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# 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.