Intertwining operators and unitary representations. I. (English) Zbl 0683.22010

The problem of classifying the irreducible unitary representations of a semisimple Lie group G comes down to finding out whether some intertwining operators are semidefinite. In their preceding five works the authors have introduced some formulas to solve the latter problem and have used them for classifying UIR’s in certain cases: SU(N,2), some other groups of real rank two, Langlands quotients obtained from maximal parabolic subgroups. But in these works there were no proofs of the formulas. In the paper reviewed the authors give the derivations of these formulas - when G has a compact Cartan subgroup.
Reviewer: V.F.Molchanov


22E46 Semisimple Lie groups and their representations
22D10 Unitary representations of locally compact groups
Full Text: DOI


[1] Baldoni-Silva, M.W; Knapp, A.W, Indefinite intertwining operators, (), 1272-1275 · Zbl 0539.22009
[2] Baldoni-Silva, M.W; Knapp, A.W, Indefinite intertwining operators, II, (), 1-46 · Zbl 0651.22008
[3] Baldoni-Silva, M.W; Knapp, A.W, Irreducible unitary representations of some groups of real rank two, (), 15-36 · Zbl 0621.22013
[4] Baldoni-Silva, M.W; Knapp, A.W, Vogan’s algorithm for computing composition series, (), 37-72 · Zbl 0631.22012
[5] Baldoni-Silva, M.W; Knapp, A.W, Unitary representations induced from maximal parabolic subgroups, J. funct. anal., 69, 21-120, (1986) · Zbl 0614.22004
[6] Blank, B.E, Knapp-wallach szegö integrals and generalized principal series representations: the parabolic rank one case, J. funct. anal., 60, 127-145, (1985) · Zbl 0576.22015
[7] Harish-Chandra, Discrete series for semisimple Lie groups, II, Acta math., 116, 1-111, (1966) · Zbl 0199.20102
[8] Helgason, S, Differential geometry and symmetric spaces, (1962), Academic Press New York · Zbl 0122.39901
[9] Humphreys, J.E, Introduction to Lie algebras and representation theory, (1972), Springer-Verlag Berlin · Zbl 0254.17004
[10] Klimyk, A.U; Gavrilik, A.M, Representation matrix elements and Clebsch-Gordan coefficients of the semisimple Lie groups, J. math. phys., 20, 1624-1642, (1979) · Zbl 0416.22017
[11] Knapp, A.W, Minimal K-type formula, (), 107-118 · Zbl 0525.22016
[12] Knapp, A.W, Representation theory of semisimple groups: an overview based on examples, (1986), Princeton Univ. Press Princeton, NJ · Zbl 0604.22001
[13] Knapp, A.W; Stein, E.M, Intertwining operators for semisimple groups, II, Invent. math., 60, 9-84, (1980) · Zbl 0454.22010
[14] Knapp, A.W; Vogan, D.A, Cohomological induction and unitary representations, (1987), unpublished manuscript · Zbl 0863.22011
[15] Knapp, A.W; Wallach, N.R; Knapp, A.W; Wallach, N.R, Szegö kernels associated with discrete series, Invent. math., Invent. math., 62, 341-346, (1980) · Zbl 0452.22016
[16] Knapp, A.W; Zuckerman, G.J; Knapp, A.W; Zuckerman, G.J, Classification of irreducible tempered representations of semisimple groups, Ann. of math., Ann. of math., 119, 639-501, (1984) · Zbl 0539.22012
[17] Langlands, R.P, On the classification of irreducible representations of real algebraic groups, (1973), Institute for Advanced Study Princeton, New Jersey, mimeographed notes · Zbl 0741.22009
[18] Midorikawa, H, Clebsch-Gordon coefficients for a tensor product representation ad ⊗ π of a maximal compact subgroup of real semisimple Lie group, (), 149-175
[19] Vogan, D.A, The algebraic structure of the representation of semisimple Lie groups, I, Ann. of math., 109, 1-60, (1979) · Zbl 0424.22010
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.