Square integrable representations and a Plancherel theorem for parabolic subgroups

Author:
Frederick W. Keene

Journal:
Trans. Amer. Math. Soc. **243** (1978), 61-73

MSC:
Primary 22E45

DOI:
https://doi.org/10.1090/S0002-9947-1978-0498983-X

MathSciNet review:
0498983

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Abstract: Let *G* be a semisimple Lie group with Iwasawa decomposition . Let be the corresponding decomposition of the Lie algebra of *G*. Then the nilpotent subgroup *N* has square integrable representations if and only if the reduced restricted root system is of type or . The Plancherel measure for *N* can be found explicitly in these cases. We then prove the Plancherel theorem in the case for the solvable subgroup *NA* by combining Mackey's ``Little Group'' method with an idea due to C. C. Moore: we find an operator *D*, defined on the functions on *NA* with compact support, such that

*e*is the identity, and is the Plancherel measure for

*NA*, and

*D*is an unbounded selfadjoint operator. In the case,

*D*involves fractional powers of the Laplace operator and hence is not a differential operator.

**[1]**Shôrô Araki,*On root systems and an infinitesimal classification of irreducible symmetric spaces*, J. Math. Osaka City Univ.**13**(1962), 1–34. MR**0153782****[2]**Louis Auslander and Calvin C. Moore,*Unitary representations of solvable Lie groups*, Mem. Amer. Math. Soc. No.**62**(1966), 199. MR**0207910****[3]**D. S. Drucker,*Non-associative algebras and hermitian symmetric spaces*, Doctoral Dissertation, Univ. of California, Berkeley, 1973.**[4]**R. Godement,*Sur les relations d'orthogonalité de V. Bargmann*. I, II, C. R. Acad. Sci. Paris**225**(1947), 521-523, 657-659. MR**9**, 134.**[5]**Harish-Chandra,*Representations of semisimple Lie groups. VI. Integrable and square-integrable representations*, Amer. J. Math.**78**(1956), 564–628. MR**0082056**, https://doi.org/10.2307/2372674**[6]**Harish-Chandra,*Harmonic analysis on semisimple Lie groups*, Bull. Amer. Math. Soc.**76**(1970), 529–551. MR**0257282**, https://doi.org/10.1090/S0002-9904-1970-12442-9**[7]**SigurÄ‘ur Helgason,*Differential geometry and symmetric spaces*, Pure and Applied Mathematics, Vol. XII, Academic Press, New York-London, 1962. MR**0145455****[8]**F. W. Keene,*Square-integrable representations of Iwasawa subgroups of a semisimple Lie group*, Doctoral Dissertation, Univ. of California, Berkeley, 1974.**[9]**Frederick W. Keene,*𝐿₂-representations and a Plancherel-type theorem for parabolic subgroups*, Bull. Amer. Math. Soc.**81**(1975), 117–120. MR**0354938**, https://doi.org/10.1090/S0002-9904-1975-13663-9**[10]**A. A. Kirillov,*Unitary representations of nilpotent Lie groups*, Uspehi Mat. Nauk**17**(1962), no. 4 (106), 57–110 (Russian). MR**0142001****[11]**Bertram Kostant,*On the existence and irreducibility of certain series of representations*, Bull. Amer. Math. Soc.**75**(1969), 627–642. MR**0245725**, https://doi.org/10.1090/S0002-9904-1969-12235-4**[12]**George W. Mackey,*Unitary representations of group extensions. I*, Acta Math.**99**(1958), 265–311. MR**0098328**, https://doi.org/10.1007/BF02392428**[13]**Calvin C. Moore,*Representations of solvable and nilpotent groups and harmonic analysis on nil and solvmanifolds*, Harmonic analysis on homogeneous spaces (Proc. Sympos. Pure Math., Vol. XXVI, Williams Coll., Williamstown, Mass., 1972) Amer. Math. Soc., Providence, R.I., 1973, pp. 3–44. MR**0385001****[14]**Calvin C. Moore and Joseph A. Wolf,*Square integrable representations of nilpotent groups*, Trans. Amer. Math. Soc.**185**(1973), 445–462 (1974). MR**0338267**, https://doi.org/10.1090/S0002-9947-1973-0338267-9**[15]**S. R. Quint,*Representations of solvable Lie groups*, Lecture Notes, Univ. of California, Berkeley, 1972.**[16]**I. E. Segal,*An extension of Plancherel’s formula to separable unimodular groups*, Ann. of Math. (2)**52**(1950), 272–292. MR**0036765**, https://doi.org/10.2307/1969470**[17]**G. B. Seligman,*Topics in Lie algebras*, Yale Univ. mimeographed notes, 1969; revised, with additions, 1973.**[18]**Elias M. Stein,*Singular integrals and differentiability properties of functions*, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR**0290095****[19]**Joseph A. Wolf,*The action of a real semisimple Lie group on a complex flag manifold. II. Unitary representations on partially holomorphic cohomology spaces*, American Mathematical Society, Providence, R.I., 1974. Memoirs of the American Mathematical Society, No. 138. MR**0393350****[20]**Joseph A. Wolf,*Representations of certain semidirect product groups*, J. Functional Analysis**19**(1975), no. 4, 339–372. MR**0422519****[21]**Frederick W. Keene, Ronald L. Lipsman, and Joseph A. Wolf,*The Plancherel formula for parabolic subgroups*, Israel J. Math.**28**(1977), no. 1-2, 68–90. MR**0507242**, https://doi.org/10.1007/BF02759782

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1978-0498983-X

Keywords:
Parabolic subgroup,
semisimple Lie group,
Plancherel theorem,
nonunimodular group,
square integrable representations,
solvable subgroups of type *AN*

Article copyright:
© Copyright 1978
American Mathematical Society