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\begin{corollary*}
  If $k = \CC$ then
  \[ \Gip\langle \chichar_V, \chichar_{\Ind_H^G W}
  \rangle_G = \langle \chichar_V|_H, \chichar_W
  \rangle_G \]
  In particular, $\chichar_{\Ind_H^G W} =
  \Gamma^*(\chichar_W)$.
\end{corollary*}

\begin{proof}[Proof of \nameref{frob_recip}]
  We'll prove $\GHom_G(V, \FXW(G, W)) \simeq
  \Hom_k(V, W)$ as \glspl{rep} of $H$ and then
  deduce the result by taking \Ginvpl[H]. Here the
  action of $H$ on the left hand side is given by
  \[ (h \cdot \theta)(v) = h \cdot \theta(v)
  \qquad \theta \in \GHom_G(V, \FXW(G, W)), v \in
  V, h \in H \]
  so $\GHom_G(V, \FXW(G, W))^H = \GHom_G(V,
  \FXW(G, W)^H) = \GHom_G(V, \Ind_H^G W)$.
  Note that this means
  \begin{align*}
    (h \cdot \theta)(v)(x)
    &= h(\theta(v)(xh)) \\
    h(\theta(h^{-1} x^{-1} v)(w))
  \end{align*}
  $\forall x \in G$ since $\theta$ is \Ginv. We
  can define a linear map
  \begin{align*}
    \label{lec15_l33_eq}
    \psi : \GHom_G(V, \FXW(G, W))
    &\to \Hom_k(V, W) \\
    \psi(\theta)(v)
    &= \theta(v)(e) \tag{$*$}
  \end{align*}
  We claim $\psi$ is an $H$-\gls{itt_map}. First
  we prove for $h \in H$, $\theta \in \GHom_G(V,
  \FXW(G, W))$, $v \in V$.
  \begin{align*}
    h \cdot (\psi(\theta))(v)
    &= h(\psi(\theta)(h^{-1} v)) \\
    &= h(\theta(h^{-1} v)(e)) \\
    &= (h \cdot \theta)(v)(e)
    &&\text{by \eqref{lec15_l33_eq} for $x = e$}
    \\
    &= \psi((h \cdot \theta))(v)
  \end{align*}
  and $\psi$ is $H$-equivariant.

  Given $\varphi \in \Hom_k(V, W)$ we can define
  \begin{align*}
    \varphi_G 
    &\in \Hom_k(V, \FXW(G, W)) \\
    \varphi_G(v)(x)
    &= \varphi(x^{-1} v)
    \qquad \forall x \in G, v \in V
  \end{align*}
  Then for all $g, x \in G$, $v \in V$,
  \[ \varphi_G(gv)(x) = \varphi(x^{-1} g v) =
  \varphi_G(v)(g^{-1} x) = (g \cdot
  \varphi_G(v))(x) \]
  i.e. $\varphi_G \in \GHom_G(V, \FXW(G, W))$. We
  can compute $\psi(\varphi_G)(v) =
  \varphi_G(v)(e) = \varphi(v)$, $\varphi \in
  \Hom_k(V, W), v \in V$ and
  \[ \psi(\theta)_G (v)(x) = \psi(\theta)(x^{-1}
  v) = \theta(x^{-1} v)(e) = x^{-1} \theta(v)(e) =
  \theta(v)(x) \]
  for $\theta \in \GHom_G(V, \FXW(G, W)), x \in G,
  v \in V$. Thus $\varphi \mapsto \varphi_G$ is an
  inverse to $\psi$.
\end{proof}

\begin{remark*}
  We could instead have computed
  $\chichar_{\Ind_H^G V}$ directly and shown that
  it is equal to $\indGamma^*(\chichar_W)$ and
  then deduced \nameref{frob_recip} from this when
  $k = \CC$.
\end{remark*}

\subsection{Mackey Theory}

This is the study of \glspl{rep} like $\Res_K^G
\Ind_H^G W$ for $H, k$ subgroups of $G$ and $W$ a
\gls{rep} of $H$. We can (and will) use it to
characterise when $\Ind_H^G W$ is \gls{irred} as a
\gls{rep} of $G$ (when $k = \CC$). If $H, K$ are
subgroups of $G$ then $H \times K$ acts on $G$ via
\[ (h, k) \cdot g = kgh^{-1} \]
An orbit of this action is called a \emph{double
coset}. We write
\[ KgH = \{kgh \st k \in K, h \in H\} \]
for the orbit containing $g$.

\begin{flashcard}[double-cosets-notation-defn]
\begin{definition*}
  $K \leftquot G / H = \cloze{\{KgH \st g \in
  G\}}$ \cloze{is the
  set of double cosets.}
\end{definition*}
\end{flashcard}
\vspace{-1em}

\glssymboldefn{leftgrep}{${}^g W, \rho, H$}{${}^g
W, \rho, H$}
For any \gls{rep} $(\rho, W)$ of $H$ and $g \in G$
we can define a \gls{rep} $({}^g \rho, {}^g W)$ bw
\begin{align*}
  \rho^g : {}^g H
  &\to \GL(W) \\
  ghg^{-1}
  &\mapsto \rho(h)
\end{align*}
where ${}^g H \defeq gHg^{-1} \le G$.

\begin{flashcard}[mackeys-restriction-formula-thm]
\begin{theorem*}[Mackey's restriction formula]
  \label{mack_rest_form}
  \cloze{If $G$ is a finite group with subgroups $H$ and
  $K$ and $W$ is a \gls{rep} of $H$ then
  \[ \Res_K^G \Ind_H^G W \simeq \bigoplus_{KgH \in
  K \leftquot H / H} \Ind_{K \cap \leftgrep {}^g
  H}^K \Res_{\leftgrep {}^g H \cap K}^H \leftgrep
  {}^g W \]
}
\end{theorem*}

\begin{proof}
  \cloze{
  Note that
  \begin{align*}
    \Ind_H^G W
    &= \FXW(G, W)^H \\
    &= \FXW \left( \coprod_{KgH \in K \leftquot G
    / H} KgH, W \right)^H \\
    &\cong \bigoplus_{KgH \in K \leftquot G / H}
    \FXW(KgH, W)^H
  \end{align*}
  as \glspl{rep} of $K$.
  
  So it suffices to show
  \[ \FXW(KgH, W)^H \simeq \FXW(K, \leftgrep {}^g
  W)^{K \cap \leftgrep {}^g H} \]
  as \glspl{rep} of $K$. We'll defer this to next
  time.
  }
\end{proof}
\end{flashcard}

\begin{corollary*}[Character version of Mackey
  restriction]
  If $\chichar$ is a \gls{char_rep} or a \gls{rep}
  of a $H$ then
  \[ (\Ind_H^G \chichar)|_K = \sum_{KgH \in K
  \leftquot G / H} \Ind_{\leftgrep {}^g H \cap
  K}^K (\leftgrep {}^g \chichar|_{\leftgrep {}^g H
  \cap K}) \]
\end{corollary*}
\vspace{-1em}

\textbf{Exercise:} Prove this corollary directly
using \glsref[char_rep]{characters}.

\begin{flashcard}[mackeys-irreducibility-criterion-coro]
\begin{corollary*}[Mackey's irreducibility
  criterion]
  \label{mack_irred_crit}
  If $H \le G$ and $W$ is a $\CC$-\gls{rep} of $H$
  then $\Ind_H^G W$ is \gls{irred} if and only if
  \begin{enumerate}[(i)]
    \item \cloze{$W$ is \gls{irred} as a \gls{rep}
      of $H$}
    \item \cloze{For each $g \in G \setminus H$, the two
      \glspl{rep} $\Res_{\leftgrep {}^g H \cap
      H}^{\leftgrep {}^g H} \leftgrep {}^g W$ and
      $\Res_{\leftgrep {}^g H \cap H}^H W$ of $H
      \cap \leftgrep {}^g H$ have no \gls{irred}
      \glspl{subrep} in common.}
  \end{enumerate}
\end{corollary*}

\begin{proof}
  \cloze{
  \begin{align*}
    \Gip\langle \chichar_{\Ind_H^G W},
    \chichar_{\Ind_H^G W} \rangle_G
    &= \langle \chichar_W, \chichar_{\Res_H^G
    \Ind_H^G W} \rangle_G
    &&\text{(\nameref{frob_recip})} \\
    &= \sum_{HgH \in H \leftquot G / H} \langle
    \chichar_W, \chichar_{\Ind_{H \cap \leftgrep
    {}^g H}^H \Res_{\leftgrep {}^g H \cap
    H}^{\leftgrep {}^g H} \leftgrep {}^g W}
    \rangle_H
    &&\text{(\nameref{mack_rest_form})} \\
    &= \sum_{HgH \in H \leftquot G / H} \langle
    \Res_{\leftgrep {}^g H \cap H}^H \chichar_W,
    \Res_{\leftgrep {}^g H \cap H}^{\leftgrep
    {}^g H} \chichar_{\leftgrep {}^g W} \rangle
    &&\text{(\nameref{frob_recip})}
  \end{align*}
  So $\Ind_H^G W$ is \gls{irred} if and only if
  $\RHS$ is $1$. The term for the double coset
  $HeH$ is $\langle \chichar_W, \chichar_W
  \rangle_H \ge 1$ and all the other terms are
  $\ge 0$ so irreducibility is equivalent to
  $\langle \chichar_W, \chichar_W \rangle_H = 1$
  and all otehr temrs are $0$.

  $\langle \chichar_W, \chichar_W \rangle_H = 1$
  if and only if condition (1).

  $\langle \Res_{H \cap \leftgrep {}^g H}^H
  \chichar_W, \Res_{\leftgrep {}^g H \cap
  H}^{\leftgrep {}^g H} \chichar_{\leftgrep {}^g
  W} \rangle = 0$ if and only if (ii) for $g$.}
\end{proof}
\end{flashcard}

Note for condition (ii) we only need to check for
a family of double cosets excluding $HeH = H$.

\begin{corollary*}
  If $H \normalsub G$ and $W$ is an \gls{irred}
  \gls{rep} of $H$ then
  \[ \text{$\Ind_H^G W$ is \gls{irred}} \iff
  \leftgrep {}^g \chichar_W \neq \chichar_W \qquad
  \forall g \in G \setminus H \]
  ($\leftgrep {}^g \chichar_W (ghg^{-1}) =
  \chichar_W (h)$).
\end{corollary*}

\begin{proof}
  Since $H \normalsub G$, $\leftgrep {}^g H = H$
  for all $g \in G$ and $\leftgrep {}^g W$ is
  \gls{irred} since $W$ is. So by
  \nameref{mack_irred_crit},
  \begin{align*}
    \text{$\Ind_H^G W$ is \gls{irred}}
    &\iff W \not\simeq \leftgrep{}^g W \qquad
    \forall g \in G \setminus H \\
    &\iff \chichar_W \neq \leftgrep{}^g \chichar_w
    \qquad \forall g \in G \setminus H \qedhere
  \end{align*}
\end{proof}

\begin{example*}
  \phantom{}
  \begin{enumerate}[(1)]
    \item $H = \langle r \rangle \simeq C_n$ the
      subgroup of rotations in $G = D_{2n}$. The
      \gls{irred} \glsref[char_rep]{characters} of
      $H$ are all of the form
      \[ \chichar_k(r^j) = e^{2\pi i j k / n} \]
      We see that $\Ind_H^G \chichar_K$ is
      \gls{irred} if and only if
      \[ \text{$\chichar_K(rj) \neq \chichar(r -
      j)$ for some $j$} \iff \text{$\chichar_K$ is
      not real valued} \]
    \item $G = S_n$, $H = A_n$. If $g \in S_n$ is
      a cycle type that splits in $A_n$ and
      $\chichar$ is an \gls{irred} \gls{char_rep}
      of $A_n$ taking different values on the two
      classes, then
      \[ \Ind_{A_n}^{S_n} \chichar \]
      is \gls{irred}.
  \end{enumerate}
\end{example*}
