Frobenius reciprocity and extensions of nilpotent Lie groups

Author:
Jeffrey Fox

Journal:
Trans. Amer. Math. Soc. **298** (1986), 123-144

MSC:
Primary 22E25; Secondary 22E27, 22E40, 22E45

DOI:
https://doi.org/10.1090/S0002-9947-1986-0857436-1

MathSciNet review:
857436

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Abstract: In we use -vector methods, essentially Frobenius reciprocity, to derive the Howe-Richardson 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.

**[**L. Auslander and C. Moore,**A-M**]*Unitary representations of solvable Lie groups*, Mem. Amer. Math. Soc. No. 62 (1966). MR**0207910 (34:7723)****[**L. Corwin and F. Greenleaf,**C-G-1**]*Character formula and spectra of compact nilmanifolds*, J. Funct. Anal.**21**(1976), 123-154. MR**0393345 (52:14155)****[**-,**C-G-2**]*Intertwining operators for representations induced from uniform subgroups*, Acta Math.**136**(1976), 275-301. MR**0425009 (54:12967)****[**L. Corwin, F. Greenleaf, and R. Penney,**C-G-P**]*A general character formula for irreducible projections on**of a nilmanifold*, Math. Ann.**225**(1977), 21-32. MR**0425021 (54:12979)****[**I. Gelfand, M. Graev, and I. Pyatetskii-Shapiro,**G**]*Representation theory and automorphic functions*, Saunders, Philadelphia, Pa., 1969. MR**0233772 (38:2093)****[**R. Howe,**H-1**]*On Frobenius reciprocity for unipotent algebraic groups over*, Amer. J. Math.**93**(1971). MR**0281842 (43:7556)****[**-,**H-2**]*Topics in harmonic analysis on solvable algebraic groups*, Pacific J. Math.**73**(1977), 383-437. MR**0480858 (58:1007)****[**-,**H-3**]*On a condition between nilpotent groups and oscillatory integrals associated to singularities*, Pacific J. Math.**73**(1977), 329-365. MR**0578891 (58:28270)****[**A. Kleppner and R. Lipsman,**K-L-1**]*The Plancherel formula for group extensions*. I, Ann. Sci. École Norm. Sup. (4)**5**(1972), 459-516. MR**0342641 (49:7387)****[**C. Moore,**M**]*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. 3-45. MR**0385001 (52:5871)****[**G. Mostow,**Mo**]*Factor spaces of solvable Lie groups*, Ann. of Math. (2)**60**(1954), 1-27. MR**0061611 (15:853g)****[**R. Penney,**P-1**]*Abstract Plancherel theorems and a Frobenius reciprocity theorem*, J. Funct. Anal.**18**1975, 177-190. MR**0444844 (56:3191)****[**-,**P-2**]*Spherical distributions on Lie groups and**-vectors*, Trans. Amer. Math. Soc.**223**(1976), 368-384. MR**0457632 (56:15837)****[**-,**P-3**]*Decomposition of**-intertwining operators for Lie groups*, Proc. Amer. Math. Soc.**54**(1976), 368-370. MR**0404531 (53:8331)****[**-,**P-4**]*Lie algebra cohomology of representations of nilpotent Lie groups and holomorphically induced representations*, Trans. Amer. Math. Soc.**261**(1980), 33-51. MR**576862 (81i:22005)****[**-,**P-5**]*Canonical objects in Kirillov theory on nilpotent Lie groups*, Proc. Amer. Math. Soc.**66**(1977), 175-178. MR**0453922 (56:12175)****[**N. Poulsen,**Po**]*On**-vectors and intertwining bilinear forms for representations of Lie groups*, J. Funct. Anal.**9**(1972), 87-120. MR**0310137 (46:9239)****[**L. Richardson,**R**]*Decomposition of the**-space of a general compact nilmanifold*, Amer. J. Math.**93**(1971). MR**0284546 (44:1771)****[**J. Rosenberg,**Ro**]*Realization of square integrable representations of unimodular Lie groups on**-cohomology spaces*, Trans. Amer. Math. Soc.**26**(1980), 1-32. MR**576861 (81k:22012)**

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DOI:
https://doi.org/10.1090/S0002-9947-1986-0857436-1

Article copyright:
© Copyright 1986
American Mathematical Society