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The principal series for a reductive symmetric space. I: H-fixed distribution vectors. (English) Zbl 0714.22009

Let G/H be a reductive symmetric space. We discuss the main topics treated in this paper under separate headings. (Unexplained notation is mostly standard and explained in the paper.)
1. The principal series. Let \(P=MAN\) be a \(\sigma\theta\)-stable parabolic subgroup with \(a=L(A)\) containing the maximal abelian \(a_{oq}\) in \(p\cap q\) where \(g=k+p=h+q\). (These \(P\in:\) \({\mathcal P}_{\sigma}(A_ q)\) are classified in terms of root and Weyl group data.) Let \(\hat M_{ps}\) be the set of (unitary) \(\xi\in \hat M\) for which there is \(w\in W=W(g,a_ 0)\), \(a_ 0\supset a_{0q}\) maximal abelian in p, so that \(w\xi\) has nontrivial (M\(\cap H)\)-distribution vectors in \({\mathcal K}^{- \infty}_{w\xi}\). By definition, the (non-unitary) principal series for G/H consists of representations \(Ind^ G_ P(\xi \otimes e^{\lambda}\otimes 1)\), \(\xi\in \hat M_{ps}\), \(\lambda \in a^*_{qc}\). It follows from results of Bruhat that \(Ind^ G_ P(\xi \otimes e^{\lambda}\otimes 1)\) is irreducible when \(\lambda \in ia^*_ q\) satisfies \(<\lambda,\alpha >\neq 0\) for all \(\alpha \in \Sigma =\Sigma (g,a_ p).\)
2. Intertwining operators. Let \(P_ 1=MAN_ 1\) and \(P_ 2=MAN_ 2\) be two P’s as above. Let \(A(P_ 2:P_ 1:\xi:\lambda): C^{\infty}(P_ 1:\xi:\lambda)\to C^{\infty}(P_ 2:\xi:\lambda)\) be the intertwining operator of the \(C^{\infty}\) induced representations defined as those of Knapp and Stein (K.-S.). Theorem. As for \(f\in C^{\infty}(K,\xi)\), the \(C^{\infty}(K,\xi)\)-valued function \(\lambda \to A(P_ 1:P_ 2:\xi:\lambda)f\), initially defined for \(\lambda\in {\mathcal A}(P_ 2| P_ 1)\) extends meromorphically to \(a^*_{qc}\). For each \(\lambda_ 0\in a^*_{qc}\) there is a neighbourhood \(N(\lambda_ 0)\) and a non- zero holomorphic \(\phi\) : N(\(\lambda\) \({}_ 0)\to {\mathbb{C}}\) such that the map \((\lambda,g)\to \phi (\lambda)A(P_ 1:P_ 2:\xi:\lambda)f\) of \(N(\lambda_ 0)\times C^{\infty}(K:\xi)\) into \(C^{\infty}(K:\xi)\) is continuous. The proof of this follows K.-S. with modifications and new arguments needed for the case at hand. The transformation properties of the K.-S. operators carry over as well.
3. H-fixed distributions. Fix \(P\in {\mathcal P}_{\sigma}(A_ q)\) and \(\xi\in \hat M_{fu}\) (finite-dimensional unitary). Let \({\mathcal O}(P)\) denote the union of the open H-orbits in \(P\setminus G\), \[ C^{\infty}({\mathcal O}(P):P:\xi:\lambda)=\{f: {\mathcal O}(P)\to {\mathcal K}_{\xi}| C^{\infty},\quad f(man\cdot x)=a^{\lambda +\rho_ P}f(x)\}, \] and r: \({\mathcal D}'(G:P:\xi:\lambda)^ H\to C^{\infty}({\mathcal O}(P):P:\xi:\lambda)\) the restriction map. Theorem. If \(k\in {\mathbb{N}}\), then for \(\lambda\) in the complement \({\mathcal B}_ k\) of a finite union of hyperplanes, r maps \({\mathcal D}'_ k(G:P:\xi:\lambda)^ H\) (distributions of order \(\leq k)\) injectively into \(C^{\infty}({\mathcal O}(P):P:\xi:\lambda)\). As a consequence, there is an injective evaluation map ev: \({\mathcal D}'(G:P:\xi:\lambda)^ H\to V(\xi)\) into a finite- dimensional space V(\(\xi\)), which leads further to a map j(P:\(\xi\) :\(\lambda\) :\(\eta\)): V(\(\xi\))\(\to {\mathcal D}'(G:P:\xi:\lambda)^ H\) that is bijective for \(\lambda\in {\mathcal B}=\cap {\mathcal B}_ k\) and with inverse ev.
4. The matrix B. It is defined by the relation \(A(P_ 2:P_ 1:\xi:\lambda)\circ j(P_ 1:\xi:\lambda)=j(P_ 2:\xi:\lambda)\circ B(P_ 2:P_ 1:\xi:\lambda)\) and is a meromorphic function of \(\lambda \in a^*_{qc}\) with values in End V(\(\xi\)). Theorem. Assume all Cartan subgroups of G are abelian. Then \(B(P_ 2:P_ 1:\xi:\lambda)^*=B(P_ 2:P_ 1:\xi:-{\bar \lambda})\), adjoint with respect to a natural unitary structure on V(\(\xi\)). The proof involves a lengthy reduction to \(\sigma\)-split-rank one.
Reviewer: W.Rossmann

MSC:

22E30 Analysis on real and complex Lie groups
22E46 Semisimple Lie groups and their representations
43A85 Harmonic analysis on homogeneous spaces
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References:

[1] E. P. van den BAN , A Convexity Theorem for Semisimple Symmetric Spaces (Pac. J. Math., Vol. 124, 1986 , pp. 21-55). Article | MR 87m:22039 | Zbl 0599.22014 · Zbl 0599.22014
[2] E. P. van den BAN , Invariant Differential Operators on a Semisimple Symmetric Space and Finite Multiplicities in a Plancherel Formula (Ark. för Mat., Vol. 25, 1987 , pp. 175-187). MR 89g:22019 | Zbl 0645.43009 · Zbl 0645.43009
[3] E. P. van den BAN , Asymptotic Behaviour of Matrix Coefficients Related to Reductive Symmetric Spaces (Proc. Kon. Ned. Akad. Wet., Ser. A, Vol. 90, 1987 , pp. 225-249). MR 89c:22025 | Zbl 0629.43008 · Zbl 0629.43008
[4] E. P. van den BAN , The Principal Series for a Reductive Symmetric Space II. Eisenstein Integrals , in preparation. · Zbl 0791.22008
[5] M. BERGER , Les espaces symétriques non-compacts (Ann. scient. Ec. Norm. Sup., Vol. 74, 1957 , pp. 85-117). Numdam | MR 21 #3516 | Zbl 0093.35602 · Zbl 0093.35602
[6] I. N. BERNSTEIN , The analytic continuation of generalized functions with respect to a parameter (Funct. Anal. and its Applic., Vol. 6, 1972 , pp. 273-285). MR 47 #9269 | Zbl 0282.46038 · Zbl 0282.46038
[7] I. N. BERNSTEIN and I. S. GELFAND , Meromorphic Property of the Function P\lambda (Funct. Anal. and its Applic., Vol. 3, 1969 , pp. 68-69). · Zbl 0208.15201
[8] F. BRUHAT , Sur les représentations induites des groupes de Lie (Bull. Soc. Math. France, Vol. 84, 1956 , pp. 97-205). Numdam | MR 18,907i | Zbl 0074.10303 · Zbl 0074.10303
[9] P. DELORME , Injection de modules sphériques pour les espaces symétriques réductifs dans certaines représentations induites (Proceedings Marseille-Luminy 1985 , LNM 1243), Springer-Verlag, 1987 , pp. 108-134. MR 89c:22026 | Zbl 0658.22003 · Zbl 0658.22003
[10] G. van DIJK and M. POEL , The Plancherel Formula for the Pseudo-Riemannian Space SL (n, \Bbb R)/G1 (n - 1, \Bbb R) (Comp. Math., Vol. 58, 1986 , pp. 371-397). Numdam | MR 87m:22022 | Zbl 0593.43009 · Zbl 0593.43009
[11] J. FARAUT , Distributions sphériques sur les espaces hyperboliques (J. Math. Pures Appl., Vol. 58, 1979 , pp. 369-444). MR 82k:43009 | Zbl 0436.43011 · Zbl 0436.43011
[12] M. FLENSTED-JENSEN , Analysis on non-Riemannian Symmetric Spaces (A.M.S. Regional Conference Series 61, 1986 ). MR 87h:43013 | Zbl 0589.43008 · Zbl 0589.43008
[13] HARISH-CHANDRA , Spherical Functions on a Semisimple Lie Group I, II (Amer. J. of Math., Vol. 80, 1958 , pp. 241-310 and pp. 553-613). MR 20 #925 | Zbl 0093.12801 · Zbl 0093.12801
[14] HARISH-CHANDRA , Harmonic Analysis on Real Reductive Groups I. The theory of the constant term (J. Funct. Anal., Vol. 19, 1975 , pp. 103-204). MR 53 #3201 | Zbl 0315.43002 · Zbl 0315.43002
[15] HARISH-CHANDRA , Harmonic Analysis on Real Reductive Groups II. Wave Packets in the Schwartz Space (Invent. Math., Vol. 36, 1979 , pp. 1-55). MR 55 #12874 | Zbl 0341.43010 · Zbl 0341.43010
[16] HARISH-CHANDRA , Harmonic Analysis on Real Reductive Groups III. The Maass-Selberg Relations and the Plancherel Formula (Ann. of Math., Vol. 104, 1976 , pp. 117-201). MR 55 #12875 | Zbl 0331.22007 · Zbl 0331.22007
[17] S. HELGASON , Groups and Geometric Analysis , Academic Press, 1984 . MR 86c:22017 | Zbl 0543.58001 · Zbl 0543.58001
[18] L. HÖRMANDER , The Analysis of Linear Partial Differential Operators I , Springer-Verlag, 1983 . · Zbl 0521.35001
[19] 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., Vol. 107, 1978 , pp. 1-39). MR 81f:43013 | Zbl 0377.43012 · Zbl 0377.43012
[20] A. W. KNAPP and E. M. STEIN , Intertwining Operators for Semisimple Groups (Ann. of Math., Vol. 93, 1971 , pp. 489-578). MR 57 #536 | Zbl 0257.22015 · Zbl 0257.22015
[21] A. W. KNAPP and E. M. STEIN , Intertwining Operators for Semisimple Groups, II (Invent. Math., Vol. 60, 1980 , pp. 9-84). MR 82a:22018 | Zbl 0454.22010 · Zbl 0454.22010
[22] A. W. KNAPP , Commutativity of Intertwining Operators for Semisimple Groups (Comp. Math., Vol. 46, 1982 , pp. 33-84). Numdam | MR 83i:22022 | Zbl 0488.22027 · Zbl 0488.22027
[23] H. KOMATSU , Projective and Injective Limits of Weakly Compact Sequences of Locally Convex Spaces (J. Math. Soc. Japan, Vol. 19, 1967 , pp. 366-383). Article | MR 36 #646 | Zbl 0168.10603 · Zbl 0168.10603
[24] B. KOSTANT and S. RALLIS , Orbits and Representations Associated with Symmetric Spaces (Amer. J. Math., Vol. 93, 1971 , pp. 753-809). MR 47 #399 | Zbl 0224.22013 · Zbl 0224.22013
[25] T. MATSUKI , The Orbits of Affine Symmetric Spaces under the Action of Minimal Parabolic Subgroups (J. Math. Soc. Japan, Vol. 31, 1979 , pp. 331-357). Article | MR 81a:53049 | Zbl 0396.53025 · Zbl 0396.53025
[26] T. MATSUKI , Orbits on Affine Symmetric Spaces under the Action of Parabolic Subgroups (Hiroshima Math. J. Vol. 12, 1982 , pp. 307-320). MR 83k:53072 | Zbl 0495.53049 · Zbl 0495.53049
[27] G. OLAFSSON , Fourier and Poisson Transformation Associated to a Semisimple Symmetric Space (Invent. Math., Vol. 90, 1987 , pp. 605-629). MR 89d:43011 | Zbl 0665.43004 · Zbl 0665.43004
[28] T. OSHIMA , Poisson Transformations on Affine Symmetric Spaces (Proc. Japan Acad., Vol. 55, Série A, 1979 , pp. 323-327). Article | MR 81k:43013 | Zbl 0485.22011 · Zbl 0485.22011
[29] T. OSHIMA and J. SEKIGUCHI , Eigenspaces of Invariant Differential Operators on an Affine Symmetric Space (Invent. Math., Vol. 57, 1980 , pp. 1-81). MR 81k:43014 | Zbl 0434.58020 · Zbl 0434.58020
[30] R. PENNEY , Abstract Plancherel Theorems and a Frobenius Reciprocity Theorem (J. of Funct. Anal., Vol. 18, 1975 , pp. 177-190). MR 56 #3191 | Zbl 0305.22016 · Zbl 0305.22016
[31] W. ROSSMANN , Analysis on Real Hyperbolic Spaces (J. Funct. Anal., Vol. 30, 1978 , pp. 448-477). MR 80f:43021 | Zbl 0395.22014 · Zbl 0395.22014
[32] W. ROSSMANN , The Structure of Semisimple Symmetric Spaces (Can. J. Math., Vol. 31, 1979 , pp. 157-180). MR 81i:53042 | Zbl 0357.53033 · Zbl 0357.53033
[33] G. SCHIFFMANN , Intégrales d’entrelacement et fonctions de Whittaker (Bull. Soc. Math., France, Vol. 99, 1971 , pp. 3-72). Numdam | MR 47 #400 | Zbl 0223.22017 · Zbl 0223.22017
[34] H. SCHLICHTKRULL , Hyperfunctions and Harmonic Analysis on Symmetric Spaces , Birkhöuser, 1984 . MR 86g:22021 | Zbl 0555.43002 · Zbl 0555.43002
[35] L. SCHWARTZ , Théorie des distributions , Tome I, Hermann, 1950 . MR 12,31d | Zbl 0037.07301 · Zbl 0037.07301
[36] R. S. STRICHARTZ , Harmonic analysis on hyperboloids (J. Funct. Anal., Vol. 12, 1973 , pp. 341-383). MR 50 #5370 | Zbl 0253.43013 · Zbl 0253.43013
[37] V. S. VARADARAJAN , Lie Groups, Lie Algebras, and their Representations , Prentice Hall, 1974 . MR 51 #13113 | Zbl 0371.22001 · Zbl 0371.22001
[38] V. S. VARADARAJAN , Harmonic Analysis on Real Reductive Groups (Lecture Notes 576), Springer-Verlag, 1977 . MR 57 #12789 | Zbl 0354.43001 · Zbl 0354.43001
[39] D. A. VOGAN , Representations of Real Reductive Lie Groups , (P.M. 15), Birkhöuser, Boston, 1981 . MR 83c:22022 | Zbl 0469.22012 · Zbl 0469.22012
[40] G. WARNER , Harmonic Analysis on Semi-Simple Lie Groups I , Springer-Verlag, 1972 . Zbl 0265.22020 · Zbl 0265.22020
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