Determination of the Harish-Chandra \(C\)-function for \(SU(n, 1)\) and its application to the construction of the composition series (Q1817244)

Let \(G\) be a reductive Lie group of the Harish-Chandra class, \(G= \text{KAN}\) its Iwasawa decomposition, \(M\) the centralizer in \(K\), \({\mathfrak A}\) the Lie algebra of \(A\), \(\widehat{K}\) (resp. \(\widehat{M})\) the set of equivalence classes of irreducible unitary representations of \(K\) (resp. \(M\)). For \(\tau\in \widehat{K}\), \(\sigma\in \widehat{M}\), \([\tau:\sigma]\) the multiplicity of \(\sigma\in \widehat{M}\) occurring in \(\tau|_M\), \(\widehat{M}(\tau)= \{\sigma\in\widehat{M}\); \([\tau:\sigma]=1\}\). For \(g\in G\), let \(g=\kappa(g) \exp H(g)n(g)\), \((\kappa(g)\in K\), \(H(g)\in{\mathfrak A}\), \(n(g)\in N)\). For \(\tau\in \widehat{K}\) and \(\lambda\in{\mathfrak A}_{\mathbb{C}}^*\) the integral \[ C_\tau(\lambda)= \int_{\overline{N}} \tau(\kappa(\overline{n}))^{-1} e^{- (\lambda+\rho) (H(\overline{n}))} d\overline{n} \] is known as the Harish-Chandra \(C\)-function associated with \(\tau\in \widehat{K}\). When \(G= \text{SU}(n,1)\), the case considered by the authors, \([\tau,\sigma]=0\) or 1 for any \(\tau\in\widehat{K}\) and \(\sigma\in \widehat{M}\). For \(\tau\in \widehat{K}\) write \(\widehat{M}(\tau)= \{\sigma\in \widehat{M}\), \([\tau,\sigma]=1\}\). Let \(\tau\in \widehat{K}\), \(\sigma\in \widehat{M}(\tau)\), and \(V_\tau\), \(H_\sigma\) the corresponding representation spaces. Choose \(P_\sigma(\tau)\in \Hom_M(V_\tau,H_\sigma)\) such that \(P_\sigma(\tau) P_\sigma(\tau)^*= \sigma(1)\). Since \(P_\sigma(\tau) C_\tau(\lambda)\in \Hom_M(V_\tau,H_\sigma)\) and \(\dim \Hom_M(V(\tau), H(\sigma))=1\), there exists a constant \(C_\tau(\sigma:\lambda)\) such that \(P_\sigma(\tau) C_\tau(\lambda)= C_\tau(\sigma:\lambda) P_\sigma(\tau)\). The meromorphic function \(C_\tau(\sigma:\lambda)\) is the Harish-Chandra \(C\)-function associated with \(\tau\) and \(\sigma\). Its importance for harmonic analysis on \(G\) is well-known. It is connected with intertwining operators. It carries information regarding irreducibility and unitarizability of induced representations. \textit{K. D. Johnson} [Advances Math. 14, 346-363 (1974; Zbl 0293.43010)] and \textit{K. D. Johnson} and \textit{N. R. Wallach} [Bull. Am. Math. Soc. 78, 1053-1059 (1972; Zbl 0257.22016); Trans. Am. Math. Soc. 229, 137-173 (1977; Zbl 0349.43010)] showed the unitarizability and square integrability of the irreducible components of the spherical principal series of rank 1 classical groups. By an analysis of the \(K\)-spectrum \textit{H. Kraljevic} [Trans. Am. Math. Soc. 221, 433-448 (1976; Zbl 0365.22012)], and A. U. Klimyk and A. M. Gavrilik have obtained the composition series of the full non-unitary principal series of \(\text{SU}(n,1)\). In the present paper the authors, in the case \(G=\text{SU} (n,1)\), obtain the explicit formula for the Harish-Chandra \(C\)-function and apply it to the description of the composition series. Its application to unitarizability and square integrability is postponed to a future publication.