zbMATH — the first resource for mathematics

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)

53C35 Differential geometry of symmetric spaces
Full Text: DOI