Knapp-Wallach Szegő integrals. II. The higher parabolic rank case

Blank, B. E.

doi:10.1090/S0002-9947-1987-0871664-1

Knapp-Wallach Szegő integrals. II. The higher parabolic rank case
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by B. E. Blank

Trans. Amer. Math. Soc. 300 (1987), 49-59

DOI: https://doi.org/10.1090/S0002-9947-1987-0871664-1

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Abstract:

Let $G$ be a connected reductive linear Lie group with compact center and real rank $l$. For each integer $k(1 \leqslant k \leqslant l)$ and each discrete series representation $\pi$ of $G$, an explicit embedding of $\pi$ into a generalized principal series representation induced from a parabolic subgroup of rank $k$ is given. The existence of such embeddings was proved by W. Schmid. In this paper an explicit integral formula with Szegö kernel is given which provides these mappings.

References

Embedding limits of discrete series of semisimple Lie groups

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, DOI 10.1016/0022-1236(85)90048-5
Harish-Chandra, Discrete series for semisimple Lie groups. I. Construction of invariant eigendistributions, Acta Math. 113 (1965), 241–318. MR 219665, DOI 10.1007/BF02391779
Harish-Chandra, Discrete series for semisimple Lie groups. II. Explicit determination of the characters, Acta Math. 116 (1966), 1–111. MR 219666, DOI 10.1007/BF02392813
Harish-Chandra, On the theory of the Eisenstein integral, Conference on Harmonic Analysis (Univ. Maryland, College Park, Md., 1971), Lecture Notes in Math., Vol. 266, Springer, Berlin, 1972, pp. 123–149. MR 0399355
Harish-Chandra, Harmonic analysis on real reductive groups. I. The theory of the constant term, J. Functional Analysis 19 (1975), 104–204. MR 0399356, DOI 10.1016/0022-1236(75)90034-8
Sigurđur Helgason, Differential geometry and symmetric spaces, Pure and Applied Mathematics, Vol. XII, Academic Press, New York-London, 1962. MR 0145455
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
A. W. Knapp, Commutativity of intertwining operators for semisimple groups, Compositio Math. 46 (1982), no. 1, 33–84. MR 660154
A. W. Knapp and N. R. Wallach, Szegö kernels associated with discrete series, Invent. Math. 34 (1976), no. 3, 163–200. MR 419686, DOI 10.1007/BF01403066
A. W. Knapp and N. R. Wallach, Correction and addition: “Szegő kernels associated with discrete series” [Invent. Math. 34 (1976), no. 3, 163–200; MR 54 #7704], Invent. Math. 62 (1980/81), no. 2, 341–346. MR 595593, DOI 10.1007/BF01389165
A. W. Knapp and Gregg J. Zuckerman, Classification of irreducible tempered representations of semisimple groups, Ann. of Math. (2) 116 (1982), no. 2, 389–455. MR 672840, DOI 10.2307/2007066
Adam Korányi and Joseph A. Wolf, Realization of hermitian symmetric spaces as generalized half-planes, Ann. of Math. (2) 81 (1965), 265–288. MR 174787, DOI 10.2307/1970616
George W. Mackey, Induced representations of locally compact groups. I, Ann. of Math. (2) 55 (1952), 101–139. MR 44536, DOI 10.2307/1969423
Wilfried Schmid, Homogeneous complex manifolds and representations of semisimple Lie groups, Proc. Nat. Acad. Sci. U.S.A. 59 (1968), 56–59. MR 225930, DOI 10.1073/pnas.59.1.56
Wilfried Schmid, On the realization of the discrete series of a semisimple Lie group, Rice Univ. Stud. 56 (1970), no. 2, 99–108 (1971). MR 277668
Wilfried Schmid, On the characters of the discrete series. The Hermitian symmetric case, Invent. Math. 30 (1975), no. 1, 47–144. MR 396854, DOI 10.1007/BF01389847
V. S. Varadarajan, Harmonic analysis on real reductive groups, Lecture Notes in Mathematics, Vol. 576, Springer-Verlag, Berlin-New York, 1977. MR 0473111, DOI 10.1007/BFb0097814