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\begin{document}

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\section*{induced representation}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{representation_theory}{}\paragraph*{{Representation theory}}\label{representation_theory}

[[!include representation theory - contents]]

\hypertarget{induced_modules}{}\section*{{Induced modules}}\label{induced_modules}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{TraditionalFormulation}{Traditional formulation}\dotfill \pageref*{TraditionalFormulation} \linebreak
\noindent\hyperlink{InducedRepresentations}{Induced representations}\dotfill \pageref*{InducedRepresentations} \linebreak
\noindent\hyperlink{TraditionalFormulationExplanation}{More exposition}\dotfill \pageref*{TraditionalFormulationExplanation} \linebreak
\noindent\hyperlink{TraditionalDetailed}{Detailed description}\dotfill \pageref*{TraditionalDetailed} \linebreak
\noindent\hyperlink{GeneralAbtractDefinition}{General abstract formulation in homotopy type theory}\dotfill \pageref*{GeneralAbtractDefinition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{BrauerInductionTheorem}{Brauer induction theorem}\dotfill \pageref*{BrauerInductionTheorem} \linebreak
\noindent\hyperlink{unitarity}{Unitarity}\dotfill \pageref*{unitarity} \linebreak
\noindent\hyperlink{adjoint_of_induced_bundle_construction}{Adjoint of induced bundle construction}\dotfill \pageref*{adjoint_of_induced_bundle_construction} \linebreak
\noindent\hyperlink{examples_and_applications}{Examples and Applications}\dotfill \pageref*{examples_and_applications} \linebreak
\noindent\hyperlink{regular_representation}{Regular representation}\dotfill \pageref*{regular_representation} \linebreak
\noindent\hyperlink{HeckeAlgebra}{Centralizer algebra / Hecke algebra}\dotfill \pageref*{HeckeAlgebra} \linebreak
\noindent\hyperlink{zuckerman_functors_coinduction_on_harishchandra_modules}{Zuckerman functors: Coinduction on Harish-Chandra modules}\dotfill \pageref*{zuckerman_functors_coinduction_on_harishchandra_modules} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{ReferencesTraditionalFormulation}{Traditional formulation}\dotfill \pageref*{ReferencesTraditionalFormulation} \linebreak
\noindent\hyperlink{general_abstract_formulation}{General abstract formulation}\dotfill \pageref*{general_abstract_formulation} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

Given a [[group]] $G$ with [[subgroup]] $H \hookrightarrow G$ and a [[representation]] of $H$, there is canonically induced a representation of $G$: the \emph{induced representation}.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

We give an exposition of the

\begin{itemize}%
\item \hyperlink{TraditionalFormulation}{Traditional formulation}

\end{itemize}
of induced representations. Then we provide a

\begin{itemize}%
\item \hyperlink{GeneralAbtractDefinition}{General abstract formulation in homotopy type theory}

\end{itemize}
that refines the notion to [[∞-action|∞-representations]] of [[∞-groups]] equipped with any additional geometric structure.

\hypertarget{TraditionalFormulation}{}\subsubsection*{{Traditional formulation}}\label{TraditionalFormulation}

\hypertarget{InducedRepresentations}{}\paragraph*{{Induced representations}}\label{InducedRepresentations}

Every [[subgroup]]-inclusion $H \overset{\iota}{\hookrightarrow} G$ induces a [[restricted representation]]-functor between the corresponding [[categories of representations]]

\begin{displaymath}
Rep(G)
    \overset{\iota^\ast }{\longrightarrow}
  Rep(H)
\end{displaymath}
which simply [[forgetful functor|forgets]] the full $G$-[[action]] on a given $G$-representation $V$ and remembers only the action of the subgroup $H$.

\begin{defn}
\label{InducedRepresentationsAsLeftAdjointToRestriction}\hypertarget{InducedRepresentationsAsLeftAdjointToRestriction}{}
\textbf{(left-induced representations as [[left adjoint]] to [[restricted representations]])}

If the [[restricted representation|restriction functor]] $\iota^\ast$ has a [[left adjoint]] (which is usually the case, but depends on which exact flavour of [[groups]] and of their [[category of representations]] one considers), then this is called the functor assigning \emph{left-induced representations}, often just \emph{induced representations}, for short:

\begin{displaymath}
Rep(G)
    \underoverset
     {\underset{\iota^\ast}{\longrightarrow}}
     {\overset{ind_H^G}{\longrightarrow}}
     {\bot}
  Rep(H)
\end{displaymath}
\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
This is directly analogous to [[extension of scalars]] $\dashv$ [[restriction of scalars]].

\end{remark}
With given flavour of [[groups]] and their [[category of representations]] specified, it is typically immediate to give explicit formulas for left induced representations:

\begin{example}
\label{InductionOfFiniteDimensionalRepresentationsOfFiniteGroups}\hypertarget{InductionOfFiniteDimensionalRepresentationsOfFiniteGroups}{}
\textbf{(induction of [[finite-dimensional vector space|finite-dimensional]] [[linear representations]] of [[finite groups]])}

In the case that $G$ (and hence $H$) is a [[finite group]] and $Rep(G)$ is the [[category of representations|category of]] [[finite-dimensional vector space|finite-dimensional]] representations over some [[ground field]] $k$., the general induced representation functor (Def. \ref{InducedRepresentationsAsLeftAdjointToRestriction}) exists and is explicitly given by forming the [[tensor product of representations]] with the $H$-[[permutation representation]] spanned by the underlying [[set]] of $G$:

\begin{displaymath}
ind_H^G
  \;\colon\;
  V 
    \mapsto 
  k[G] \otimes_{H} V
  \,.
\end{displaymath}
For example, if $V = \mathbf{1}$ is the [[trivial representation]] of [[dimension]] 1 then its induced representation is the basic [[permutation representation]] spanned by the [[coset]]-space $G/H$:

\begin{displaymath}
ind_H^G
  \left( 
    \mathbf{1}
  \right)
  \;=\;
  k[G/H]
  \,.
\end{displaymath}
See at \emph{[[induced representation of the trivial representation]]} for more.

\end{example}
See e.g. \hyperlink{tomDieck09}{tomDieck 09, Chapter 4}.



\hypertarget{TraditionalFormulationExplanation}{}\paragraph*{{More exposition}}\label{TraditionalFormulationExplanation}

Suppose a [[Lie group]] $G$ acts smoothly and transitively on a [[smooth manifold]] $M$. The [[stabilizer subgroup]] of a given point $x \in M$ is then a Lie subgroup $H \subseteq G$, and

\begin{displaymath}
M \cong G/H
  \,,
\end{displaymath}
is the [[coset]] space.

Starting from this, there's a recipe taking any [[representation]] $s$ of $H$ on a [[vector space]] $V$ and turns it into a [[vector bundle]] $E$ over $M$ --- called the \emph{induced bundle}. Moreover, the group $G$ [[action|acts]] on this [[bundle]], and the [[projection]]

\begin{displaymath}
\pi : E \to M
\end{displaymath}
is compatible with the [[action]] of $G$:

\begin{displaymath}
\pi(g e) = g \pi(e) .
\end{displaymath}
Hence $E$ is a \textbf{$G$-equivariant} vector bundle over $M$.

The `process' described is actually a [[functor]], the \textbf{induction functor}.

There's a [[category]]

\begin{displaymath}
Rep(H)
\end{displaymath}
of linear representations of $H$, and a category

\begin{displaymath}
Vect(M,G)
\end{displaymath}
of $G$-equivariant vector bundles over $M$. The induced bundle construction gives a functor

\begin{displaymath}
L: Rep(H) \to Vect(M,G)
\end{displaymath}
But, if you think about it, you'll notice there's also a functor going back the other way:

\begin{displaymath}
R: Vect(M,G) \to Rep(H)
\end{displaymath}
If you give me a $G$-equivariant vector bundle $E$ over $M$, I can take its fiber over your favorite point $x$, and I get a vector space --- and this becomes a representation of the stabilizer group $H$, thanks to how $G$ acts on $E$.

This functor is  than the induced bundle construction!

Whenever we have functors going both ways between two categories, we should suspect that they're [[adjoint functor|adjoints]]. The simpler functor often amounts to `forgetting' something. This forgetful functor is usually the  adjoint. It's partner going the other way, the  adjoint, usually involves `constructing' something instead of `forgetting' something.

And indeed, that's what's happening here! Technically, this is to say that

\begin{displaymath}
hom(L V, F) \cong hom(V, R F)
\end{displaymath}
Here $V$ is a representation of $H$ --- note abuse of notation in calling it $V$, which is the name for the vector space on which $G$ acts, instead of the more pedantic full name for a representation, which is something like $s: G \to GL(V)$.

Similarly, $F$ is a $G$-equivariant vector bundle over $M$ --- and this should be something like $\pi : F \to M$, or something even more long-winded that gives a name to how $G$ acts on $F$ and $M$.

$L V$ is the induced bundle corresponding to $V$.

$R F$ is the fiber of $F$ over your favorite point $x$, which becomes a representation of $G$.

And this:

\begin{displaymath}
hom(L V, F) \cong hom(V, R F)
\end{displaymath}
says that $G$-equivariant vector bundle maps from $L V$ to $F$ are in natural 1-1 correspondence with intertwining operators from $V$ to $R F$.

Now, whenever you see any sort of `forgetful' process, you should wonder if it has a left adjoint, a construction which in some loose sense is the `reverse' of forgetting. Why? Because these left adjoints tend to be important.

Endowed with this heuristic, as soon as you see there's a rather obvious `forgetful' process that takes a $G$-equivariant vector bundle over $M$ and gives a representation of $H$ on the fiber over $x \in M$, you will seek the `reverse' process --- and then you'll rediscover the induced bundle construction!

And why is this so great? Well, there's also a process that takes any representation of $G$ and [[restricted representation|restricts]] it to a representation of $H$:

\begin{displaymath}
R': Rep(G) \to Rep(H)
\end{displaymath}
And this too, has a [[left adjoint]]:

\begin{displaymath}
L' : Rep(H) \to Rep(G)
\end{displaymath}
which is called the \emph{induced representation}.

\hypertarget{TraditionalDetailed}{}\paragraph*{{Detailed description}}\label{TraditionalDetailed}

Given a [[group]] $G$ with a subgroup $H$, and a [[representation]] $s$ of $H$ on a vector space $V$, we define a left [[action]] of $H$ on the [[product]] $G\times V$ by $h\cdot (g, v) = (g h^{-1}, s(h)v)$. We write $[(g,v)]$ for the orbit, or equivalence class, that contains $(g,v)$.

We then define $E = (G\times V)/H$ as the set of orbits of that [[action]] of $H$, $M = G/H$ as the set of left cosets of $H$, and the projection $\pi: E\to M$ by $\pi ([(g,v)]) = g H$, where of course it makes no difference if we re-describe the orbit $[(g,v)]$ as $[(g h^{-1}, s(h)v]$ for any $h\in H$ because $(g h^{-1}) H = g H$.

For each $x\in M$, choose $g$ to be any element of $G$ such that $x = g H$. Define $E_x = \pi^{-1}(x)$, and $\phi_g:V\to E_x$, $\phi_g(v) = [(g,v)]$.

The map $\phi_g$ is onto: for any $[(k,w)]\in E_{(g H)} = \pi^{-1}(g H)$, we have $k=g h_1^{-1}$ for some $h_1\in H$, so $k^{-1} g\in H$, $(k^{-1} g)\cdot (g, s(g^{-1} k)w) = (k,w)$, so $\phi_g(s(g^{-1} k)w) = [(g, s(g^{-1} k)w)] = [(k,w)]$.

The map $\phi_g$ is one-to-one: if $\phi_g(v) = \phi_g(w)$, then $[(g,v)]=[(g,w)]$, so for some $h_1\in H$, we have $h_1\cdot (g,v) = (g,w)$, or $(g h_1^{-1}, s(h_1)v) = (g,w)$; equating the first coordinates requires $h_1=e$, and $s$ is a representation so $s(e)=1_V$, and $v=w$.

Since $\phi_g$ is a bijection between $E_x$ and the vector space $V$, we can make $E_x$ into a vector space by defining $\alpha p + \beta q \equiv \phi_g(\alpha \phi_g^{-1}(p) + \beta \phi_g^{-1}(q))$, for all $\alpha, \beta \in \mathbb{R}, p, q \in E_x$. But is this independent of our choice of $g$? If we chose $g h$ instead of $g$, we'd have $\phi_{g h}(v) = [(g h,v)] = [(g, s(h)v)] = \phi_g(s(h)v)$, so $\phi_{g h}=\phi_g\circ s(h)$, and $\phi_{g h}^{-1}=s(h^{-1})\circ \phi_g^{-1}$. Then:

$\phi_{g h}(\alpha \phi_{g h}^{-1}(p) + \beta \phi_{g h}^{-1}(q)) =
(\phi_g\circ s(h))(\alpha (s(h^{-1})\circ \phi_g^{-1})(p) + \beta (s(h^{-1})\circ \phi_g^{-1})(q)) = \phi_g(\alpha \phi_g^{-1}(p) + \beta \phi_g^{-1}(q))$

in agreement with our original definition.

We define the action of $G$ on $E$ by $g_1\cdot [(g,v)] = [(g_1 g,v)]$, or in other words $g_1\cdot \phi_g(v) = \phi_{g_1 g}(v)$. We then have:

$\pi(g_1\cdot [(g,v)]) = \pi[(g_1 g,v)] = (g_1 g) H = g_1\cdot (g H) = g_1\cdot \pi([(g,v)])$

That is, $\pi$ is a $G$-morphism. This also means that the action maps fibers to fibers, $g_1:E_{(g H)}\to E_{g_1\cdot (g H)}$. What's more, the action of $g_1$ restricted to the fiber $E_{(g H)}$ is $\phi_{g_1 g}\circ \phi_g^{-1}$, passing from $E_{(g H)}\to V \to E_{g_1\cdot (g H)}$, and this is linear simply by virtue of the way we've defined the vector space operations on the $E_x$.

We get a representation $r$ of $G$ on the vector space $\Gamma(E)$ of sections of the bundle $E$ by:

$(r(g_1)f)(x) = g_1\cdot f(g_1^{-1}\cdot x)$

\hypertarget{GeneralAbtractDefinition}{}\subsubsection*{{General abstract formulation in homotopy type theory}}\label{GeneralAbtractDefinition}

We formulate induction and coinduction of representations abstractly in [[homotopy type theory]]. (Hence the following is automatically the [[(∞,1)-category theory]]-version, which in parts is sometimes referred to as \emph{[[cohomological induction]]}.)

Let $\mathbf{H}$ be an ambient [[(∞,1)-topos]]. By the discussion at \emph{[[∞-action]]}, for $G \in Grp(\mathbf{H})$ a [[group object in an (∞,1)-category|group object]] in $\mathbf{H}$, hence an [[∞-group]], the [[slice (∞,1)-topos]] $\mathbf{H}_{/\mathbf{B}G}$ over its [[delooping]] is the [[(∞,1)-category]] of $G$-[[∞-actions]]

\begin{displaymath}
Act(G) \simeq \mathbf{H}_{/\mathbf{B}G}
  \,.
\end{displaymath}
(A genuine \emph{[[∞-representation]]/[[∞-module]]} over $G$ may be taken to be a an abelian $\infty$-group object in $Act(G)$, but we can just as well work in the more general context of possibly non-linear representations, hence of actions.)

Accordingly, for $f \colon H \to G$ a homomorphism of [[∞-groups]], hence for a morphism $\mathbf{B}f \colon \mathbf{B}H \to \mathbf{B}G$ of their [[deloopings]], there is the corresponding [[base change geometric morphism]]

\begin{displaymath}
(\sum_f \dashv f^* \dashv \prod_f)
  \colon
  Act(H)
   \stackrel{\overset{\sum_f}{\to}}{\stackrel{\overset{f^*}{\leftarrow}}{\underset{\prod_f}{\to}}}
  Act(G)
  \,.
\end{displaymath}
Here

\begin{itemize}%
\item the [[inverse image]]/[[(∞,1)-pullback]] functor $f^*$ produces the ``[[restricted representation|restricted]]'' $\infty$-representations along $f$;


\item the [[dependent sum]] $\sum_f$ is the \emph{induced representation} ∞-functor;


\item the [[dependent product]] $\prod_f$ is the \emph{coinduced representation} ∞-functor.



\end{itemize}
 For the case of [[permutation representations]] of [[discrete groups]] this perspective is made explicit in (\hyperlink{Lawvere69}{Lawvere 69, p. 14}, \hyperlink{Lawvere70}{Lawvere 70, p. 5}).

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{BrauerInductionTheorem}{}\subsubsection*{{Brauer induction theorem}}\label{BrauerInductionTheorem}

The [[Brauer induction theorem]] says that over the [[complex numbers]] the [[virtual representations]] of a [[finite group]] are all virtual combinations of [[induced representations]] of 1-dimensional representations.

\hypertarget{unitarity}{}\subsubsection*{{Unitarity}}\label{unitarity}

Beware! The chain of reasoning in this subsection is not complete, and I'm not confident that it's entirely correct. I'm posting it half-finished in the hope that many hands will make lighter (and more accurate) work.

We discuss that [[unitary representations]] induce again unitary representations.

(This is for instance relevant in applications to [[physics]], such as in the study of [[unitary representation of the Poincaré group]].)

Let's say $V$ has an inner product, $\lang \cdot, \cdot \rang$, and $s$ is a unitary representation. We can define an inner product on $E_x$ by $\lang \lang p, q \rang \rang \equiv \lang \phi_g^{-1}(p), \phi_g^{-1}(q) \rang$. This definition is independent of our choice of $g$: if we chose $g h$ instead, we'd have

\begin{displaymath}
\lang \lang p, q \rang \rang = \lang \phi_{g h}^{-1}(p), \phi_{g h}^{-1}(q) \rang = \lang s(h^{-1}) \circ \phi_g^{-1}(p), s(h^{-1}) \circ \phi_g^{-1}(q) \rang = \lang \phi_g^{-1}(p), \phi_g^{-1}(q) \rang.
\end{displaymath}
To be really thorough, we should verify that $\lang \lang \cdot, \cdot \rang \rang$ is in fact an inner product, but this should follow directly from our definition of the vector space operations on $E_x$.

Now we need to show that the action of any $g_1 \in G$ on the fiber $E_{(g H)}$ is unitary:

\begin{displaymath}
\lang \lang g_1 \cdot p, g_1 \cdot q \rang \rang = \lang \lang \phi_{g_1 g} \circ \phi_g^{-1}(p), \phi_{g_1 g} \circ \phi_g^{-1}(q) \rang \rang = \lang \phi_{g_1 g}^{-1} \circ \phi_{g_1 g} \circ \phi_g^{-1}(p), \phi_{g_1 g}^{-1} \circ \phi_{g_1 g} \circ \phi_g^{-1}(q) \rang = \lang \phi_g^{-1}(p), \phi_g^{-1}(q) \rang = \lang \lang p, q \rang \rang.
\end{displaymath}
Finally, we need to define an inner product on $\Gamma(E)$, and show that the representation $r$ is unitary. If we had a $G$-invariant measure $\mu$ on $G/H$, we could define the inner product of two sections of $f$ and $f'$ of $E$ to be

\begin{displaymath}
\int \lang \lang f(x), f'(x) \rang \rang \; d\mu(x).
\end{displaymath}
We would then have

\begin{displaymath}
\int \lang \lang (r(g_1)f)(x), (r(g_1)f')(x) \rang \rang \; d\mu(x) = \int \lang \lang g_1 \cdot f(g_1^{-1} \cdot x), g_1 \cdot f'(g_1^{-1} \cdot x) \rang \rang \; d\mu(x) = \int \lang \lang f(g_1^{-1} \cdot x), f'(g_1^{-1} \cdot x) \rang \rang \; d\mu(x)
\end{displaymath}
(because $g_1$ acts unitarily on each fiber)

\begin{displaymath}
= \int \lang \lang f(x), f'(x) \rang \rang \; d\mu(g_1 \cdot x)
\end{displaymath}
(because $G$ acts transitively on $G/H$)

\begin{displaymath}
= \int \lang \lang f(x), f'(x) \rang \rang \; d\mu(x)
\end{displaymath}
(because $\mu$ is $G$-invariant). This shows that $r$ is unitary.

But where do we get a $G$-invariant measure on $G/H$?

\hypertarget{adjoint_of_induced_bundle_construction}{}\subsubsection*{{Adjoint of induced bundle construction}}\label{adjoint_of_induced_bundle_construction}

The induced bundle construction described above is a [[functor]] that takes representations of the [[stabilizer subgroup]] $H$ to $G$-equivariant vector bundles over $M$:

\begin{displaymath}
L: Rep(H) \to Vect(M,G)
\end{displaymath}
There is a related functor going the other way:

\begin{displaymath}
R: Vect(M,G) \to Rep(H)
\end{displaymath}
which [[restriction|restricts]] the action of $G$ on the whole bundle to the action of the stabilizer subgroup $H$ on the fiber over the chosen point $x$. The existence of this [[adjunction]] is known as \textbf{[[Frobenius reciprocity]]}.

We now wish to show that $L$ and $R$ are adjoint functors.

[[!include induced representation {\tt \symbol{62}} adjoint]]

In the diagram above, on the top left we have a generic $G$-equivariant vector bundle over $M$, $F\in Vect(M,G)$, with projection $\pi_1:F\to M$, and a chosen point $x\in M$ whose stabilizer subgroup is $H$. The functor $R$ maps $F$ to a representation of $H$ on the fiber over $x$, $\pi_1^{-1}(x)$, shown on the top right.

On the bottom right, we have a generic representation of $H$ on a vector space $V$. The morphisms of $Rep(H)$ are intertwiners, so we are interested in intertwiners such as $i:V\to \pi_1^{-1}(x)$. The functor $L$, the induced bundle construction, maps a generic representation of $H$ to a $G$-equivariant vector bundle $(G\times V)/H$, shown on the bottom left. This bundle has a projection $\pi_2: (G\times V)/H \to G/H$, $\pi_2([(g,v)])=g H$. Since $M \cong G/H$, this bundle is in $Vect(M,G)$. And we are interested in the morphisms of $Vect(M,G)$, such as $(f,m)$ where $f:L(V)\to F$ and $m:G/H\to M$.

In fact, we need to work with a \emph{subcategory} of $Vect(M,G)$ in which all morphisms preserve the point $x\in M$. When we deal with bundles over $G/H \cong M$, we will use the obvious bijection $g H \to g\cdot x$, and accordingly restrict ourselves to vector bundle morphisms that map $x$ to the coset $e H$ or \emph{vice versa}.

We are assuming that $G$ acts transitively on $M$, so given any $y\in M$ there exists at least one element of $G$, say $k(y)$, such that $k(y)\cdot x = y$. We will now assume that some definite function $k:M\to G$ has been chosen with this property, and for convenience we will further assume that $k(x)=e$, the identity element in $G$. The group element $k(y)$ gives us a specific way to use the action of $G$ on $M$ to get from our chosen point $x$ to some other point $y$ --- and equally, to use the action of $G$ on the whole bundle $F$ to get from the fiber over $x$ to the fiber over $y$.

Now, to show that $L$ and $R$ are adjoint functors, we need to construct a bijection between the intertwiners $i:V\to \pi_1^{-1}(x)$ and the $G$-equivariant vector bundle morphisms $(f,m)$, where $f:L(V)\to F$ and $m:G/H\to M$.

Given an intertwiner $i:V\to \pi_1^{-1}(x)$, we start by defining $m:G/H\to M$ by:

\begin{displaymath}
m(g H)=g\cdot x
\end{displaymath}
which is independent of $i$, and is just the obvious bijection between $G/H$ and $M$. Next, we define $f:L(V)\to F$ by:

\begin{displaymath}
f([(g,v)]) = g\cdot i(v)
\end{displaymath}
In other words, given the equivalence class $[(g,v)]$ we use the intertwiner $i$ to take $v\in V$ to $\pi_1^{-1}(x)$, and then the action of $G$ on $F$ to take the result to the fiber $\pi_1^{-1}(g\cdot x)$. This satisfies the compatibility condition on the projections:

\begin{displaymath}
\pi_1(f([(g,v)])) = g\cdot x = m(g H) = m(\pi_2([(g,v)]))
\end{displaymath}
We also need to check that $f$ commutes with the actions of $G$ on the respective bundles:

\begin{displaymath}
f(g_1\cdot [(g,v)]) = f([(g_1 g,v)]) = (g_1 g)\cdot i(v) = g_1\cdot f([(g,v)])
\end{displaymath}
Next, given a $G$-equivariant vector bundle morphism $(f,m)$, where $f:L(V)\to F$ and $m:G/H\to M$ with $m(e H)=x$, we define an intertwiner $i:V\to \pi_1^{-1}(x)$ by:

\begin{displaymath}
i(v)=f([(e,v)])
\end{displaymath}
We know $i$ will map to $\pi_1^{-1}(x)$ because $f$ must map $[(e,v)]$ to a point in the fiber over $m(\pi_2([(e,v)]))=m(e H)=x$.

We check that this is an intertwiner for the representations of $H$ on the respective vector spaces:

\begin{displaymath}
i(s(h)v)=f([(e,s(h)v)])=f([(h,v)])=f(h\cdot[(e,v)])=h\cdot i(v)
\end{displaymath}
We can also demonstrate a bijection between intertwiners and $G$-equivariant vector bundle morphisms in the other direction: intertwiners $i^*:\pi_1^{-1}(x)\to V$ and vector bundle morphisms $(f^*,m^*)$, where $f^*:F\to L(V)$ and $m^*:M\to G/H$.

Given an intertwiner $i^*:\pi_1^{-1}(x)\to V$, we define $m^*:M\to G/H$ as:

\begin{displaymath}
m^*(y) = k(y) H
\end{displaymath}
We define the map $f^* : F \to L(V)$ by:

\begin{displaymath}
f^*(w) = [(k(\pi_1(w)), i^*(k(\pi_1(w))^{-1}\cdot w) )]
\end{displaymath}
for each $w\in F$. Because $k(\pi_1(w))\cdot x = \pi_1(w)$, $k(\pi_1(w))^{-1}$ will map the entire fiber to which $w$ belongs to $\pi_1^{-1}(x)$, the domain of the intertwiner $i^*$. And we have:

\begin{displaymath}
\pi_2(f^*(w)) = k(\pi_1(w)) H = m^*(\pi_1(w))
\end{displaymath}
The map $f^*$ is a linear map between the fibers $\pi_1^{-1}(y)$ and $\pi_2^{-1}(m^*(y))$, because, along with the linearity of $i^*$, the vector space structure on the fibers of $L(V)$ is \emph{defined} so all maps of the form $v\to [(g,v)]$ are linear. So, $m^*$ and $f^*$ together give us a vector bundle morphism from $F$ to $L(V)$.

In order to be a morphism in the category of $G$-equivariant vector bundles, $f^*$ should also commute with the action of $G$. We have:

\begin{displaymath}
f^*(g\cdot w) = [(k(\pi_1(g\cdot w)), i^*(k(\pi_1(g\cdot w))^{-1} g\cdot w) )] = [(k(g\cdot \pi_1(w)), i^*(k(g\cdot \pi_1(w))^{-1} g\cdot w) )]
\end{displaymath}
Let's abbreviate $\pi_1(w)$ as $y$ and define $h=k(g\cdot y)^{-1} g k(y)$, which takes $x$ to $x$ and so must lie in $H$. Then we have:

\begin{displaymath}
f^*(g\cdot w) = [(k(g\cdot y), i^*(h k(y)^{-1}\cdot w) )] = [(k(g\cdot y), s(h) i^*(k(y)^{-1}\cdot w) )] = [(k(g\cdot y) h, i^*(k(y)^{-1}\cdot w) )]
\end{displaymath}
\begin{displaymath}
= [(g k(y) , i^*(k(y)^{-1}\cdot w) )] = g\cdot [(k(y) , i^*(k(y)^{-1}\cdot w) )] = g\cdot f^*(w)
\end{displaymath}
Suppose we're given a $G$-invariant vector bundle morphism $(f^*,m^*)$, where $f^*:F\to L(V)$ and $m^*:M\to G/H$, with $m^*(x)=e H$.

We make use of the linear bijection $\phi_e:V\to E_{e H}$, defined by $\phi_e(v)=[(e,v)]$. We introduced these linear bijections $\phi_g$ when initially describing the induced bundle construction. We define $i^*:\pi_1^{-1}(x)\to V$ by:

\begin{displaymath}
i^*(w) = \phi_e^{-1}(f^*(w))
\end{displaymath}
We check that this is an intertwiner between the relevant representations of $H$:

\begin{displaymath}
i^*(h\cdot w) = \phi_e^{-1}(f^*(h\cdot w))= \phi_e^{-1}(h\cdot f^*(w))
\end{displaymath}
Suppose $f^*(w)=[(e,v)]$ for some $v\in V$. Then $i^*(w) = v$, and :

\begin{displaymath}
h\cdot f^*(w) = [(h,v)] = [(e,s(h)v)]
\end{displaymath}
\begin{displaymath}
i^*(h\cdot w) = \phi_e^{-1}([(e,s(h)v)]) = s(h)v = s(h) i^*(w)
\end{displaymath}
\hypertarget{examples_and_applications}{}\subsection*{{Examples and Applications}}\label{examples_and_applications}

\hypertarget{regular_representation}{}\subsubsection*{{Regular representation}}\label{regular_representation}

The [[regular representation]] of a group $G$, as a linear representation, is the induced representation of the trivial representation along the trivial subgroup inclusion $Ind_1^G(1)$.

\hypertarget{HeckeAlgebra}{}\subsubsection*{{Centralizer algebra / Hecke algebra}}\label{HeckeAlgebra}

Let

\begin{displaymath}
i \colon H \hookrightarrow G
\end{displaymath}
be a group homomorphism (often assumed to be a [[subgroup]] inclusion, and sometimes with $G$ assumed to be a [[finite group]]). For $E \in H Rep$ some $H$-representation (often taken to be the trivial $H$-representation), let $Ind_i E \in G Rep$ be the induced $G$-representation. Then the [[endomorphism ring]] $End_G(Ind_i E)$ of $Ind_i E$ in $G Rep$ is called the \emph{centralizer algebra} or also the \emph{[[Hecke algebra]]} or \emph{[[Iwahori?Hecke algebra]]} $Hecke(E,i)$ of the induced representation. (Basics are in (\hyperlink{Woit}{Woit, def. 2}), details are in (\hyperlink{CR}{Curtis-Reiner, section 67}), a quick survey of related theory is in (\href{http://homepages.math.uic.edu/~srinivas/bfggpaper.pdf}{Srinivasan})).

In terms of the notation in \emph{\hyperlink{GeneralAbtractDefinition}{General abstract formulation}} above and for $i \colon H \to G$ any homomorphism of $\infty$-groups, we have the [[∞-monoid]]

\begin{displaymath}
Hecke(E,i)
  \coloneqq
\underset{\mathbf{B}G}{\prod}\left[\underset{\mathbf{B}i}{\sum} E, \underset{\mathbf{B}i}{\sum} E \right]
  \,,
\end{displaymath}
where $[-,-]$ is the [[internal hom]] in the [[slice (∞,1)-topos]] $\mathbf{H}_{/\mathbf{B}G} \simeq G Act(\mathbf{H})$.

For $V \in Act(G)$ any other representation, there is a canonical [[∞-action]] of $Hecke(E,i)$ on $\underset{\mathbf{B}G}{\prod} \left[\underset{\mathbf{B}i}{\sum} E , V \right]$. If here $E$ is the trivial representation then by adjointness this is the [[invariants]] $V^G$ of $V$ and hence the Hecke algebra acts on the invariants. (See for instance (\hyperlink{Woit}{Woit, def. 2})). This is sometimes called the \emph{Hecke algebra action on the Iwahori fixed vectors} (\hyperlink{http://sporadic.stanford.edu/bump/bbf.pdf}{e.g. here, p. 9})

\hypertarget{zuckerman_functors_coinduction_on_harishchandra_modules}{}\subsubsection*{{Zuckerman functors: Coinduction on Harish-Chandra modules}}\label{zuckerman_functors_coinduction_on_harishchandra_modules}

Coinduction on [[Harish-Chandra modules]] is referred to as \emph{[[Zuckerman induction]]}. See there for more details.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item The identification of representation induction as the extra left adjoint in a [[base change]] morphism, as discussed in the \emph{\hyperlink{GeneralAbtractDefinition}{General abstract discussion}} above, puts induced representations in the same general abstract framework as [[existential quantification]] in [[logic]] and generally of [[dependent sum]] in [[dependent type theory]] (see there for more details). This relation has first been amplified in (\hyperlink{Lawvere}{Lawvere}).


\item If the modules over a group are considered as comodules over the function Hopf algebra over the group, then one can instead consider the induction for comodules. See [[cotensor product]].


\item The [[derived functor]] of the representation induction functor is often referred to as \emph{[[cohomological induction]]}.


\item [[Dirac induction]]


\item The [[character]] of an induced representation is an [[induced character]].



\end{itemize}
[[!include homotopy type representation theory -- table]]

\hypertarget{references}{}\subsection*{{References}}\label{references}

\hypertarget{ReferencesTraditionalFormulation}{}\subsubsection*{{Traditional formulation}}\label{ReferencesTraditionalFormulation}

Original articles includes

\begin{itemize}%
\item [[George Mackey]], \emph{Induced Representations of Locally Compact Groups I}, Annals of Mathematics, 55 (1952) 101--139;


\item [[George Mackey]], \emph{Induced Representations of Locally Compact Groups II}, Annals of Mathematics, 58 (1953) 193--221;


\item [[George Mackey]], \emph{Induced Representations of Groups and Quantum Mechanics}, W. A. Benjamin, New York, 1968



\end{itemize}
Textbook accounts include

\begin{itemize}%
\item C. Curtis and I. Reiner, \emph{Methods of Representation Theory with applications to finite groups and orders}, Wiley (1987)

\end{itemize}
Lecture note with standard material on induced representations and [[Frobenius reciprocity]] include

\begin{itemize}%
\item [[Tammo tom Dieck]], Chapter 4 of \emph{Representation theory}, 2009 (\href{http://www.uni-math.gwdg.de/tammo/rep.pdf}{pdf})

\end{itemize}
MO discussion includes

\begin{itemize}%
\item \href{http://mathoverflow.net/questions/132272/wrong-way-frobenius-reciprocity-for-finite-groups-representations}{Wrong-way Frobenius reciprocity for finite groups representations}

\end{itemize}
The \hyperlink{TraditionalFormulation}{exposition of the Traditional formulation} in the above entry is in parts taken from

\begin{itemize}%
\item [[Greg Egan]], \emph{\href{http://golem.ph.utexas.edu/category/2009/03/unitary_representations_of_the.html#c023252}{Induced representations}}


\item [[John Baez]] \emph{\href{http://golem.ph.utexas.edu/category/2009/03/unitary_representations_of_the.html#c023258}{Reply}}



\end{itemize}
and related discussion is in

\begin{itemize}%
\item [[John Baez]], \href{http://golem.ph.utexas.edu/category/2009/03/unitary_representations_of_the.html#comments}{Unitary Representations of the Poincare Group}

\end{itemize}
\hypertarget{general_abstract_formulation}{}\subsubsection*{{General abstract formulation}}\label{general_abstract_formulation}

The \hyperlink{GeneralAbtractDefinition}{general abstract formulation} above is mentioned (for [[discrete groups]] and their [[permutation representations]]) in

\begin{itemize}%
\item [[Bill Lawvere]], \emph{[[Adjointness in Foundations]]}, Dialectica 23 (1969), 281-296, Reprints in Theory and Applications of Categories, No. 16, 2006, pp. 1--16. (\href{http://www.tac.mta.ca/tac/reprints/articles/16/tr16.pdf}{pdf})


\item [[Bill Lawvere]], \emph{[[Equality in hyperdoctrines and comprehension schema as an adjoint functor]]}, Proceedings of the AMS Symposium on Pure Mathematics XVII (1970), 1-14. (\href{https://ncatlab.org/nlab/files/LawvereComprehension.pdf}{pdf})



\end{itemize}
The general case of $\infty$-groups in $\infty$-toposes is further discussed in

\begin{itemize}%
\item [[Urs Schreiber]], \emph{[[schreiber:differential cohomology in a cohesive topos]]}

\end{itemize}
[[!redirects induced module]] [[!redirects induced representations]] [[!redirects induction functor]] [[!redirects induction functors]]

[[!redirects coinduced representation]] [[!redirects coinduced representations]] [[!redirects coinduction functor]] [[!redirects coinduction functors]]



\end{document}