Setting Let $V$ transform according to the direct representation of the unitary group $U(d)$. I have a polynomial on $P:V^k \times (V^\star)^l\rightarrow \mathbb R$ where $V^\star$ is the conjugate representation.
Assume that $f$ is invariant such that $f(uv,u^\dagger w)=f(v,w)$ can I deduce that $f$ is a function of the elementary invariants $(v_i|w_i)$ ?
This corresponds to the first fundamental theorem of invariants, which is true for the groups $GL(d)$, $Sp(d)$, $O(d)$ [C. Procesi, Lie Groups]. Is this also true for $U(d)$ ?
If instead I assume that $f(uv,u^\dagger v)=f(v,v)$ for all $v\in V$ and $u\in U(d)$, does this imply that $f$ is a function of the invariants $(v_i|v_j)$ ?