geometric representation theory
representation, 2-representation, ∞-representation
Grothendieck group, lambda-ring, symmetric function, formal group
principal bundle, torsor, vector bundle, Atiyah Lie algebroid
Eilenberg-Moore category, algebra over an operad, actegory, crossed module
Given a group $G$ with subgroup $H \hookrightarrow G$ and a representation of $H$, there is canonically induced a representation of $G$: the induced representation.
We give an exposition of the
of induced representations. Then we provide a
that refines the notion to ∞-representations of ∞-groups equipped with any additional geometric structure.
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
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 induced bundle. Moreover, the group $G$ acts on this bundle, and the projection
is compatible with the action of $G$:
Hence $E$ is a $G$-equivariant vector bundle over $M$.
The ‘process’ described is actually a functor, the induction functor.
There’s a category
of linear representations of $H$, and a category
of $G$-equivariant vector bundles over $M$. The induced bundle construction gives a functor
But, if you think about it, you’ll notice there’s also a functor going back the other way:
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 simpler than the induced bundle construction!
Whenever we have functors going both ways between two categories, we should suspect that they’re adjoints. The simpler functor often amounts to ‘forgetting’ something. This forgetful functor is usually the right adjoint. It’s partner going the other way, the left adjoint, usually involves ‘constructing’ something instead of ‘forgetting’ something.
And indeed, that’s what’s happening here! Technically, this is to say that
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:
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 restricts it to a representation of $H$:
And this too, has a left adjoint:
which is called the induced representation.
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:
$(r(g_1)f)(x) = g_1\cdot f(g_1^{-1}\cdot x)$
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 cohomological induction.)
Let $\mathbf{H}$ be an ambient (∞,1)-topos. By the discussion at ∞-action, for $G \in Grp(\mathbf{H})$ a 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
(A genuine ∞-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
Here
the inverse image/(∞,1)-pullback functor $f^*$ produces the “restricted” $\infty$-representations along $f$;
the dependent sum $\sum_f$ is the induced representation ∞-functor;
the dependent product $\prod_f$ is the coinduced representation ∞-functor.
For the case of permutation representations of discrete groups this perspective is made explicit in (Lawvere, p. 14).
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
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:
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
We would then have
(because $g_1$ acts unitarily on each fiber)
(because $G$ acts transitively on $G/H$)
(because $\mu$ is $G$-invariant). This shows that $r$ is unitary.
But where do we get a $G$-invariant measure on $G/H$?
The induced bundle construction described above is a functor that takes representations of the stabilizer subgroup $H$ to $G$-equivariant vector bundles over $M$:
There is a related functor going the other way:
which 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 Frobenius reciprocity.
We now wish to show that $L$ and $R$ are adjoint functors.
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 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 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:
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:
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:
We also need to check that $f$ commutes with the actions of $G$ on the respective bundles:
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:
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:
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:
We define the map $f^* : F \to L(V)$ by:
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:
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 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:
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:
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:
We check that this is an intertwiner between the relevant representations of $H$:
Suppose $f^*(w)=[(e,v)]$ for some $v\in V$. Then $i^*(w) = v$, and :
The regular representation of a group $G$, as a linear representation, is the induced representation of the trivial representation along the trivial subgroup inclusion $Ind_1^G(1)$.
Let
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 centralizer algebra or also the Hecke algebra or Iwahori–Hecke algebra $Hecke(E,i)$ of the induced representation. (Basics are in (Woit, def. 2), details are in (Curtis-Reiner, section 67), a quick survey of related theory is in (Srinivasan)).
In terms of the notation in General abstract formulation above and for $i \colon H \to G$ any homomorphism of $\infty$-groups, we have the ∞-monoid
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 (Woit, def. 2)). This is sometimes called the Hecke algebra action on the Iwahori fixed vectors (e.g. here, p. 9)
Coinduction on Harish-Chandra modules is referred to as Zuckerman induction. See there for more details.
The identification of representation induction as the extra left adjoint in a base change morphism, as discussed in the 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 (Lawvere).
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.
The derived functor of the representation induction functor is often referred to as cohomological induction.
representation theory and equivariant cohomology in terms of (∞,1)-topos theory/homotopy type theory:
homotopy type theory | representation theory |
---|---|
pointed connected context $\mathbf{B}G$ | ∞-group $G$ |
dependent type | ∞-action/∞-representation |
dependent sum along $\mathbf{B}G \to \ast$ | coinvariants/homotopy quotient |
context extension along $\mathbf{B}G \to \ast$ | trivial representation |
dependent product along $\mathbf{B}G \to \ast$ | homotopy invariants/∞-group cohomology |
dependent product of internal hom along $\mathbf{B}G \to \ast$ | equivariant cohomology |
dependent sum along $\mathbf{B}G \to \mathbf{B}H$ | induced representation |
context extension along $\mathbf{B}G \to \mathbf{B}H$ | |
dependent product along $\mathbf{B}G \to \mathbf{B}H$ | coinduced representation |
spectrum object in context $\mathbf{B}G$ | spectrum with G-action (naive G-spectrum) |
Textbook accounts include
Lecture note with standard material on induced representations and Frobenius reciprocity include
MO discussion includes
The exposition of the Traditional formulation in the above entry is in parts taken from
and related discussion is in
The general abstract formulation above is mentioned (for discrete groups and their permutation representations) in
The general case of $\infty$-groups in $\infty$-toposes is further discussed in