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In this paper we give the construction of the adjoint and the co-adjoint of the restriction functor in the category of differentiable G-modules, where G is a Lie group. Stated in terms of representation theory this means that two types of induced representations are introduced, both differing from the classical definition of differentiably induced representation given by Bruhat. The Frobenius reciprocity theorem is shown to hold. The main part of the paper is devoted to obtaining suitable realizations of the spaces of the induced representations. It turns out that they may be given as EK(G, E) and E'(G, E) respectively, i.e. as certain spaces of K-invariant differentiable functions or distributions on G. This makes it possible to establish a rather complete duality theory. In the last part we consider the relationship to some of Bruhat's work. In particular his Frobenius theorem is shown to be a direct consequence of the tensor-product machinery we employ. We also offer a result on inducing in stages.

Differentiable representations. I. Induced representations and Frobenius reciprocity