Let $G$ be a group and $\pi:G\to B(H)$ be its irreducible unitary representation (one can endow $G$ with topology and claim that $\pi$ is continuous in some sense, this doesn't matter). For a given $x\in H$ the function $$ f(t)=\langle\pi(t)x,x\rangle,\qquad t\in G $$ is positive definite, but not all positive definite functions can be represented in this form (with irreducible $\pi$).
Question: is there a name for these functions (generated by irreducible representations)? What do people call them?