Let $K = SU(2) = \{ k[\alpha ,\beta] | \alpha ,\beta \in \mathbb{C}, |\alpha |^2 + |\beta |^2 = 1 \} $ with $$ k [ \alpha , \beta ] = \begin{pmatrix} \alpha & \beta \\ - \overline{\beta} & \overline{\alpha} \end{pmatrix}$$ Let $V_n$ denote the space of polynomials in one variable of degree at most $n$. We choose the set $\{ z^{l-q} : q\in\mathbb{Z},\ |q|\leq l\}$ as a basis for $V_{2l}$. Then the (irreducible and unitary) representations of $K$ are given by the formula $$ \sigma_l(k [\alpha , \beta]) z^{l-q} = ( \alpha z - \overline{\beta} )^{l-q} ( \beta z + \overline{\alpha})^{l+q} .$$ Let $ \Phi_{p,q}^l $ denote the coefficient of $z^{l-p}$ in the polynomial expansion of $\sigma_l(k [\alpha , \beta]) z^{l-q}$. i.e, $$ \sum_{|p|\leq l} \Phi_{p,q}^l(k[\alpha,\beta]) z^{l-p} = ( \alpha z - \overline{\beta} )^{l-q} ( \beta z + \overline{\alpha})^{l+q} ,\quad for \ |q|\leq l. $$

There is an explicit formula of $$ \int_K \Phi_{p,q}^l(k) \overline{\Phi_{p_1,q_1}^{l_1}(k)} dk$$ in the book 'Sum Formula for SL2 over Imaginary Quadratic Number Fields' by Lokvenec-Guleska. (https://dokumen.tips/documents/sum-formula-for-sl2-over-imaginary-quadratic-number-fields-za-sumirae-za-sl-2.html?page=23)

I have to solve the triple product $$ \int_K \Phi_{p,q}^l(k) \Phi_{p_1,q_1}^{l_1}(k) \Phi_{p_2,q_2}^{l_2}(k) dk.$$

I know how to compute this equation through brute force. However, my advisor doesn’t want me to include the complex calculation process in my paper. Is there a place where I can directly find this formula? Any suggestion would be helpful. Thanks!

In fact I have obtained the following formula: For any $a,b,c,d \in\mathbb{Z}$ and $k = k[\alpha,\beta]$, we have $$\int_K \alpha^{a} \bar{\alpha} ^{b} \beta^{c} \bar{\beta} ^{d} d k = \delta ( a=b, c=d ) \frac{a!c!}{(a+c+1)!},$$ where $\delta$ denotes the Kronecker symbol.