Left and Right Morphism Spaces
One more explicit parallel between left and right representations: we have morphisms between right -modules just like we had between left -modules. I won’t really go into the details — they’re pretty straightforward — but it’s helpful to see how our notation works.
In the case of left modules, we had a vector space . Now in the case of right modules we also have a vector space . We use the same notation in both cases, and rely on context to tell us whether we’re talking about right or left module morphisms. In a sense, applying “eats up” an action of on each module, and on the same side.
We can see this even more clearly when we add another action to one of the modules. Let’s say that carries a left action of — we write — and carries commuting left actions of both and another group — we write . I say that there is a “residual” left action of on the space of left -module morphisms. That is, the space “eats up” an action of on each module, and it leaves the left action of behind.
So, how could act on the space of morphisms? Well, let be an intertwinor of the actions, and let any act on by sending it to , defined by . That is, first sends a vector to a vector , and then acts on the left to give a new vector . We must check that this is an intertwinor, and not just a linear map from to . For any , we calculate
using first the fact that is an intertwinor, and then the fact that the action of commutes with that of to pull all the way out to the left.
Similarly, if is an extra action on the right of , we have a “residual” right action on the space . And the same goes for right -modules: we have a residual left action of on , and a residual right action on .
It’s a little more complicated when we have extra commuting actions on . The complication is connected to the fact that the hom functor is contravariant in its first argument, which if you don’t know much about category theory you don’t need to care about. The important thing is that if has an extra left action of , then the space will have a residual right action of .
In this case, given a map intertwining the actions, we define by . We should verify that this is, indeed, a right action:
using the fact that acts on the left on . Again, we must verify that is actually another intertwinor:
using the fact that is an intertwinor and the actions of and on commute.
Similarly, we find a residual right action on the space , and residual left actions on the spaces and .