Boundary behavior of limits of discrete series representations of real rank one semisimple groups (Q1077555)

Let G be a real rank one connected semisimple Lie group having a compact Cartan subgroup \(T\subset K\) where K is a maximal compact subgroup of G and \(\Delta\) the roots of (\({\mathfrak g}_{{\mathbb{C}}},{\mathfrak t}_{{\mathbb{C}}})\). Each nonsingular integral form \(\Lambda\) on t decides a positive root \(\Delta^+\) for which \(\Lambda\) is \(\Delta^+\)-dominant and corresponds to a discrete series \((\pi_{\Lambda},H^{\Lambda})\) whose lowest K-type is given by the Blattner parameter \(\lambda =\Lambda -\delta_ k+\delta_ n\). When the integral form is singular with respect to noncompact roots \(\{\pm \alpha \}\), there are two choices of positive roots and the Blattner parameters, say \(\lambda\) and \(\lambda\) '. Then \textit{A. W. Knapp} and \textit{N. R. Wallach} [Invent. Math. 34, 163- 200 (1976; Zbl 0332.22015)] showed that the reducible unitary principal series \((U(\sigma_{\lambda}:\rho)\), \(L^ 2(K,\sigma_{\lambda}))\) is infinitesimally equivalent with the direct sum of K-finite images of \(U(\sigma_{\lambda}:\rho)\) and \(U(\rho_{\lambda '}:\rho)\) under the Szegö maps \(S_{\lambda}\) and \(S_{\lambda '}\), respectively. Here \((\tau_{\lambda},V_{\lambda})\) and \((\sigma_{\lambda},H_{\lambda})\) \((H_{\lambda}\subset V_{\lambda})\) are irreducible representations with highest weight \(\lambda\) of K and \(M=Z_ K(A)\), respectively. In this paper, for each \(S_{\lambda}f\) \((f\in C^{\infty}(K,\sigma_{\lambda}))\) the author introduces a boundary value defined by \[ {\mathcal L}(S_{\lambda}f)(k)=\lim_{a\to \infty}E_{\lambda} e^{\rho (\log (a))} S_{\lambda} U(\sigma_{\lambda}:\rho:w^{-1}k) f(a)\quad (k\in K), \] where w is a nontrivial Weyl group element and \(E_{\lambda}\) the projection of \(V_{\lambda}\) onto \(H_{\lambda}\). Then \({\mathcal L}\) maps \(S_{\lambda}(C^{\infty}(K,\sigma_{\lambda}))\) into \(L^ 2(K,\sigma_{\lambda})\) in a G-equivariant manner. This means that the decomposition of \(U(\sigma_{\lambda}:\rho)\) is established in a global analytic fashion.