On special values of Selberg type zeta functions on \(SU(1,q+1)\) (Q1184406)

The aim of the paper under review is to show that special values of Selberg type zeta functions split into a product of a suitable ``regulator'' times a ``period''. \S1 recalls the necessary facts from the theory of representations of \(G=SU(1,q+1)\) \((q>0)\). In particular, the connection between the Harish-Chandra parametrization of square integrable representations of \(G\) and the parametrization of Kraljevic is established. The latter parametrization enables the author to prove a Paley-Wiener theorem for the spherical Fourier transform on \(G\) (see \S2). Selberg type zeta functions \(Z_{\Gamma,r}(\chi,s)\) for \(G\) with one- dimensional \(K\)-type are considered in \S3. The underlying discrete subgroup \(\Gamma\subset G\) is assumed to be torsion-free and cocompact, \(\chi\) is a finite-dimensional unitary representation of \(\Gamma\), and the integer \(r\) describes the \(K\)-type. The Selberg trace formula yields the meromorphic continuation of \(Z_{\Gamma,r}(\chi,s)\), the location and residues of its poles and its functional equation. The main results of the author state that \(Z_{\Gamma,r}(\chi,n)\) and a certain residue of \(Z_{\Gamma,0}(\chi,s)\) split into a product \(R\cdot P\) of a suitable regulator times a period (see Theorems 4.4, 4.5). In \S5, Dedekind zeta functions are considered. The residue of \(\zeta_ K(s)\) at \(s\equiv 1\) factors into a product \(R\cdot P\) which is analogous to the formula for the Selberg zeta function. The analogy is made visible by the appearance of certain functional determinants which form the essential parts of the regulators. The striking analogy is discussed more fully in \S6. In addition, the special case of the Selberg zeta function for a cocompact subgroup of \(SL_ 2(\mathbb{R})\) is written out in \S6 and the corresponding factorization theorems for the special value and residues are formulated in Theorems 6.1, 6.2. (The reviewer wants to point out that these results are closely related with the following work: \textit{A. A. Bejlinson} and \textit{Yu. I. Manin}: The values of the Selberg zeta function at integral points, Funkts. Anal. Prilozh. 21, 68-69 (1987; Zbl 0601.30049), transl. in Funct. Anal. Appl. 21, 58-60 (1987)).