Completeness of the \(3j\)-symbols for the group \(\mathrm{SL}(2, \mathbb{C} )\)
The problem of decomposing the tensor product of two unitary principal series representations into irreducible representations was solved in a classical work of \textit{M. A. Naĭmark} [Transl., Ser. 2, Am. Math. Soc. 36, 101--136 (1964; Zbl 0132.11202); translation from Tr. Mosk. Mat. Obshch. 8, 121--153 (1959)]. The goal of the paper under review is to simplify some proofs given there: the authors offer a new proof of formula (2), see below. We use the following notations. For complex numbers, the overline means complex conjugation, for complex \(s\) and \(z\) we write \[ \widehat s=-1-\overline s, \ \ \ [z]^s=z^s \, \overline z ^{ \, \widehat s} \] The group \(\mathrm{SL}(2, \mathbb{C})\) consists of complex matrices \[ g=\left( \begin{array} {ll} a & b\\ c & d \end{array} \right), \ \ \ ad-bc=1. \] A unitary principal series representation \(T^{(s)}\) of the group \(\mathrm{SL}(2, \mathbb{C})\) acts on the functions \(\Phi(z)\) on the complex plane by \[ (T^{(s)}(g)\Phi)(z) = [d-bz]^{2s} \Phi\left(\frac{-c+az}{d-bz}\right), \eqno(1) \] here \[ s=-\frac{1}{2} +\frac{n}{4} +\frac{i\sigma}{2} \, , \, \, n \in \mathbb Z, \, \, \sigma \in \mathbb R, \ \ \text{so that} \ \ \widehat s=-\frac{1}{2} -\frac{n}{4} +\frac{i\sigma}{2} \, . \] The tensor product of such representations is reducible and decomposes into direct integral of them: \[ T^{(s_1)}\otimes T^{(s_2)} = \frac{1}{2}\sum \int_{\mathbb R} \, \rho(s) T^{(s)} d\sigma, \] where the summation is taken over \(n\) such that \(n_1+n_2+n\) is even, \[ \rho(s) = - \frac{(2s + 1)(2\widehat s + 1)}{4\pi^4} = \frac{1}{16\pi^4} \left( n^2+4\sigma^2 \right) \geqslant 0. \] The components \(\Phi_s(z)\) of this decomposition transformed by \(T^{(s)}\) are given by \[ \Phi_s(z)=\int W\left( \begin{array} {lll} s_1, & s_2, & s\\ z_1, & z_2, & z \end{array} \right) \Phi(z_1, z_2) \, dz_1 \, dz_2 \] where \[ W\left( \begin{array} {lll} s_1, & s_2, & s\\ z_1, & z_2, & z \end{array} \right)=[z_2-z_1]^{-2-s_1-s_2-s} [z-z_1]^{-s_1+s_2+s} [z-z_2]^{s_1-s_2+s} \] These functions \(W\) satisfy the following orthogonality relation \begin{align*} &\frac{1}{2}\sum \rho(s) d\sigma \int_{\mathbb C} \, \overline W\left( \begin{array} {lll} s_1, & s_2, & s\\ z'_1, & z'_2, & z \end{array} \right) W\left( \begin{array} {lll} s_1, & s_2, & s\\ z_1, & z_2, & z \end{array} \right) \, dz \nonumber \\ &=\delta(z_1-z'_1) \, \delta(z_2-z'_2)\tag{1} \end{align*} The authors offer a new proof of formula (2) -- only for \(n_1+n_2\) even. For that, they reduce the function \(W\) to its simplest variant, namely, to the function \(W\) with \(s_1=s_2=-1/2\) by an integral transform: \begin{align*} &W\left( \begin{array} {lll} s_1, & s_2, & s\\ z_1, & z_2, & z \end{array} \right)=\frac{(-1)^{s-\widehat s}}{a(-s, 1-s_1+s_2+s)} \times \\ \nonumber &\times \int_{\mathbb C} U\left( \begin{array} {lll} \ \ s_1, & \ \ s_2 \\ z_1, & z_2, & w \end{array} \right) W\left( \begin{array} {rrr} -1/2, & -1/2, & s\\ w, & z_2, & z \end{array} \right) \, dw \end{align*} where \begin{align*} &U\left( \begin{array} {lll} \ \ s_1, & \ \ s_2 \\ z_1, & z_2, & w \end{array} \right)= \frac{1}{\pi a(1+s_1-s_2)} \times \\ &\times [z_2-z_1]^{-1-s_1-s_2} [w-z_1]^{-1-s_1+s_2} [z_2-w]^{s_1-s_2} \end{align*} and \(a(\alpha)=\Gamma(1-\overline\alpha)/\Gamma(\alpha)\), \(a(\alpha,\beta, \ldots) = a(\alpha)a(\beta) \ldots\). Then they insert this formula into relation (2) and use some orthogonality for \(U\). It gives for \(W\) with \(s_1=s_2=-1/2\) the same formula as (2), let us denote it by (2'). Further this equation (2') is proving with help of suitable change of variables, Fourier transforms and so on. The text is accompanied by diagrams (planar oriented graphs), they correspond to products of functions of kind \([z - w]^{-\alpha}\), namely, this function \([z - w]^{-\alpha}\) moves to the line labelled by \(\alpha\) and going from the point \(w\) to the point \(z\), the function \(W\) moves to the triangle with vertices \(z_1, z_2, z\) and so on. Reviewer's remark. The authors use one and the same notation in two different senses: equally with the usual complex conjugate numbers (say, \(\overline \alpha\)), there are two complex parameters \(s\) and \(\overline s\), not necessarily complex conjugate. Therefore some formulas are unclear. I was forced to introduce the notation \(\widehat s\) for the parameter to avoid an ambiguity.
representations of the group \(\mathrm{SL}(2, \mathbb{C})\)
unitary principal series representations
orthogonality relations, the decomposition of the tensor product of representations
scientific article; zbMATH DE number 7390404