Boundary value maps, Szegö maps and intertwining operators

Author:
L. Barchini

Journal:
Trans. Amer. Math. Soc. **349** (1997), 1877-1900

MSC (1991):
Primary 22C05, 22E45, 22E46

DOI:
https://doi.org/10.1090/S0002-9947-97-01834-5

MathSciNet review:
1401761

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider one series of unitarizable representations, the cohomological induced modules with dominant regular infinitesimal character. The minimal -type of determines a homogeneous vector bundle . The derived functor modules can be realized on the solution space of a first order differential operator on . Barchini, Knapp and Zierau gave an explicit integral map from the derived functor module, realized in the Langlands classification, into the space of smooth sections of the vector bundle . In this paper we study the asymptotic behavior of elements in the image of . We obtain a factorization of the standard intertwining opeartors into the composition of the Szegö map and a passage to boundary values.

**[Bar]**V. Bargman,*Irreducible unitary representations of the Lorentz group*, Ann. of Math.**48**(1947), 586-640. MR**9:133a****[Bl-1]**B. Blank,*Embedding limits of discrete series of semisimple Lie groups*, Canadian Math. Conf. Proc.**1**(1981), 55-64. MR**83h:22001****[Bl-2]**B. Blank,*Knapp-Wallach Szegö integrals and the P-induced continuous representation: The parabolic rank one case*, J.Func.Anal.**60**(1985), 127-145. MR**86i:22032****[Bl-3]**B. Blank,*Boundary behavior of limits of discrete series representations of real rank one groups*, Pacific Jour. of Math.**122**(1986), 299-318. MR**87e:22027****[Bl-4]**B. Blank,*Knapp-Wallach Szegö integrals II, The higher parabolic rank case*, Trans. A.M.S**300**(1987), 49-59. MR**88f:22041****[BKZ]**L. Barchini, A.W. Knapp and R. Zierau,*InterTwining operators into Dolbeault cohomology*, Jour. Func. Anal.**107**(1992), 302- 341. MR**93e:22026****[B-1]**L. Barchini,*Szegö mappings, harmonic forms and Dolbeault cohomology*, Jour. Func. Anal.**118**(1993), 351-406. MR**94k:22033****[B-2]**L. Barchini,*Szegö kernels associated with Zuckerman modules*, J. Func. Anal.**131**(1995), 145-181. MR**96i:22033****[BW]**A. Borel and N. Wallach,*Continuous cohomology, discrete subgroups, and representations of reductive groups*, Annals of Math. Studies, no. 94, Princeton Univ. Press, Princeton, NJ, 1980. MR**83c:22018****[C]**W. Casselman,*The differential equations satisfied by matrix coefficients*, manuscript (1975).**[GKST]**J.E. Gilbert, A. Kunze, R.J. Stanton, and P. Thomas,*Higher gradients and representations of Lie groups*, Conference on Harmonic Analysis in honor of Antoni Zygmund, vol. II, Wadsworth, Belmont, CA, 1983, pp. 416-436. MR**85k:22032****[HC-1]**Harish-Chandra,*Harmonic Analysis on Real Reductive Groups I, the theory of the constant term*, J.Func.Anal.**19**(1975), 104-204. MR**53:3201****[HC-2]**Harish-Chandra,*Harmonic Analysis on Real Reductive Groups II*, Invent. Math.**36**(1976), 1-55. MR**55:12874****[HP]**R. Hotta and R. Parthasarathy,*Multiplicity formulae for discrete series*, Invent. Math.**26**(1974), 133-178. MR**50:539****[KKMOOT]**M. Kashiwara, A. Kowata, K. Minemura, K. Okamoto, T. Oshima and M. Tanaka,*Eigenfunctions of invariant differential operators on a symmetric space*, Ann. of Math.**107**(1978), 1-39. MR**81f:43013****[Ka]**T. Kawazoe,*On a global realization of a discrete series for as applications of Szegö operators and limits of discrete series*, Tokyo J.Math.**12**(1989). MR**91b:22020****[K-O]**A.W. Knapp and K. Okamoto,*Limits of holomorphic discrete series*, J. Func.Anal.**9**(1972), 375-409. MR**45:8774****[K-1]**A.W. Knapp,*A Szegö kernel for discrete series*, Proceedings International Congress of Mathematicians. Canadian Mathematical Congress.**2**(1975), 99-104. MR**55:10606****[KW]**A.W. Knapp and N. Wallach,*Szegö kernels associated to discrete series*, Invent. Math.**34, 62**(1976, 1980), 163-200, 341-346. MR**54:7704**; MR**82i:22016****[K-2]**A.W. Knapp,*Commutativity of Intertwining Operators for Semisimple Lie Groups*, Compositio Math.**46**(1982), 33-84. MR**83i:22022****[KS]**A.W. Knapp and E. Stein,*Intertwining operators for semisimple groups II*, Invent. Math.**60**(1980), 9-84. MR**82a:22018****[KZ]**A.W. Knapp and G. Zuckerman,*Classification of irreducible tempered representations of semisimple groups*, Ann. of Math.**116**(1982), 389-455. MR**84h:22034a****[K-3]**A.W. Knapp,*Representation Theory of Semisimple Groups : An overview based on examples*, Princeton Univ. Press, Princeton, NJ, 1986. MR**87j:22022****[L]**R. Langlands,*On the classification of irreducible representations of real algebraic groups*, Mimeographed notes, Inst. Adv. Stud., Princeton, NJ, 1973; reprinted in*Representation theory and harmonic analysis on semisimple Lie groups*(P.J. Sally, Jr., and D. A. Vogan, eds.), Math. Surveys and Monographs, vol. 31, Amer. Math. Soc., Providence, RI, 1989, pp. 101-170. MR**91e:22017****[Ma]**H. Matumoto,*Cohomological Hardy spaces for*, Adv. Studies in Pure Math.**14**(1988), 469-497. MR**91c:22032****[Mi]**H. Midorikawa,*On certain irreducible representations of real rank one classical groups*, J.Fac.of Science, The Univ. Tokyo, Sec IA.**27**, no. 3 (1974), 435-459. MR**51:809****[M]**D. Milicic,*Asymptotic behavior of matrix coefficients of discrete series*, Duke Math.J.**44**(1977), 59-88. MR**55:3171****[S]**W. Schmid,*Homogeneous complex manifolds and representations of semisimple Lie groups*, Ph.D. thesis, Univ. of California, Berkeley, CA, 1967; reprinted in*Representation theory and harmonic analysis on semisimple Lie groups*(P. J. Sally, Jr., and D. A. Vogan, Jr., eds.), Math. Surveys and Monographs, vol. 31, Amer. Math. Soc., Providence, RI, 1989, pp. 223-286. MR**90i:22025****[Ta]**R.Takashashi,*Sur les fonctions spheriques et la formulae de Plancherel dans le groupe hyperbolique*, Jap. J.Math.**31**(1961), 55-90. MR**27:2809****[VZ]**D. Vogan and G.J. Zuckerman,*Unitary representations with continuous cohomology*, Compositio Math.**53**(1984), 51-90. MR**86k:22040****[W-1]**N. Wallach,*Asymptotic expansion of generalized matrix entries of representations of real reductive groups*, Lie Group Representations, I (College Park, MD, 1982/83), Lecture Notes in Math., vol. 1024, Springer-Verlag, Berlin, 1983, pp. 287-369. MR**85g:22029****[W-2]**N. Wallach,*Real Reductive Groups I*, Academic Press, 1988. MR**89i:22029****[W-3]**N. Wallach,*Real Reductive Groups II*, Academic Press, 1992. MR**93m:22018****[Wo]**H-W. Wong,*Dolbeault cohomologies and Zuckerman modules associated with finite rank representations*, Ph.D. thesis, Harvard Univ., Cambridge, MA, 1991. See also J. Funct. Anal.**129**(1995), 428-454. MR**96c:22024**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (1991):
22C05,
22E45,
22E46

Retrieve articles in all journals with MSC (1991): 22C05, 22E45, 22E46

Additional Information

**L. Barchini**

Affiliation:
Department of Mathematics, Temple University, Philadelphia, Pennsylvania 19122

Address at time of publication:
Department of Mathematics, Oklahoma State University, Stillwater, Oklahoma 74078

Email:
leticiz@math.okstate.edu

DOI:
https://doi.org/10.1090/S0002-9947-97-01834-5

Received by editor(s):
February 17, 1995

Received by editor(s) in revised form:
October 9, 1995

Article copyright:
© Copyright 1997
American Mathematical Society