Primer of spherical harmonic analysis on SL 2 ( C )

[2] If we knew g ∈ SL2(C) had an expression g = kark, then gg∗ = ka2rk, where g∗ is conjugate transpose. This suggests how to determine components k, ar, k ′: by the spectral theorem for positive-definite hermitian operators gg∗, we can find k ∈ SU(2) and a diagonal matrix a2r of positive real eigenvalues, so that gg∗ = ka2rk. We claim that k′ = (kar) −1g ∈ SU(2). Indeed, ( (kar) −1g )( (kar) −1g )∗ = a−1 r k −1 (gg∗) k a−1 r = a −1 r k −1 ka2rk ∗ k a−1 r = 1