= Solution
The <Mackey's theorem>, or Mackey restriction formula, says that if $H,K\leq G$, $W$ is a complex $K$-representation, and $D$ is a set of representatives for the <double cosets> $H\backslash G/K$, then
$$
\boxed{
\operatorname{Res}_H^G\operatorname{Ind}_K^G W
\cong
\bigoplus_{x\in D}
\operatorname{Ind}_{H\cap xKx^{-1}}^H
\operatorname{Res}_{H\cap xKx^{-1}}^{xKx^{-1}}({}^xW).}
$$
Here ${}^xW$ is the representation of the <conjugate subgroup> $xKx^{-1}$ defined by
$$
{}^xW(xkx^{-1})=W(k).
$$
For the proof, use the group-algebra model
$$
\operatorname{Ind}_K^G W
=\mathbb C[G]\otimes_{\mathbb C[K]}W.
$$
The partition of $G$ into double cosets gives the direct-sum decomposition of $(H,K)$-bimodules
$$
\mathbb C[G]
=\bigoplus_{x\in D}\mathbb C[HxK].
$$
Put $L_x=H\cap xKx^{-1}$. For each $x$, define
$$
T_x:
\mathbb C[H]\otimes_{\mathbb C[L_x]}{}^xW
\longrightarrow
\mathbb C[HxK]\otimes_{\mathbb C[K]}W,
\qquad
h\otimes w\longmapsto hx\otimes w.
$$
If $\ell=xkx^{-1}\in L_x$, then
$$
T_x(h\ell\otimes w)
=h\ell x\otimes w
=hxk\otimes w
=hx\otimes kw
=T_x(h\otimes\ell w),
$$
so $T_x$ is well defined. It is visibly $H$-equivariant and surjective, and choosing representatives for $H/L_x$ shows that both sides have dimension $[H:L_x]\dim W$; hence it is an isomorphism. Summing the $T_x$ proves the formula.
The <Frobenius reciprocity> theorem states that for an $H$-representation $U$ and a $G$-representation $V$,
$$
\boxed{
\operatorname{Hom}_G(\operatorname{Ind}_H^G U,V)
\cong
\operatorname{Hom}_H(U,\operatorname{Res}_H^G V).}
$$
For the corresponding <character of a representation>, this is
$$
\boxed{
\langle\operatorname{Ind}_H^G\chi,\psi\rangle_G
=\langle\chi,\operatorname{Res}_H^G\psi\rangle_H.}
$$
Back to article page
