Solution
= Solution
By <Schur lemma>, $z\in Z$ acts on irreducible $V$ by a <scalar>. <Induced representation> shows that $V$ occurs in $\operatorname{Ind}_Z^G\Theta$ iff that <scalar> character is $\Theta$, and then with multiplicity $\dim V$.
= Solution
By <Schur lemma>, $z\in Z$ acts on irreducible $V$ by a <scalar>. <Induced representation> shows that $V$ occurs in $\operatorname{Ind}_Z^G\Theta$ iff that <scalar> character is $\Theta$, and then with multiplicity $\dim V$.