The completeness relation for the spherical harmonics is:
$$\sum_{l=0}^{\infty} \sum_{m=-l}^{l} Y_{lm}^*\left(\theta_1,\phi_1\right)Y_{lm}\left(\theta_2,\phi_2\right) = \delta\left(\phi_1-\phi_2\right)\delta\left(\theta_1-\theta_2\right)$$
Any ideas what the same sum without the conjugation would evaluate to? I.e. what is:
$$\sum_{l=0}^{\infty} \sum_{m=-l}^{l} Y_{lm}\left(\theta_1,\phi_1\right)Y_{lm}\left(\theta_2,\phi_2\right) =\ ?$$