Cauchy-Szegő maps, invariant differential operators and some representations of

Author:
Christopher Meaney

Journal:
Trans. Amer. Math. Soc. **313** (1989), 161-186

MSC:
Primary 22E46

MathSciNet review:
930080

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Fix an integer . Let be the semisimple Lie group and be the subgroup . For each finite dimensional representation of there is the space of smooth -covariant functions on , denoted by and equipped with the action of by right translation. Now take to be , the representation of on the space of harmonic polynomials on which are bihomogeneous of degree . For a real number there is the corresponding spherical principal series representation of , denoted by . In this paper we show that, as a -module, the irreducible quotient of can be realized as the space of the -finite elements of the kernel of a certain invariant first order differential operator acting on . Johnson and Wallach had shown that these representations are not square-integrable. Thus, some exceptional representations of are realized in a manner similar to Schmid's realization of the discrete series. The kernels of the differential operators which we use here are the intersection of kernels of some Schmid operators and quotient maps, which we call Cauchy-Szegö maps, a generalization the Szegö maps used by Knapp and Wallach. We also identify this representation of with an end of complementary series representation.

**[BBS]**M. W. Baldoni Silva and D. Barbasch,*The unitary spectrum for real rank one groups*, Invent. Math.**72**(1983), no. 1, 27–55. MR**696689**, 10.1007/BF01389128**[BLANK]**B. E. Blank,*Boundary behaviour of limits of discrete series representations*, Ph.D Thesis, Cornell Univ., 1980.**[BLANK 2]**B. E. Blank,*Knapp-Wallach Szegő integrals and generalized principal series representations: the parabolic rank one case*, J. Funct. Anal.**60**(1985), no. 2, 127–145. MR**777234**, 10.1016/0022-1236(85)90048-5**[BLANK 3]**Brian Blank,*Boundary behavior of limits of discrete series representations of real rank one semisimple groups*, Pacific J. Math.**122**(1986), no. 2, 299–318. MR**831115****[BW]**Armand Borel and Nolan R. Wallach,*Continuous cohomology, discrete subgroups, and representations of reductive groups*, Annals of Mathematics Studies, vol. 94, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1980. MR**554917****[GI]**J. E. Gilbert,*Cauchy-Szegö and reproducing kernels*, handwritten notes.**[G2]**-,*Invariant differential operators in harmonic analysis*, One-hour address, AMS meeting, Mobile, Alabama, May 1985.**[GKST:Cort.]**J. E. Gilbert, R. A. Kunze, R. J. Stanton, and P. A. Tomas,*A kernel for generalized Cauchy-Riemann systems*, Harmonic analysis (Cortona, 1982) Lecture Notes in Math., vol. 992, Springer, Berlin, 1983, pp. 402–415. MR**729366**, 10.1007/BFb0069171**[GKST:Zyg]**J. E. Gilbert, P. A. Tomas, R. A. Kunze, and R. J. Stanton,*Higher gradients and representations of Lie groups*, Conference on harmonic analysis in honor of Antoni Zygmund, Vol. I, II (Chicago, Ill., 1981) Wadsworth Math. Ser., Wadsworth, Belmont, CA, 1983, pp. 416–436. MR**730082****[GKT:Clev]**John E. Gilbert, Ray A. Kunze, and Peter A. Tomas,*Intertwining kernels and invariant differential operators in representation theory*, Probability theory and harmonic analysis (Cleveland, Ohio, 1983) Monogr. Textbooks Pure Appl. Math., vol. 98, Dekker, New York, 1986, pp. 91–112. MR**830233****[HO]**Ryoshi Hotta,*Elliptic complexes on certain homogeneous spaces*, Osaka J. Math.**7**(1970), 117–160. MR**0265519****[HP]**R. Hotta and R. Parthasarathy,*Multiplicity formulae for discrete series*, Invent. Math.**26**(1974), 133–178. MR**0348041****[HU]**James E. Humphreys,*Introduction to Lie algebras and representation theory*, Springer-Verlag, New York-Berlin, 1972. Graduate Texts in Mathematics, Vol. 9. MR**0323842****[JW]**Kenneth D. Johnson and Nolan R. Wallach,*Composition series and intertwining operators for the spherical principal series. I*, Trans. Amer. Math. Soc.**229**(1977), 137–173. MR**0447483**, 10.1090/S0002-9947-1977-0447483-0**[K]**A. W. Knapp,*A Szegő kernel for discrete series*, Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974) Canad. Math. Congress, Montreal, Que., 1975, pp. 99–104. MR**0437682****[KS]**A. W. Knapp and E. M. Stein,*Intertwining operators for semisimple groups*, Ann. of Math. (2)**93**(1971), 489–578. MR**0460543****[KW]**A. W. Knapp and N. R. Wallach,*Szegö kernels associated with discrete series*, Invent. Math.**34**(1976), no. 3, 163–200. MR**0419686****[KR]**Hrvoje Kraljević,*On representations of the group 𝑆𝑈(𝑛,1)*, Trans. Amer. Math. Soc.**221**(1976), no. 2, 433–448. MR**0409725**, 10.1090/S0002-9947-1976-0409725-6**[SC]**Wilfried Schmid,*On the realization of the discrete series of a semisimple Lie group.*, Rice Univ. Studies**56**(1970), no. 2, 99–108 (1971). MR**0277668****[SW]**E. M. Stein and G. Weiss,*Generalization of the Cauchy-Riemann equations and representations of the rotation group*, Amer. J. Math.**90**(1968), 163–196. MR**0223492****[TOMAS]**P. A. Tomas,*Operators of Stein-Weiss and Schmid, and representations of Lie groups*, handwritten notes, 1984-1985.**[Z]**D. P. Želobenko,*Compact Lie groups and their representations*, American Mathematical Society, Providence, R.I., 1973. Translated from the Russian by Israel Program for Scientific Translations; Translations of Mathematical Monographs, Vol. 40. MR**0473098**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
22E46

Retrieve articles in all journals with MSC: 22E46

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1989-0930080-6

Article copyright:
© Copyright 1989
American Mathematical Society