Is it true that $\left(H^\dagger H\right)^2$ is invariant under $U\left(1\right) \times SU\left(2\right)$ where $H$ is the Higgs field $(1,2,1/2)$?
Does this invariance imply that its hypercharge is invariant under $U\left(1\right)$ and its spin is invariant under $SU\left(2\right)$? .
$$H = [H_+, H_0]$$
$$H_+ = [H_-]$$
$$H_0 = [?]$$
$$H^\dagger H = [H_-][H_+] + [?][H_0]$$