of -equivariant vector bundles over . 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 -equivariant vector bundle over , I can take its fiber over your favorite point , and I get a vector space — and this becomes a representation of the stabilizer group , thanks to how acts on .
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 is a representation of — note abuse of notation in calling it , which is the name for the vector space on which acts, instead of the more pedantic full name for a representation, which is something like .
Similarly, is a -equivariant vector bundle over — and this should be something like , or something even more long-winded that gives a name to how acts on and .
is the induced bundle corresponding to .
is the fiber of over your favorite point , which becomes a representation of .
says that -equivariant vector bundle maps from to are in natural 1-1 correspondence with intertwining operators from to .
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 -equivariant vector bundle over and gives a representation of on the fiber over , 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 and restricts it to a representation of :
Given a group with a subgroup , and a representation of on a vector space , we define a left action of on the product by . We write for the orbit, or equivalence class, that contains .
We then define as the set of orbits of that action of , as the set of left cosets of , and the projection by , where of course it makes no difference if we re-describe the orbit as for any because .
For each , choose to be any element of such that . Define , and , .
The map is onto: for any , we have for some , so , , so .
The map is one-to-one: if , then , so for some , we have , or ; equating the first coordinates requires , and is a representation so , and .
Since is a bijection between and the vector space , we can make into a vector space by defining , for all . But is this independent of our choice of ? If we chose instead of , we’d have , so , and . Then:
in agreement with our original definition.
We define the action of on by , or in other words . We then have:
That is, is a -morphism. This also means that the action maps fibers to fibers, . What’s more, the action of restricted to the fiber is , passing from , and this is linear simply by virtue of the way we’ve defined the vector space operations on the .
We get a representation of on the vector space of sections of the bundle by:
General abstract formulation in homotopy type theory
(A genuine ∞-representation/∞-module over may be taken to be a an abelian -group object in , but we can just as well work in the more general context of possibly non-linear representations, hence of actions.)
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 now wish to show that and are adjoint functors.
In the diagram above, on the top left we have a generic -equivariant vector bundle over , , with projection , and a chosen point whose stabilizer subgroup is . The functor maps to a representation of on the fiber over , , shown on the top right.
On the bottom right, we have a generic representation of on a vector space . The morphisms of are intertwiners, so we are interested in intertwiners such as . The functor , the induced bundle construction, maps a generic representation of to a -equivariant vector bundle , shown on the bottom left. This bundle has a projection , . Since , this bundle is in . And we are interested in the morphisms of , such as where and .
In fact, we need to work with a subcategory of in which all morphisms preserve the point . When we deal with bundles over , we will use the obvious bijection , and accordingly restrict ourselves to vector bundle morphisms that map to the coset or vice versa.
We are assuming that acts transitively on , so given any there exists at least one element of , say , such that . We will now assume that some definite function has been chosen with this property, and for convenience we will further assume that , the identity element in . The group element gives us a specific way to use the action of on to get from our chosen point to some other point — and equally, to use the action of on the whole bundle to get from the fiber over to the fiber over .
Now, to show that and are adjoint functors, we need to construct a bijection between the intertwiners and the -equivariant vector bundle morphisms , where and .
Given an intertwiner , we start by defining by:
which is independent of , and is just the obvious bijection between and . Next, we define by:
In other words, given the equivalence class we use the intertwiner to take to , and then the action of on to take the result to the fiber . This satisfies the compatibility condition on the projections:
We also need to check that commutes with the actions of on the respective bundles:
Next, given a -equivariant vector bundle morphism , where and with , we define an intertwiner by:
We know will map to because must map to a point in the fiber over .
We check that this is an intertwiner for the representations of on the respective vector spaces:
We can also demonstrate a bijection between intertwiners and -equivariant vector bundle morphisms in the other direction: intertwiners and vector bundle morphisms , where and .
Given an intertwiner , we define as:
We define the map by:
for each . Because , will map the entire fiber to which belongs to , the domain of the intertwiner . And we have:
The map is a linear map between the fibers and , because, along with the linearity of , the vector space structure on the fibers of is defined so all maps of the form are linear. So, and together give us a vector bundle morphism from to .
In order to be a morphism in the category of -equivariant vector bundles, should also commute with the action of . We have:
Let’s abbreviate as and define , which takes to and so must lie in . Then we have:
Suppose we’re given a -invariant vector bundle morphism , where and , with .
We make use of the linear bijection , defined by . We introduced these linear bijections when initially describing the induced bundle construction. We define by:
We check that this is an intertwiner between the relevant representations of :
Suppose for some . Then , and :
Examples and Applications
The regular representation of a group , as a linear representation, is the induced representation of the trivial representation along the trivial subgroup inclusion .
For any other representation, there is a canonical ∞-action of on . If here is the trivial representation then by adjointness this is the invariants of 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-Chandra modules