopted systematically, namely, that a continuous group can be regarded as a global object. To develop this

Homologie et cohomologie des algèbres de Lie

© Bulletin de la S. M. F., 1950, tous droits réservés. L’accès aux archives de la revue « Bulletin de la S. M. F. » (http://smf. emath.fr/Publications/Bulletin/Presentation.html) implique l’accord

Sur certaines formes Riemanniennes remarquables des géométries à groupe fondamental simple

© Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 1927, tous droits réservés. L’accès aux archives de la revue « Annales scientifiques de l’É.N.S. » (http://www.