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Vector valued Poisson transforms on Riemannian symmetric spaces of rank one. (English) Zbl 0813.53034
Let $$G$$ be a connected real semisimple Lie group with finite center and $$K$$ be a maximal compact subgroup of $$G$$. Let $$G= KAN$$ be an Iwasawa decomposition of $$G$$, $${\mathfrak g}$$ and $${\mathfrak a}$$ be the Lie algebra of $$G$$ and $$A$$ respectively. Let $$\rho$$ be the half sum of the positive roots of $${\mathfrak a}$$ in $${\mathfrak g}$$. Define $$M= Z_ K (A)$$, then $$P=MAN$$ is a minimal parabolic subgroup of $$G$$. The Riemannian symmetric space of the noncompact type $$G/K$$ has the Furstenberg boundary $$K/M$$. Helgason conjectured that all eigenfunctions on $$G/K$$ are Poisson integrals of hyperfunctions on the boundary. Kashiwara et al. proved in its full generality the conjecture. Let $$G\times_ K W$$ be the associated vector bundle to the $$K$$-representation $$(\sigma, W)$$ over the Riemannian symmetric space $$G/K$$ of rank one, this paper completely describes the eigensections of the Casimir operator by using a generalization of the Poisson transform. More precisely we have:
(1) If $$(\sigma, W)$$ is a continuous representation of $$K$$ in a finite- dimensional complex vector space, the smooth sections of the associated vector bundle $$G\times_ K W$$ can be identified with $C^ \infty \text{ Ind}_ K^ G (\sigma)= \{f: G\to W\mid f(xk)= \sigma(k)^{-1} f(x),\;x\in G,\;k\in K,\;f \text{ smooth}\}.$ For $$\mu\in C$$, $$C^ \infty_ \mu \text{ Ind}^ G_ K (\sigma)$$ denotes the subspace of eigensections of the Casimir with eigenvalue $$\mu$$. If $$\tau$$ is a finite-dimensional representation of $$P$$ on the vector space $$V_ \tau$$. Let $C^ \infty \text{ Ind}^ G_ P (\tau)= \{f: G\to V_ \tau\mid f(x man)= a^{-\rho} \tau(man)^{-1} f(x),\;x\in G,\;man\in MAN\}.$ If $$(\delta, V_ \delta)$$ is a finite-dimensional representation of $$M$$. Let $C^ \infty \text{ Ind}^ K_ M (\delta)=\{f: K\to V_ \delta\mid f(km)= \delta(m)^{-1} f(k),\;k\in K,\;m\in M,\;f \text{ smooth}\}.$ Then the Poisson transform on $$C^ \infty \text{ Ind}^ G_ P (\tau)$$ is defined as a continuous linear, $$G$$-equivariant map from $$C^ \infty \text{ Ind}^ G_ P (\tau)$$ to $$C^ \infty \text{ Ind}^ G_ K (\sigma)$$.
(2) Using the idea of E. P. van den Ban [ibid. 109, No. 2, 331-441 (1992; Zbl 0791.22008)], E. P. van den Ban and H. Schlichtkrull [J. Reine Angew. Math. 380, 108-165 (1987; Zbl 0631.58028)] and N. R. Wallach [Asymptotic expansions of generalized matrix entries of representations of real reductive groups, Lect. Notes Math. 1024, 287-369 (1983; Zbl 0553.22005)], the author of the present paper proves the existence of asymptotic expansions of eigensections satisfying certain growth conditions. By mapping an eigenfunction in $$C^ \infty_ \mu \text{ Ind}^ G_ K (\sigma)$$ to the sum of some of the coefficients in its expansion, one can define a $$K$$-equivariant boundary value operator $\beta_{\mu, \mu_ 0}: C^ \infty_ \mu \text{ Ind}^ G_ K (\sigma)\to C^ \infty \text{ Ind}^ K_ M (\sigma_{| M}).$ For a certain direct sum $$C^ \infty \text{ Ind}^ G_ P (\sigma_ \mu)$$ of principal series representations, for which $$(\sigma_ \mu)_{| M}= \sigma_{| M}$$, the Poisson transform $${\mathcal P}_ \mu$$ on $$C^ \infty \text{ Ind}^ G_ P (\sigma_ \mu)$$ is defined by ${\mathcal P}_ \mu f(x)= \int_ K \sigma(k) f(xk) dk$ which maps into $$C^ \infty_ \mu \text{ Ind}^ G_ K (\sigma)$$ and we obtain ${\mathcal P}_ \mu: C \text{ Ind}^ K_ M (\sigma_{| M})\to C^ \infty_ \mu \text{ Ind}^ G_ K (\sigma).$ For generic $$\mu_ 0$$ the transform $${\mathcal P}_{\mu_ 0}$$ is an isomorphism and in this case it is essentially inverted by $$\beta_{\mu, \mu_ 0}$$. For a degenerate $$\mu_ 0$$ one only has that the composite map $\beta_{\mu, \mu_ 0}\circ {\mathcal P}_ \mu: C^ \infty \text{ Ind}^ K_ M (\sigma_{| M})\to C^ \infty \text{ Ind}^ G_ K (\sigma_{| M})$ is invertible for $$\mu\neq \mu_ 0$$ in a neighbourhood of $$\mu_ 0$$.
(3) The main theorem of this paper is that the possible poles of $$(\beta_{\mu, \mu_ 0} \circ{\mathcal P}_ \mu)^{-1}$$ at $$\mu= \mu_ 0$$ are precisely cancelled by the zeroes of $${\mathcal P}_ \mu$$ at $$\mu= \mu_ 0$$. Moreover, the transform $$[{\mathcal P}_ \mu\circ (\beta_{\mu, \mu_ 0}\circ {\mathcal P}_ \mu)^{-1} ]_{\mu= \mu_ 0}$$ is an isomorphism onto $$C^ \infty_{\mu_ 0} \text{ Ind}^ G_ K (\sigma)$$.
(4) At last the author gives two applications of the main theorem. The first is the computation of the composition factors of $$C^ \infty_ \mu \text{ Ind}^ G_ K (\sigma)$$. The second gives an embedding of the $$G$$-representation on $$C^ \infty \text{ Ind}^ K_ M (\sigma_{| M})$$ defined by the isomorphism $${\mathcal P}_ \mu\circ (\beta_{\mu, \mu_ 0}\circ {\mathcal P}_ \mu)^{-1}$$ into an induced representation of $$G/P$$, and so one gets a $$G$$-equivariant embedding from $$C^ \infty_{\mu_ 0} \text{ Ind}^ G_ K (\sigma)$$ into an induced representation space over $$G/P$$.
Reviewer: Zhu Fuliu (Hubei)

##### MSC:
 53C35 Differential geometry of symmetric spaces
##### Keywords:
Furstenberg boundary; Casimir operator; Poisson transform
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