Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 324 / 1 / b / ii / Solution

Put $x=g_{0}g$ in the definition of $U(g_0)|\alpha ij⟩$ and useOne obtainsThus every $d_{\alpha }^2$-dimensional subspace ${\mathcal{H}}_{\alpha }$ is invariant; the left regular representation acts as $M_{\alpha }(g_0)$ on the first matrix index and as the identity on the second.

Let $P_{\alpha }$ project onto ${\mathcal{H}}_{\alpha }$. Since the subspace is invariant under every $U(g)$, $P_{\alpha }$ commutes with those unitaries. Also $|gH⟩=U(g)|H⟩$. Thereforewhich is independent of the coset representative $g$ and depends only on $H$.