nLab
induced representation

Induced modules

Idea

Given a group GG with subgroup HGH \hookrightarrow G and a representation of HH, there is canonically induced a representation of GG: the induced representation.

Definition

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.

Traditional formulation

Explanation

Suppose a Lie group GG acts smoothly and transitively on a smooth manifold MM. The stabilizer subgroup of a given point xMx \in M is then a Lie subgroup HGH \subseteq G, and

MG/H, M \cong G/H \,,

is the coset space.

Starting from this, there’s a recipe taking any representation ss of HH on a vector space VV and turns it into a vector bundle EE over MM — called the induced bundle. Moreover, the group GG acts on this bundle, and the projection

π:EM \pi : E \to M

is compatible with the action of GG:

π(ge)=gπ(e). \pi(g e) = g \pi(e) .

Hence EE is a GG-equivariant vector bundle over MM.

The ‘process’ described is actually a functor, the induction functor.

There’s a category

Rep(H)Rep(H)

of linear representations of HH, and a category

Vect(M,G)Vect(M,G)

of GG-equivariant vector bundles over MM. The induced bundle construction gives a functor

L:Rep(H)Vect(M,G)L: Rep(H) \to Vect(M,G)

But, if you think about it, you’ll notice there’s also a functor going back the other way:

R:Vect(M,G)Rep(H)R: Vect(M,G) \to Rep(H)

If you give me a GG-equivariant vector bundle EE over MM, I can take its fiber over your favorite point xx, and I get a vector space — and this becomes a representation of the stabilizer group HH, thanks to how GG acts on EE.

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

hom(LV,F)hom(V,RF)hom(L V, F) \cong hom(V, R F)

Here VV is a representation of HH — note abuse of notation in calling it VV, which is the name for the vector space on which GG acts, instead of the more pedantic full name for a representation, which is something like s:GGL(V)s: G \to GL(V).

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

LVL V is the induced bundle corresponding to VV.

RFR F is the fiber of FF over your favorite point xx, which becomes a representation of GG.

And this:

hom(LV,F)hom(V,RF)hom(L V, F) \cong hom(V, R F)

says that GG-equivariant vector bundle maps from LVL V to FF are in natural 1-1 correspondence with intertwining operators from VV to RFR 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 GG-equivariant vector bundle over MM and gives a representation of HH on the fiber over xMx \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 GG and restricts it to a representation of HH:

R:Rep(G)Rep(H)R': Rep(G) \to Rep(H)

And this too, has a left adjoint:

L:Rep(H)Rep(G)L' : Rep(H) \to Rep(G)

which is called the induced representation trick.

Detailed description

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

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

For each xMx\in M, choose gg to be any element of GG such that x=gHx = g H. Define E x=π 1(x)E_x = \pi^{-1}(x), and ϕ g:VE x\phi_g:V\to E_x, ϕ g(v)=[(g,v)]\phi_g(v) = [(g,v)].

The map ϕ g\phi_g is onto: for any [(k,w)]E (gH)=π 1(gH)[(k,w)]\in E_{(g H)} = \pi^{-1}(g H), we have k=gh 1 1k=g h_1^{-1} for some h 1Hh_1\in H, so k 1gHk^{-1} g\in H, (k 1g)(g,s(g 1k)w)=(k,w)(k^{-1} g)\cdot (g, s(g^{-1} k)w) = (k,w), so ϕ g(s(g 1k)w)=[(g,s(g 1k)w)]=[(k,w)]\phi_g(s(g^{-1} k)w) = [(g, s(g^{-1} k)w)] = [(k,w)].

The map ϕ g\phi_g is one-to-one: if ϕ g(v)=ϕ g(w)\phi_g(v) = \phi_g(w), then [(g,v)]=[(g,w)][(g,v)]=[(g,w)], so for some h 1Hh_1\in H, we have h 1(g,v)=(g,w)h_1\cdot (g,v) = (g,w), or (gh 1 1,s(h 1)v)=(g,w)(g h_1^{-1}, s(h_1)v) = (g,w); equating the first coordinates requires h 1=eh_1=e, and ss is a representation so s(e)=1 Vs(e)=1_V, and v=wv=w.

Since ϕ g\phi_g is a bijection between E xE_x and the vector space VV, we can make E xE_x into a vector space by defining αp+βqϕ g(αϕ g 1(p)+βϕ g 1(q))\alpha p + \beta q \equiv \phi_g(\alpha \phi_g^{-1}(p) + \beta \phi_g^{-1}(q)), for all α,β,p,qE x\alpha, \beta \in \mathbb{R}, p, q \in E_x. But is this independent of our choice of gg? If we chose ghg h instead of gg, we’d have ϕ gh(v)=[(gh,v)]=[(g,s(h)v)]=ϕ g(s(h)v)\phi_{g h}(v) = [(g h,v)] = [(g, s(h)v)] = \phi_g(s(h)v), so ϕ gh=ϕ gs(h)\phi_{g h}=\phi_g\circ s(h), and ϕ gh 1=s(h 1)ϕ g 1\phi_{g h}^{-1}=s(h^{-1})\circ \phi_g^{-1}. Then:

ϕ gh(αϕ gh 1(p)+βϕ gh 1(q))=(ϕ gs(h))(α(s(h 1)ϕ g 1)(p)+β(s(h 1)ϕ g 1)(q))=ϕ g(αϕ g 1(p)+βϕ g 1(q))\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 GG on EE by g 1[(g,v)]=[(g 1g,v)]g_1\cdot [(g,v)] = [(g_1 g,v)], or in other words g 1ϕ g(v)=ϕ g 1g(v)g_1\cdot \phi_g(v) = \phi_{g_1 g}(v). We then have:

π(g 1[(g,v)])=π[(g 1g,v)]=(g 1g)H=g 1(gH)=g 1π([(g,v)])\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, π\pi is a GG-morphism. This also means that the action maps fibers to fibers, g 1:E (gH)E g 1(gH)g_1:E_{(g H)}\to E_{g_1\cdot (g H)}. What’s more, the action of g 1g_1 restricted to the fiber E (gH)E_{(g H)} is ϕ g 1gϕ g 1\phi_{g_1 g}\circ \phi_g^{-1}, passing from E (gH)VE g 1(gH)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 xE_x.

We get a representation rr of GG on the vector space Γ(E)\Gamma(E) of sections of the bundle EE by:

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

General abstract formulation in homotopy type theory

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

Act(G)H /BG. Act(G) \simeq \mathbf{H}_{/\mathbf{B}G} \,.

(A genuine ∞-representation/∞-module over GG may be taken to be a an abelian \infty-group object in Act(G)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:HGf \colon H \to G a homomorphism of ∞-groups, hence for a morphism Bf:BHBG\mathbf{B}f \colon \mathbf{B}H \to \mathbf{B}G of their deloopings, there is the corresponding base change geometric morphism

( ff * f):Act(H) ff * fAct(H). (\sum_f \dashv f^* \dashv \prod_f) \colon Act(\mathbf{H}) \stackrel{\overset{\sum_f}{\to}}{\stackrel{\overset{f^*}{\leftarrow}}{\underset{\prod_f}{\to}}} Act(\mathbf{H}) \,.

Here

For the case of permutation representations of discrete groups this perspective is made explicit in (Lawvere, p. 14).

Properties

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 VV has an inner product, ,\lang \cdot, \cdot \rang, and ss is a unitary representation. We can define an inner product on E xE_x by p,qϕ g 1(p),ϕ g 1(q)\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 gg: if we chose ghg h instead, we’d have

p,q=ϕ gh 1(p),ϕ gh 1(q)=s(h 1)ϕ g 1(p),s(h 1)ϕ g 1(q)=ϕ g 1(p),ϕ g 1(q).\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.

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 xE_x.

Now we need to show that the action of any g 1Gg_1 \in G on the fiber E (gH)E_{(g H)} is unitary:

g 1p,g 1q=ϕ g 1gϕ g 1(p),ϕ g 1gϕ g 1(q)=ϕ g 1g 1ϕ g 1gϕ g 1(p),ϕ g 1g 1ϕ g 1gϕ g 1(q)=ϕ g 1(p),ϕ g 1(q)=p,q.\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.

Finally, we need to define an inner product on Γ(E)\Gamma(E), and show that the representation rr is unitary. If we had a GG-invariant measure μ\mu on G/HG/H, we could define the inner product of two sections of ff and ff' of EE to be

f(x),f(x)dμ(x).\int \lang \lang f(x), f'(x) \rang \rang \; d\mu(x).

We would then have

(r(g 1)f)(x),(r(g 1)f)(x)dμ(x)=g 1f(g 1 1x),g 1f(g 1 1x)dμ(x)=f(g 1 1x),f(g 1 1x)dμ(x)\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)

(because g 1g_1 acts unitarily on each fiber)

=f(x),f(x)dμ(g 1x)= \int \lang \lang f(x), f'(x) \rang \rang \; d\mu(g_1 \cdot x)

(because GG acts transitively on G/HG/H)

=f(x),f(x)dμ(x)= \int \lang \lang f(x), f'(x) \rang \rang \; d\mu(x)

(because μ\mu is GG-invariant). This shows that rr is unitary.

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

Adjoint of induced bundle construction

The induced bundle construction described above is a functor that takes representations of the stabilizer subgroup HH to GG-equivariant vector bundles over MM:

L:Rep(H)Vect(M,G)L: Rep(H) \to Vect(M,G)

There is a related functor going the other way:

R:Vect(M,G)Rep(H)R: Vect(M,G) \to Rep(H)

which restricts the action of GG on the whole bundle to the action of the stabilizer subgroup HH on the fiber over the chosen point xx. The existence of this adjunction is known as Frobenius reciprocity.

We now wish to show that LL and RR are adjoint functors.

M x G-equivariant vector bundle F, π1: FM Representation R(F) of H on π1–1(x) h π1 h π1 h π1 h G / H G-equivariant vector bundle L(V), π2: (G×V) / HG / H Representation s of H on V h π2 h π2 h π2 h R Restrict to action of H on a single fiber π1–1(x) L Construct induced bundle h ⋅ (g, v) = (g h–1, s(h)v) π2: (G×V) / HG / H π2([(]g, v)]) = g H g1[(]g, v)] = [(]g1g, v)] Intertwiner i* i* : π1–1(x) → V i* h = h i* Intertwiner i i : V →π1–1(x) i h = h i Vector bundle morphism (f*, m*) f* : FL(V) m* : MG / H m* π1 = π2 f* f* g = g f* Vector bundle morphism (f, m) f : L(V) → F m : G / HM m π2 = π1 f f g = g f

In the diagram above, on the top left we have a generic GG-equivariant vector bundle over MM, FVect(M,G)F\in Vect(M,G), with projection π 1:FM\pi_1:F\to M, and a chosen point xMx\in M whose stabilizer subgroup is HH. The functor RR maps FF to a representation of HH on the fiber over xx, π 1 1(x)\pi_1^{-1}(x), shown on the top right.

On the bottom right, we have a generic representation of HH on a vector space VV. The morphisms of Rep(H)Rep(H) are intertwiners, so we are interested in intertwiners such as i:Vπ 1 1(x)i:V\to \pi_1^{-1}(x). The functor LL, the induced bundle construction, maps a generic representation of HH to a GG-equivariant vector bundle (G×V)/H(G\times V)/H, shown on the bottom left. This bundle has a projection π 2:(G×V)/HG/H\pi_2: (G\times V)/H \to G/H, π 2([(g,v)])=gH\pi_2([(g,v)])=g H. Since MG/HM \cong G/H, this bundle is in Vect(M,G)Vect(M,G). And we are interested in the morphisms of Vect(M,G)Vect(M,G), such as (f,m)(f,m) where f:L(V)Ff:L(V)\to F and m:G/HMm:G/H\to M.

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

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

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

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

m(gH)=gxm(g H)=g\cdot x

which is independent of ii, and is just the obvious bijection between G/HG/H and MM. Next, we define f:L(V)Ff:L(V)\to F by:

f([(g,v)])=gi(v)f([(g,v)]) = g\cdot i(v)

In other words, given the equivalence class [(g,v)][(g,v)] we use the intertwiner ii to take vVv\in V to π 1 1(x)\pi_1^{-1}(x), and then the action of GG on FF to take the result to the fiber π 1 1(gx)\pi_1^{-1}(g\cdot x). This satisfies the compatibility condition on the projections:

π 1(f([(g,v)]))=gx=m(gH)=m(π 2([(g,v)]))\pi_1(f([(g,v)])) = g\cdot x = m(g H) = m(\pi_2([(g,v)]))

We also need to check that ff commutes with the actions of GG on the respective bundles:

f(g 1[(g,v)])=f([(g 1g,v)])=(g 1g)i(v)=g 1f([(g,v)])f(g_1\cdot [(g,v)]) = f([(g_1 g,v)]) = (g_1 g)\cdot i(v) = g_1\cdot f([(g,v)])

Next, given a GG-equivariant vector bundle morphism (f,m)(f,m), where f:L(V)Ff:L(V)\to F and m:G/HMm:G/H\to M with m(eH)=xm(e H)=x, we define an intertwiner i:Vπ 1 1(x)i:V\to \pi_1^{-1}(x) by:

i(v)=f([(e,v)])i(v)=f([(e,v)])

We know ii will map to π 1 1(x)\pi_1^{-1}(x) because ff must map [(e,v)][(e,v)] to a point in the fiber over m(π 2([(e,v)]))=m(eH)=xm(\pi_2([(e,v)]))=m(e H)=x.

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

i(s(h)v)=f([(e,s(h)v)])=f([(h,v)])=f(h[(e,v)])=hi(v)i(s(h)v)=f([(e,s(h)v)])=f([(h,v)])=f(h\cdot[(e,v)])=h\cdot i(v)

We can also demonstrate a bijection between intertwiners and GG-equivariant vector bundle morphisms in the other direction: intertwiners i *:π 1 1(x)Vi^*:\pi_1^{-1}(x)\to V and vector bundle morphisms (f *,m *)(f^*,m^*), where f *:FL(V)f^*:F\to L(V) and m *:MG/Hm^*:M\to G/H.

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

m *(y)=k(y)Hm^*(y) = k(y) H

We define the map f *:FL(V)f^* : F \to L(V) by:

f *(w)=[(k(π 1(w)),i *(k(π 1(w)) 1w))]f^*(w) = [(k(\pi_1(w)), i^*(k(\pi_1(w))^{-1}\cdot w) )]

for each wFw\in F. Because k(π 1(w))x=π 1(w)k(\pi_1(w))\cdot x = \pi_1(w), k(π 1(w)) 1k(\pi_1(w))^{-1} will map the entire fiber to which ww belongs to π 1 1(x)\pi_1^{-1}(x), the domain of the intertwiner i *i^*. And we have:

π 2(f *(w))=k(π 1(w))H=m *(π 1(w))\pi_2(f^*(w)) = k(\pi_1(w)) H = m^*(\pi_1(w))

The map f *f^* is a linear map between the fibers π 1 1(y)\pi_1^{-1}(y) and π 2 1(m *(y))\pi_2^{-1}(m^*(y)), because, along with the linearity of i *i^*, the vector space structure on the fibers of L(V)L(V) is defined so all maps of the form v[(g,v)]v\to [(g,v)] are linear. So, m *m^* and f *f^* together give us a vector bundle morphism from FF to L(V)L(V).

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

f *(gw)=[(k(π 1(gw)),i *(k(π 1(gw)) 1gw))]=[(k(gπ 1(w)),i *(k(gπ 1(w)) 1gw))]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) )]

Let’s abbreviate π 1(w)\pi_1(w) as yy and define h=k(gy) 1gk(y)h=k(g\cdot y)^{-1} g k(y), which takes xx to xx and so must lie in HH. Then we have:

f *(gw)=[(k(gy),i *(hk(y) 1w))]=[(k(gy),s(h)i *(k(y) 1w))]=[(k(gy)h,i *(k(y) 1w))]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) )]
=[(gk(y),i *(k(y) 1w))]=g[(k(y),i *(k(y) 1w))]=gf *(w) = [(g k(y) , i^*(k(y)^{-1}\cdot w) )] = g\cdot [(k(y) , i^*(k(y)^{-1}\cdot w) )] = g\cdot f^*(w)

Suppose we’re given a GG-invariant vector bundle morphism (f *,m *)(f^*,m^*), where f *:FL(V)f^*:F\to L(V) and m *:MG/Hm^*:M\to G/H, with m *(x)=eHm^*(x)=e H.

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

i *(w)=ϕ e 1(f *(w))i^*(w) = \phi_e^{-1}(f^*(w))

We check that this is an intertwiner between the relevant representations of HH:

i *(hw)=ϕ e 1(f *(hw))=ϕ e 1(hf *(w))i^*(h\cdot w) = \phi_e^{-1}(f^*(h\cdot w))= \phi_e^{-1}(h\cdot f^*(w))

Suppose f *(w)=[(e,v)]f^*(w)=[(e,v)] for some vVv\in V. Then i *(w)=vi^*(w) = v, and :

hf *(w)=[(h,v)]=[(e,s(h)v)]h\cdot f^*(w) = [(h,v)] = [(e,s(h)v)]
i *(hw)=ϕ e 1([(e,s(h)v)])=s(h)v=s(h)i *(w)i^*(h\cdot w) = \phi_e^{-1}([(e,s(h)v)]) = s(h)v = s(h) i^*(w)

Examples and Applications

Centralizer algebra / Hecke algebra

Let

i:HG i \colon H \hookrightarrow G

be a group homomorphism (often assumed to be a subgroup inclusion, and sometimes with GG assumed to be a finite group). For EHRepE \in H Rep some HH-representation (often taken to be the trivial HH-representation), let Ind iEGRepInd_i E \in G Rep be the induced GG-representation. Then the endomorphism ring End G(Ind iE)End_G(Ind_i E) of Ind iEInd_i E in GRepG Rep is called the centralizer algebra or also the Hecke algebra or Iwahori–Hecke algebra Hecke(E,i)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:HGi \colon H \to G any homomorphism of \infty-groups, we have the ∞-monoid

Hecke(E,i)BG[BiE,BiE], Hecke(E,i) \coloneqq \underset{\mathbf{B}G}{\prod}\left[\underset{\mathbf{B}i}{\sum} E, \underset{\mathbf{B}i}{\sum} E \right] \,,

where [,][-,-] is the internal hom in the slice (∞,1)-topos H /BGGAct(H)\mathbf{H}_{/\mathbf{B}G} \simeq G Act(\mathbf{H}).

For VAct(G)V \in Act(G) any other representation, there is a canonical ∞-action of Hecke(E,i)Hecke(E,i) on BG[BiE,V]\underset{\mathbf{B}G}{\prod} \left[\underset{\mathbf{B}i}{\sum} E , V \right]. If here EE is the trivial representation then by adjointness this is the invariants V GV^G of VV 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)

Zuckerman functors: Coinduction on Harish-Chanfra modules

Coinduction on Harish-Chandra modules is referred to as Zuckerman induction. See there for more details.

representation theory and equivariant cohomology in terms of (∞,1)-topos theory/homotopy type theory:

homotopy type theoryrepresentation theory
pointed connected context BG\mathbf{B}G∞-group GG
dependent type∞-action/∞-representation
dependent sum along BG*\mathbf{B}G \to \astcoinvariants/homotopy quotient
context extension along BG*\mathbf{B}G \to \asttrivial representation
dependent product along BG*\mathbf{B}G \to \asthomotopy invariants/∞-group cohomology
dependent product of internal hom along BG*\mathbf{B}G \to \astequivariant cohomology
dependent sum along BGBH\mathbf{B}G \to \mathbf{B}Hinduced representation
context extension along BGBH\mathbf{B}G \to \mathbf{B}H
dependent product along BGBH\mathbf{B}G \to \mathbf{B}Hcoinduced representation
spectrum object in context BG\mathbf{B}Gspectrum with G-action (naive G-spectrum)

References

Traditional formulation

Textbook accounts include

  • C. Curtis and I. Reiner, Methods of Representation Theory with applications to finite groups and orders, Wiley (1987)

Lecture note with standard material on induced representations and Frobenius reciprocity include

  • Peter Woit, Induced representations and Frobenius Reciprocity (pdf)

MO discussion includes

The exposition of the Traditional formulation in the above entry is in parts taken from

and related discussion is in

General abstract formulation

The general abstract formulation above is mentioned (for discrete groups and their permutation representations) in

  • Bill Lawvere, Adjointness in foundations, Reprints in Theory and Applications of Categories, No. 16, 2006, pp. 1–16. (pdf)

The general case of \infty-groups in \infty-toposes is further discussed in sections 3.3.11-3.3.13 of

Revised on November 16, 2013 07:44:10 by Urs Schreiber (89.204.155.122)