# Induction and Coinduction of Representations

I'd like to understand the general framework of induction and coinduction of representations. If G is a finite group and H a subgroup, I know that there is a restriction functor from representations of G (over an algebraically closed field of characteristic zero, say) to representations of H. We call the left adjoint to this functor induction, and the right adjoint coinduction. In fact, it can be shown that both adjoints coincide.

1. This story should work even if my field isn't algebraically closed (or even if I'm just working over Z). What happens in characteristic p?

2. For representations of compact groups, the induced vector bundle construction describes the induced representation. Is this also the coinduced representation?

3. For non-compact locally compact groups (in which the subgroup is not cocompact), there is a variation in which we take only sections with compact support. Is it true that one of these is induction and one of these is coinduction? Which is which? Also, this is where I get a bit hazy on the category-theoretic aspects of all of this. What exactly is the right notion of "category of representations" that exhibits restriction and induction/coinduction as adjoint functor pairs in this setting?

4. For non-compact locally compact groups, the usual procedure does not produce unitary representations, so there's some modification of it involving the line bundle of tensor densities of weight 1/2. What exactly is this procedure, and is there a nice "geometric" interpretation of it? How does this construction relate to Mackey's systems of imprimitivity?

5. Is there a nice theory of induced and coinduced representations for algebraic groups?

6. How does all of this work for Lie algebras? How much of the above theory is recoverable from just looking at Lie algebra representations?

Apologies for the overly detailed question; I'm just looking for an idea of how to think about induction and coinduction in general, and also for somebody to point out if I have some horrible misconceptions about how these things work.

For 1, you don't need H to be a subgroup of G. If you have a morphism of finite groups, f:H->G, then this gives rise to induction, coinduction and restriction functors between the categories of finite dimensional representations. You can try to write down a natural isomorphism between the induction and coinduction functors and you will find that it involves inverting |ker(f)|, the order of the kernel of f. So provided this order is invertible in your ring then induction and coinduction are isomorphic functors.

In particular if the order of the kernel is 1, meaning that f is an inclusion, then you get isomorphic functors regardless of the ring. Also, if the characteristic of the ring is zero then the order of the kernel is automatically invertible and you get isomorphic induction and coinduction functors.

I'm afraid I can't answer all of your questions. This is just a brain dump.

If you have any homomorphism f: H -> G, you get a restriction functor f* on representations, but the existence of adjoints depends on the existence of certain limits, in particular, how large you allow representations to be. Formally, you get an induction f! that takes V to the tensor product over group algebras (or a fiber product with G if you're working with G-sets or G-spaces) which is infinite dimensional if G/f(H) is infinite. Similarly, the coinduction f* takes V to H-module homomorphisms from the group algebra of G to V (or H-equivariant maps from G). This also works for Lie algebras and algebraic groups without significant alteration - when working with Lie algebras you use the universal enveloping algebras in place of group rings. If your groups, algebras, and representations come with smooth/algebraic/topological structure, then the definition gets more complicated due to completions, etc.

There is a fancy way to view representations as sheaves on a classifying stack BG, so the homomorphism f describes a pointed map BH -> BG, and the upper and lower stars and shrieks correspond to the usual sheaf operations. Sometimes people use derived versions of these functors, and apparently get useful results (although I don't recall any of them in particular). The induction operation can be thought of as a pushforward with proper (compact) supports while coinduction is a plain pushforward. A coinduced module tends to be "bigger", e.g., if you start with the trivial representation of the trivial group, the coinduced representation is the dual of the induced representation. Induction may use half-forms on fibers of a map, since the product of two half-forms is a volume and can be integrated along fibers to get functions on the target.

• I'm trying to understand the sheafy framework for representation functors in a not too hand wavy way. Do you know a source where the constructions of induction/coinduction are carried out in a bit more detail? It can get pretty confusing keeping track of all the different group actions equivariant stuff... Jul 21, 2016 at 13:37
• @SaalHardali Sorry, I haven't seen such a reference. I think the standard procedure is to look up the usual sheaf operations on stacks, specialize to classifying stacks, and compare with the representation theory definitions. Jul 21, 2016 at 23:01

For 3, here is a partial answer, because I'm not sure how general the correct statement is. Certainly for the case of smooth representations of p-adic groups, the right adjoint to restriction (coinduction)* is given by arbitrary sections while the left adjoint (called compact induction in the book by Bushnell and Henniart, which is my source here (Chapter 2)) is given by taking sections with compact support.

For 4, the problem of preserving unitaricity comes from the failure of the left and right Haar measures to be equal. For any group H, you define the modular quasicharacter δ_H by δ_H(b)d_L(b)=d_R(b) where d_L and d_R are the left and right Haar measures. In inducing from H to G, one then needs to twist the induction functor by the ratio of the square roots of these quasicharacters for G and H. (In most interesting cases G has trivial quasicharacter, but H often does not (eg upper-triangular intertible matrices over the reals)). The word "needs" here describes what is necessary to preserve unitaricity, though even if you don't care about unitary representations, this twisting can produce cleaner combinatorics.

*edited w.r.t Evan's correction.

• I think you may have induction and coinduction reversed (the "co" here having nothing to do with compactness, as far as I know). I'm almost certain the left adjoint is known as induction and the right as coinduction, at least in the homological algebra literature, but I don't know how standard this terminology is on the representation theory side of things. I've often seen the noncompact one written as "Ind" and the compact one written as "ind," which is probably another source of confusion seeing as "Ind" is what one normally sees written for induction in the case of finite groups. Oct 24, 2009 at 5:31
• Just to clarify: are you claiming that compact induction is left adjoint to the restriction functor, for smooth representations of p-adic groups? I am not sure if this is true in general. Jul 4, 2010 at 20:30

There is a nice answer to question 5 for algebraic groups in arbitrary characteristic over algebraically closed fields, although one needs to consider a larger category. Given an algebraic subgroup H of an algebraic group G, the hyperalgebra U(H) of H is naturally a subalgebra of the hyperalgebra U(G) of G. (In characteristic 0 the hyperalgebra of an algebraic group is just the enveloping algebra of its Lie algebra, but this is not the case in positive characteristic).

One now has induction and coinduction functors from modules for U(H) to U(G). In the case that M is a finite-dimensional H-module, although the G-module Ind M induced from M is not the same as the U(G)-module M' induced from the U(H)-module M, it is true that Ind M is a natural U(G)-submodule of M'. In my opinion this is the "right" standpoint because on the other hand, the coinduced module doesn't always exist in the category of G-modules, but rather in the category of U(G)-modules.

Here's an example: In the case that G is semisimple, H=B is a Borel subgroup, and M is a 1-dimensional B-module, hyperalgebra coinduction of M gives you the associated Verma module and hyperalgebra induction of M* gives you the associated dual Verma (well, this is a white lie -- this almost gives you the dual Verma, but one now has to apply another duality functor to actually get the dual Verma; there are two duality functors in the category of U(G)-modules, which can make things confusing). Further, the G-module Ind (M*) is a U(G)-submodule of the dual Verma. On the other hand, there is no U(G)-subspace of the Verma module that "integrates" to a G-module (i.e., no Verma module has a U(G)-locally finite subspace).

Here's a somewhat more abstract answer, which hopefully clarifies the "general framework" for induced and coinduced representations.

In general, if $R$ is a commutative ring, then representations of a group $G$ on $R$-modules are the same thing as $RG$-modules, where $RG$ is the group $R$-algebra. Thus, if $\phi: H \to G$ is a group homomorphism, then we have an $R$-algebra homomorphism $R\phi: RH \to RG$ and the restricted, induced and coinduced representations are really just special cases of the change of rings operations: restriction, extension, and coextension of scalars.

Likewise, representations of a Lie $R$-algebra $\mathfrak{g}$ on $R$-modules are the same as $U\mathfrak{g}$-modules where $U$ is the universal enveloping algebra functor. Again, restricted, induced, and coinduced representations are just special cases of the corresponding operations on ring modules: restriction, extension, and coextension of scalars.

However, note that the convention for which is called the "induced" module and which is the "coinduced" module often differ. I usually see the extension of scalars being called the "induced" module when dealing with Lie algebras, whereas for Lie groups this is often called the "coinduced" module.

Also, in general the extension and coextension of scalars will differ; there is a theorem of Morita that says that it is sufficient and necessary that a ring homomorphism $f: R \to S$ be Frobenius in order for the extension and coextension along $f$ to be the same (up to unique natural isomorphism).