TITLE: Correspondence of representation theory between $\mathrm{GL}_n(\mathbb C)$ and $\mathrm U_n(\mathbb C)$
QUESTION [8 upvotes]: If I know something about the representation theory of the general linear group $\mathrm{GL}_n(\mathbb C)$, what can I say about the representation theory of the unitary group $\mathrm U_n(\mathbb C)$? E.g. branching rules
Similarly for the groups $\mathrm{SO}_n(\mathbb C) \subset \mathrm O_n(\mathbb C)$?
Is module over ${\mathbb C\mathrm{GL}_n(\mathbb C)}$ that is not semi-simple necessarily infinite-dimensional? How does it typically look like? See new Question here
If I have information about the restriction of representations of the general linear group, can I make any statements about the induction (by Frobenius reciprocity)? E.g. I know $$\mathrm{res}^{\mathrm{GL}_n}_{\mathrm{GL}_k\times \mathrm{GL}_{n-k}} V(\lambda)_n \cong \bigoplus_{\alpha, \beta} c_{\alpha, \beta}^\lambda V(\alpha)_k \otimes V(\beta)_{n-k}$$
where $V(\lambda)_n$ is the irreducible polynomial representations corresponding to a partition (or Young diagram) $\lambda$ of $\mathrm{GL}_n(\mathbb C)$ and $c^\lambda_{\alpha,\beta}$ are the Littlewood-Richardson numbers. Is it true that
$$ \mathrm{ind}_{\mathrm{GL}_k\times \mathrm{GL}_{n-k}}^{\mathrm{GL}_n} V(\alpha)_k \otimes V(\beta)_{n-k} \cong \bigoplus_\lambda c_{\alpha,\beta}^\lambda V(\lambda)_n ?$$
(I know it is not but true but it should be true up to being semi-simple.) See new question here

REPLY [5 votes]: The "nice" representations of $GL_n(\mathbb{C})$ are equivalent to the "nice" representations of $U_n$; the question is just how you should define nice. Let $\rho: GL_n(\mathbb{C}) \to GL_N(\mathbb{C})$ be a map of groups. Then the following are equivalent:

The entries of the matrix $\rho(g)$ are complex analytic functions of the entries of the matrix $g$.
The entries of the matrix $\rho(g)$ are polynomials in the entries of the matrix $g$ and $\det(g)^{-1}$.

Let $\sigma: U_n \to GL_N(\mathbb{C})$ be a map of groups. The following are equivalent:

The entries of the matrix $\rho(g)$ are continuous functions of the entries of the matrix $g$.
The entries of the matrix $\rho(g)$ are smooth functions of the entries of the matrix $g$.
The entries of the matrix $\rho(g)$ are polynomials in the entries of the matrix $g$ and their complex conjugates.

If we use any of the conditions in these lists to define "nice" representations, the restriction from $GL_n(\mathbb{C})$ to $U_n$ is a functor from nice $GL_N$ representations to nice $U_n$ representations. 
Theorem This functor is an equivalence of categories. In particular, every $U_n$ representation lifts to a $GL_N$ representation, $\dim Hom_{GL_N}(V,W) = \dim Hom_{U_n}(V|_{U_n}, W|_{U_n})$ and $V$ is irreducible if and only if $V|_{GL_n}$ is irreducible.
The easy parts of this are that  $\dim Hom_{GL_N}(V,W) = \dim Hom_{U_n}(V|_{U_n}, W|_{U_n})$ and $V$ is irreducible if and only if $V|_{GL_n}$ is irreducible. I don't know an easy way to see that every nice $U_n$ representation extends to a nice $GL_n$ representation. I covered this material on October 10 of my representation theory course. See Chapter 5 of Joel Kamnitzer's lecture notes for a more general discussion.
I don't know of any good results about non-nice representations of $U_n$. Non-nice representations of $GL_n$, however, are interesting for two reasons. First of all, you might want to study $GL_n(\mathbb{C})$ as a topological group, ignoring its complex structure. For example, $SL_2(\mathbb{C})$ is a double cover of $SO(3,1)$. Physicists are very interested in the representation theory of $SO(3,1)$, but they don't particularly care that the representations be analytic.
Second, there are interesting infinite dimensional representations of $GL_n$. In contrast, by the Peter-Weyl theorem, any continuous Hilbert space representation of $U_n$ is a direct sum of finite dimensional representations, so passing to infinite dimensions doesn't make the $U_n$ theory deeper.