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I'd like to learn the representation theory of Lie groups. I have a good knowledge of semisimple Lie algebras and their representation theory as well as the basics of Lie groups.
To what extent are the representation theories of the groups and the corresponding algebras interrelated?
Is there a book you would recommend for someone who knows a lot about Lie algebras representations, but has only rather basic knowledge of Lie groups?
The representation theory of Lie groups and Lie algebras are very related. In fact, in the case of Simply-connected Lie groups, the irreducible representations of these Lie groups are in bijection with the irreducible representations of its corresponding Lie algebra. In the case of a connected Lie group, the irreducible representations of its corresponding Lie algebra are in bijection with the irreducible representations of its universal covering space ( which is a Lie group as well ). If your Lie group is not connected, one can still use this correspondence by considering the connected component of the identity ( your group modulo this component will only be a discrete/dimension 0 group ).
Off the top of my head, I can give 2 good books that illustrate this correspondence very well: Representation theory: a first course (Fulton and Harris) and Introduction to Lie Groups and Lie algebras ( Krilliov Jr ). You can also learn a good deal by reading the first couple of chapters of Representations of Compact Lie groups by Bockner. However, the rest of the book develops the theory almost entirely without any reference to Lie algebras.
There is an important watershed between finite-dimensional and infinite-dimensional representations of both/either Lie algebras and Lie groups, especially for non-compact Lie groups, most of whose irreducible unitary representations are definitely not finite-dimensional.
Thus, @Elliot's recommendation of Fulton-Harris seems to be a popular one, connecting to finite-dimensional representation theory...
But all but the trivial finite-dimensional repns of infinite-dimensional [Edit! non-compact! :)] Lie groups, like $SL_2(\mathbb R)$, are not unitary. The unitary repns of $SL_2(\mathbb R)$ and such are infinite-dimensional.
For the latter, one of the most congenial paper books is Varadarajan's Cambridge-Press "Harmonic analysis on semi-simple Lie groups". Formal treatments are Knapp's (Princeton Press) and Wallach's (Academic Press).
Harish-Chandra discovered in the 1950s that the repn theory of semi-simple or reductive Lie groups, not only compact ones as considered by Weyl 20+ years earlier, was well-delineated by Lie algebra repns, maybe remembering also the repn theory restricted to a maximal compact. Thus, $\frak{g}$,$K$-modules.
Perhaps the questioner can clarify their needs...
I would suggest Knapp, Lie groups beyond and introduction and Knapp, Representation theory of semisimple groups (an overview based on examples). If needed for a very simple introduction on very basics fact about Lie Groups is Hall, Lie Groups, Lie Algebras, and Representations
