The character of the exceptional series of representations of SU(1,1) (Q2737831)

Let \((T,{\mathcal H}_\sigma(U))\) \((0<\sigma<1/2)\) denote the supplementary series of representations of \(G=SU(1,1)\), where \({\mathcal H}_\sigma(U)\) is the space of functions defined on the unit circle \(U\) with inner product \((\cdot,\cdot)_\sigma\). Let \((\cdot,\cdot)=(\cdot,\cdot)_{1/2}\) denote the inner product in \(L^2(U)\). We define the operator \(T_f\) of the group ring by \(T_fg=\int_Gf(u)T_ugdu\), and the operator \(\tilde T\) by \((T_fg,h)_\sigma=({\tilde T}_fg,h)\). In this paper the kernels of these operators are explicitly calculated by reducing the argument in the space of Fourier coefficients, and the character of \(T\) is determined. This approach is well-understood in the harmonic analysis of \(SU(1,1)\), for example, see \textit{M. Sugiura} [Unitary Representations and Harmonic Analysis, North-Holland (1990; Zbl 0697.22001)].