I am searching for a integration formula
$ \int\limits_{GL_2(\mathbb{C})} \phi(g) d g = \int\limits_{SU(2)} \int\limits_{SU(2)} \int\limits_{M} \phi(k_1mk_2) \omega(m) d m d k_1 d k_2 ,$ where $M$ is the subgroup of diagonal matrices. I am in particular interested in an explicit form for the weight $\omega$ preferably with reference.