I am currently reading the book Introduction to Unconventional Superconductivity by V. P. Mineev and K. V. Samokhin. In Equation (1.6), the interaction potential $V$ is expanded in terms of spherical harmonics as follows:
$$V\left(\mathbf{k}-\mathbf{k}^{\prime}\right)=\sum_{l=0}^{\infty}V_{l}\left(k,k^{\prime}\right)\sum_{m=-l}^{l}Y_{lm}\left(\hat{\mathbf{k}}\right)Y^{*}_{lm}\left(\hat{\mathbf{k}}^{\prime}\right).$$
This appears to be a general expression, assuming the potential is a function of the difference of momenta $\mathbf{q}=\mathbf{k}-\mathbf{k}^{\prime}$. However, I am struggling to understand how this equation is derived.
From my understanding, the potential could be expanded as
$$V\left(\mathbf{k},\mathbf{k}^{\prime}\right)=\sum_{lm,l^{\prime}m^{\prime}}V_{lm,l^{\prime}m^{\prime}}\left(k,k^{\prime}\right)Y_{lm}\left(\hat{\mathbf{k}}\right)Y_{l^{\prime}m^{\prime}}\left(\hat{\mathbf{k}}^{\prime}\right),$$
where
$$V_{lm,l^{\prime}m^{\prime}}=\int d\Omega\int d\Omega^{\prime}Y_{lm}^{*}\left(\hat{\mathbf{k}}\right)Y_{l^{\prime}m^{\prime}}^{*}\left(\hat{\mathbf{k}}^{\prime}\right)V\left(\mathbf{k}-\mathbf{k}^{\prime}\right).$$
In order to arrive at the equation given in the book, it seems necessary that
$$V_{lm,l^{\prime}m^{\prime}} = \delta_{ll^{\prime}}\delta_{mm^{\prime}}V_{l}(k,k^{\prime}).$$
However, I am unsure how to proceed from here. I would appreciate any guidance or insights into this derivation.
