# Correspondence of representation theory between $\mathrm{GL}_n(\mathbb C)$ and $\mathrm U_n(\mathbb C)$

1. 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

2. Similarly for the groups $\mathrm{SO}_n(\mathbb C) \subset \mathrm O_n(\mathbb C)$?

3. 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

4. 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

• While these questions are certainly all about representation theory, they really aren't related (except 1 and 2), are they? I think these 3 and 4 should should each get their own question, and a more descriptive title would then be appropriate. – RghtHndSd Mar 10 '14 at 23:46

The representations theory of $U_n$ is simpler than that of $GL_n(\mathbb{C})$: every continuous representation of $U_n$ is semisimple, and every continuous irreducible representation is finite dimensional and is the restriction of an irreducible algebraic representation of $GL_n(\mathbb{C})$. In fact, restriction from finite dimensional algebraic representations of $GL_n(\mathbb{C})$ to $U_n$ is a $\mathbb{C}$-linear equivalence of tensor categories. The basic idea of the proof is to observe that $U_n$ is Zariski dense in $GL_n$ and therefore irreducible algebraic representations remain irreducible upon restriction (this sometimes goes by the name Weyl's unitary trick and can be used to prove his eponymous character formula). Of course, the representation theory of the non-compact Lie group $GL_n(\mathbb{C})$ is much more complicated than this!

The relation between $SO_n$ and $O_n$ is even simpler: since every matrix in $O_n$ has determinant $\pm 1$, $SO_n$ is an index $2$ normal subgroup of $O_n$. Clifford theory then completely describes the relationship between their irreducible representations: each irreducible $O_n$-module either remains irreducible or splits into two irreducibles upon restriction, and which of these happens is determined by the behavior of representations upon tensoring with the determinant representation.

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.

• Nice answer, David (+1). Personally, I think the infinite dimensional representations are "nicer". But I admit that it's a matter of taste. As for how to prove that every continuous irreducible $U_n$-rep lifts to $GL_n(\mathbb{C})$, one possible strategy is to observe that they are classified by highest weights and that you actually obtain all possible highest weights by restriction. Then by semisimplicity every continuous rep lifts. In fact, although I have never studied the proof I always imagined this was how it went for arbitrary maximal compacts in reductive groups. – Stephen Nov 4 '14 at 17:17
• Thanks. I tend to over use judgmental adjectives. How about this: The fact that infinite dimensional irreps exist at all is fascinating, and the basic theory of their classification is deeper. At the same time, they don't have as many convenient theorems, and as a result people tend to get less far into intricate combinatorial formulas than they do in the finite dimensional case. – David E Speyer Nov 4 '14 at 17:32
• As for proving $K$-reps lift to $G$-reps ($G$ is an affine algebraic group, $K$ is a maximal compact with $\mathfrak{g} =\mathfrak{k} \otimes \mathbb{C}$, I think there are two routes. One way is to do roughly what you suggest with weight theory. If you look at my lecture notes that I linked, you'll see that argument paired down to what I think is the minimum prerequisites. – David E Speyer Nov 4 '14 at 17:34
• The other way is to start with $K$ and define $\mathcal{O}(K)$ to be the set of continuous functions $f: K \to \mathbb{C}$ such that the vector space $\mathrm{span}_{k \in K} (x \mapsto f(kx))$ is finite dimensional. (In terms of the Peter-Weyl decomposition, these are the functions with finitely many Fourier terms.) It isn't too hard to show that $\mathcal{O}(K)$ is a commutative $\mathbb{C}$ algebra. One can define $G$ as $\mathrm{Hom}_{\mathrm{algebra}}(\mathcal{O}(K), \mathbb{C})$ and define the complex structure and group structure on $G$ from that. The troubles are: – David E Speyer Nov 4 '14 at 17:37
• (1) There is an obvious map $K \to G$: Send $k$ to evaluation at $k$. But, to prove that it is injective seems to need something roughly as strong as Peter-Weyl. This isn't a problem for a concrete choice of $K$ like $U_n$, where you can concretely write down lots of elements of $\mathcal{O}(K)$. – David E Speyer Nov 4 '14 at 17:39