## 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 · doi:10.2140/pjm.1986.124.21 [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 · doi:10.1007/BF02384442 [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 · doi:10.1016/0022-1236(92)90021-A [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 · doi:10.1007/BF01077645 [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 · doi:10.2307/2372786 [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 · doi:10.1016/0022-1236(75)90034-8 [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 · doi:10.1007/BF01390004 [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 · doi:10.2307/1971058 [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 · doi:10.2307/1971253 [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 · doi:10.2307/1970887 [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 · doi:10.1007/BF01389898 [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 · doi:10.2969/jmsj/01930366 [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 · doi:10.2307/2373470 [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 · doi:10.2969/jmsj/03120331 [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 · doi:10.1007/BF01389180 [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 · doi:10.3792/pjaa.55.323 [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 · doi:10.1007/BF01389818 [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 · doi:10.1016/0022-1236(75)90023-3 [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 · doi:10.1016/0022-1236(78)90065-4 [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 · doi:10.4153/CJM-1979-017-6 [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 · doi:10.1016/0022-1236(73)90001-3 [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 · doi:10.1007/BFb0097814 [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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