CauchySzegő maps, invariant differential operators and some representations of
Author:
Christopher Meaney
Journal:
Trans. Amer. Math. Soc. 313 (1989), 161186
MSC:
Primary 22E46
MathSciNet review:
930080
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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 squareintegrable. 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 CauchySzegö maps, a generalization the Szegö maps used by Knapp and Wallach. We also identify this representation of with an end of complementary series representation.
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 [BBS]
 M. W. Baldoni Silva and D. Barbasch, The unitary spectrum for real rank one groups, Invent. Math. 72 (1983), 2755. MR 696689 (84k:22022)
 [BLANK]
 B. E. Blank, Boundary behaviour of limits of discrete series representations, Ph.D Thesis, Cornell Univ., 1980.
 [BLANK 2]
 , KnappWallach Szegö integrals and generalized principal series representations: the parabolic rank one case, J. Funct. Anal. 60 (1985), 127145. MR 777234 (86i:22032)
 [BLANK 3]
 , Boundary behaviour of limits of discrete series representations of real rank one semisimple groups, Pacific J. Math. 122 (1986), 299318. MR 831115 (87e:22027)
 [BW]
 A. Borel and N. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive groups, Ann. of Math. Studies, no. 94, Princeton Univ. Press, Princeton, N.J., 1980. MR 554917 (83c:22018)
 [GI]
 J. E. Gilbert, CauchySzegö and reproducing kernels, handwritten notes.
 [G2]
 , Invariant differential operators in harmonic analysis, Onehour 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 CauchyRiemann systems, Lecture Notes in Math., vol. 992, SpringerVerlag, Berlin and New York, 1983, pp. 402415. MR 729366 (85j:22026)
 [GKST:Zyg]
 , Higher gradients and representations of Lie groups, Conference on Harmonic Analysis in Honour of Antoni Zygmund, Wadsworth, Belmont, Calif., 1983, pp. 416436. MR 730082 (85k:22032)
 [GKT:Clev]
 J. E. Gilbert, R. A. Kunze and P. A. Tomas, Intertwining kernels and invariant differential operators in analysis, Probability Theory and Harmonic Analysis (Cleveland, Ohio, 1983), Monographs and Textbooks in Pure and Appl. Math., no. 98, Dekker, New York, 1986, pp. 91112. MR 830233 (87g:22013)
 [HO]
 R. Hotta, Elliptic complexes on certain homogeneous spaces, Osaka J. Math. 7 (1970), 117160. MR 0265519 (42:428)
 [HP]
 R. Hotta and R. Parthasarthy, Multiplicity formulae for discrete series, Invent. Math. 26 (1974), 133178. MR 0348041 (50:539)
 [HU]
 J. E. Humphreys Introduction to Lie algebras and representation theory, Graduate Texts in Math., no. 9, SpringerVerlag, Berlin and New York, 1972. MR 0323842 (48:2197)
 [JW]
 K. D. Johnson and N. R. Wallach, Composition series and intertwining operators for the spherical principal series. I, Trans. Amer. Math. Soc. 229 (1971), 137173. MR 0447483 (56:5794)
 [K]
 A. W. Knapp, A Szegö kernel for discrete series, Proc. Internat. Congr. Math., Vancouver 1974, pp. 99104. MR 0437682 (55:10606)
 [KS]
 A. W. Knapp and E. M. Stein, Intertwining operators for semisimple Lie groups, Ann. of Math. (2)93 (1971), 489578. MR 0460543 (57:536)
 [KW]
 A. W. Knapp and N. R. Wallach, Szegö kernels associated with discrete series, Invent. Math. 34 (1976), 163200. MR 0419686 (54:7704)
 [KR]
 H. Kraljević, On representations of the group , Trans. Amer. Math. Soc. 221 (1976), 433448. MR 0409725 (53:13477)
 [SC]
 W. Schmid, On the realization of the discrete series of a semisimple Lie groups, Rice Univ. Stud. 56 (1970), 99108. MR 0277668 (43:3401)
 [SW]
 E. M. Stein and G. Weiss, Generalization of the CauchyRiemann equations and representations of the rotation groups, Amer. J. Math. 90 (1968), 163196. MR 0223492 (36:6540)
 [TOMAS]
 P. A. Tomas, Operators of SteinWeiss and Schmid, and representations of Lie groups, handwritten notes, 19841985.
 [Z]
 D. P. Želobenko, Compact Lie groups and their representations, Transl. Math. Monographs, no. 40, Amer. Math. Soc., Providence, R.I., 1973. MR 0473098 (57:12776b)
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DOI:
http://dx.doi.org/10.1090/S00029947198909300806
PII:
S 00029947(1989)09300806
Article copyright:
© Copyright 1989
American Mathematical Society
