The integral kernel in the Kuznetsov sum formula for \(\text{SU}(n+1,1)\). II: The case of one dimensional \(K\)-types (Q1884536)

In the context of connected noncompact semisimple Lie groups of rank one, a Kuznetsov sum formula was determined by \textit{R. J. Miatello} and \textit{N. R. Wallach} [J. Funct. Anal. 93, 171--206 (1990; Zbl 0711.11023)]. It relates spectral data concerning automorphic forms to geometric data concerning the intersection of a discrete subgroup of finite covolume with the big cell in the Bruhat decomposition. Test functions on the spectral and on the geometric sides are linked by an integral transformation. The \(\tau\)-function appears in the kernel of this transformation. In the case of trivial \(K\)-types, explicit formulas for the \(\tau\)-function were determined by Miatello and Wallach [loc. cit.] for \(SO(n+1,1)\) and \(SU(2,1)\), and by the author in Part I of this paper [Indag. Math. 12, 83--101 (2001; Zbl 1039.11032)] for \(SU(n+1,1)\). The paper under review determines an explicit formula for the \(\tau\)-function in the case of the group \(SU(n+1,1)\) and one-dimensional \(K\)-types. The computation relies on the solution of complicated recurrence relations for the coefficients appearing in a series expansion of the \(\tau\)-function.