Frobenius reciprocity and extensions of nilpotent Lie groups
Author:
Jeffrey Fox
Journal:
Trans. Amer. Math. Soc. 298 (1986), 123144
MSC:
Primary 22E25; Secondary 22E27, 22E40, 22E45
MathSciNet review:
857436
Fulltext PDF Free Access
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Abstract: In we use vector methods, essentially Frobenius reciprocity, to derive the HoweRichardson multiplicity formula for compact nilmanifolds. In we use Frobenius reciprocity to generalize and considerably simplify a reduction procedure developed by Howe for solvable groups to general extensions of nilpotent Lie groups. In we give an application of the previous results to obtain a reduction formula for solvable Lie groups.
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Leonard
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(1971), 173–190. MR 0284546
(44 #1771)
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Jonathan
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representations of unimodular Lie groups on 𝐿²cohomology
spaces, Trans. Amer. Math. Soc.
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1–32. MR
576861 (81k:22012), http://dx.doi.org/10.1090/S00029947198005768614
 [AM]
 L. Auslander and C. Moore, Unitary representations of solvable Lie groups, Mem. Amer. Math. Soc. No. 62 (1966). MR 0207910 (34:7723)
 [CG1]
 L. Corwin and F. Greenleaf, Character formula and spectra of compact nilmanifolds, J. Funct. Anal. 21 (1976), 123154. MR 0393345 (52:14155)
 [CG2]
 , Intertwining operators for representations induced from uniform subgroups, Acta Math. 136 (1976), 275301. MR 0425009 (54:12967)
 [CGP]
 L. Corwin, F. Greenleaf, and R. Penney, A general character formula for irreducible projections on of a nilmanifold, Math. Ann. 225 (1977), 2132. MR 0425021 (54:12979)
 [G]
 I. Gelfand, M. Graev, and I. PyatetskiiShapiro, Representation theory and automorphic functions, Saunders, Philadelphia, Pa., 1969. MR 0233772 (38:2093)
 [H1]
 R. Howe, On Frobenius reciprocity for unipotent algebraic groups over , Amer. J. Math. 93 (1971). MR 0281842 (43:7556)
 [H2]
 , Topics in harmonic analysis on solvable algebraic groups, Pacific J. Math. 73 (1977), 383437. MR 0480858 (58:1007)
 [H3]
 , On a condition between nilpotent groups and oscillatory integrals associated to singularities, Pacific J. Math. 73 (1977), 329365. MR 0578891 (58:28270)
 [KL1]
 A. Kleppner and R. Lipsman, The Plancherel formula for group extensions. I, Ann. Sci. École Norm. Sup. (4) 5 (1972), 459516. MR 0342641 (49:7387)
 [M]
 C. Moore, Representations of solvable and nilpotent groups and harmonic analysis on nil and solvmanifolds, Proc. Sympos. Pure Math., vol. 26, Amer. Math. Soc., Providence, R. I., 1973, pp. 345. MR 0385001 (52:5871)
 [Mo]
 G. Mostow, Factor spaces of solvable Lie groups, Ann. of Math. (2) 60 (1954), 127. MR 0061611 (15:853g)
 [P1]
 R. Penney, Abstract Plancherel theorems and a Frobenius reciprocity theorem, J. Funct. Anal. 18 1975, 177190. MR 0444844 (56:3191)
 [P2]
 , Spherical distributions on Lie groups and vectors, Trans. Amer. Math. Soc. 223 (1976), 368384. MR 0457632 (56:15837)
 [P3]
 , Decomposition of intertwining operators for Lie groups, Proc. Amer. Math. Soc. 54 (1976), 368370. MR 0404531 (53:8331)
 [P4]
 , Lie algebra cohomology of representations of nilpotent Lie groups and holomorphically induced representations, Trans. Amer. Math. Soc. 261 (1980), 3351. MR 576862 (81i:22005)
 [P5]
 , Canonical objects in Kirillov theory on nilpotent Lie groups, Proc. Amer. Math. Soc. 66 (1977), 175178. MR 0453922 (56:12175)
 [Po]
 N. Poulsen, On vectors and intertwining bilinear forms for representations of Lie groups, J. Funct. Anal. 9 (1972), 87120. MR 0310137 (46:9239)
 [R]
 L. Richardson, Decomposition of the space of a general compact nilmanifold, Amer. J. Math. 93 (1971). MR 0284546 (44:1771)
 [Ro]
 J. Rosenberg, Realization of square integrable representations of unimodular Lie groups on cohomology spaces, Trans. Amer. Math. Soc. 26 (1980), 132. MR 576861 (81k:22012)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198608574361
PII:
S 00029947(1986)08574361
Article copyright:
© Copyright 1986
American Mathematical Society
