Schur orthogonality relations and invariant sesquilinear forms

Donley, Robert, Jr.

doi:10.1090/S0002-9939-01-06227-X

Schur orthogonality relations and invariant sesquilinear forms
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by Robert W. Donley Jr. PDF

Proc. Amer. Math. Soc. 130 (2002), 1211-1219 Request permission

Abstract:

Important connections between the representation theory of a compact group $G$ and $L^{2}(G)$ are summarized by the Schur orthogonality relations. The first part of this work is to generalize these relations to all finite-dimensional representations of a connected semisimple Lie group $G.$ The second part establishes a general framework in the case of unitary representations $(\pi , V)$ of a separable locally compact group. The key step is to identify the matrix coefficient space with a dense subset of the Hilbert-Schmidt endomorphisms on $V$.

References

Garrett Birkhoff and Morgan Ward, A characterization of Boolean algebras, Ann. of Math. (2) 40 (1939), 609–610. MR 9, DOI 10.2307/1968945
A. L. Carey, Square-integrable representations of non-unimodular groups, Bull. Austral. Math. Soc. 15 (1976), no. 1, 1–12. MR 430146, DOI 10.1017/S0004972700036728
Jacques Dixmier, $C^*$-algebras, North-Holland Mathematical Library, Vol. 15, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. Translated from the French by Francis Jellett. MR 0458185
R. W. Donley, Jr., Orthogonality relations and harmonic forms for semisimple Lie groups, J. Funct. Anal. 170 (2000), 141–160.
M. Duflo and Calvin C. Moore, On the regular representation of a nonunimodular locally compact group, J. Functional Analysis 21 (1976), no. 2, 209–243. MR 0393335, DOI 10.1016/0022-1236(76)90079-3
Garrett Birkhoff and Morgan Ward, A characterization of Boolean algebras, Ann. of Math. (2) 40 (1939), 609–610. MR 9, DOI 10.2307/1968945
Anthony W. Knapp, Representation theory of semisimple groups, Princeton Mathematical Series, vol. 36, Princeton University Press, Princeton, NJ, 1986. An overview based on examples. MR 855239, DOI 10.1515/9781400883974
Anthony W. Knapp, Lie groups beyond an introduction, Progress in Mathematics, vol. 140, Birkhäuser Boston, Inc., Boston, MA, 1996. MR 1399083, DOI 10.1007/978-1-4757-2453-0
George W. Mackey, The theory of unitary group representations, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, Ill.-London, 1976. Based on notes by James M. G. Fell and David B. Lowdenslager of lectures given at the University of Chicago, Chicago, Ill., 1955. MR 0396826
Hisaichi Midorikawa, On certain irreducible representations for the real rank one classical groups, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 21 (1974), 435–459. MR 364555
Hisaichi Midorikawa, Schur orthogonality relations for certain non square integrable representations of real semisimple Lie groups, Tokyo J. Math. 8 (1985), no. 2, 303–336. MR 826991, DOI 10.3836/tjm/1270151217
Hisaichi Midorikawa, Schur orthogonality relations for nonsquare integrable representations of real semisimple linear group and its application, Representations of Lie groups, Kyoto, Hiroshima, 1986, Adv. Stud. Pure Math., vol. 14, Academic Press, Boston, MA, 1988, pp. 257–287. MR 1039840, DOI 10.2969/aspm/01410257
F. Peter and H. Weyl, Die Vollständigkeit der primitiven Darstellungen einer geschlossenen kontinuierlichen Gruppe, Math. Annalen 97 737–755.
I. Schur, Neue Begründung der Theorie der Gruppencharaktere, Sitzungsber. Preuss. Akad., 1905, 406-432.