Non-semisimple Lie groups are so weird. Weyl's unitarian trick does not work for them. So I need to constantly remind myself that:
representations of GL(n,ℂ) are not determined by their character,
not every finite-dimensional representation of GL(n,ℂ) is completely reducible,
Finite-dimensional GL(n,ℂ)-representations are not in 1:1-correspondence with finite-dimensional U(n)-representations.
However these work when you look only at irreducible representations, or when you replace GL by SL (and U by SU). The archetypical counterexample is given by the (reducible but indecomposable) representation
[\rho: \mathrm{GL}(1,\mathbb{C})=\mathbb{C}^\times\to\mathrm{GL}(2,\mathbb{C}):\quad z\mapsto\begin{pmatrix}1&\log |z|\0&1\end{pmatrix}.]
(Example shamelessly stolen from: https://math.stackexchange.com/questions/2392313/irreducible-finite-dimensional-complex-representation-of-gl-2-bbb-c)
"An important example is that in which G is the complex general linear group, and K the unitary group acting on vectors of the same size. From the fact that the representations of K are completely reducible, the same is concluded for those of G, at least in finite dimensions."
So it claims that all finite-dimensional representations of GL(n,ℂ) are completely reducible?
Surely I must have misunderstood something??
@pschwahn I'm trying to fix that article, but I'm confused.
For example, the article says that Weyl's unitary trick works for reductive groups, of which GL(n,C) is one [is at true?], but you say above that the trick doesn't apply to GL(n,C).
Your Fulton-Harris exercise also seems to contradict a statement about reductive groups made in Wikipedia, namely that all finite-dimensional representations of a reductive group in characteristic 0 are completely reducible.
@AxelBoldt Thanks for chiming in! I'm going to reply here since I'm not familiar with the Wikipedia rules/etiquette.
GL(n,ℂ) is certainly reductive (see Fulton-Harris, Ex. 9.25). Perhaps there is a version of the Weyl trick which does apply to reductive groups, but I am not aware of its formulation.
I think your suspicion in the Wikipedia talk about algebraic groups is correct: The (finite-dimensional) RATIONAL representations ρ of GL(n,ℂ), i.e. those where the entries of ρ(g) are rational functions of the entries of g∈GL(n,ℂ), are indeed determined by their character, completely reducible, and in 1:1-correspondence with finreps of U(n). And the counterexample I wrote down is not a rational representation!
Unfortunately I don't have a reference for the converse (that every completely reducible representation of GL(n,ℂ) is rational), but I strongly suspect this is the case.
@pschwahn OK, let's get to the bottom of this. Is it possible that there are two versions of Weyl's trick, one for semisimple Lie groups and their finite-dimensional analytic representations, and one for reductive linear algebraic groups and their finite-dimensional rational representations?
Does the Fulton-Harris book describe Weyl's trick, or do we have some other reference? I assume that book deals with analytic representations, correct?
@AxelBoldt Fulton-Harris describes Weyl's trick in Chapter 9 and it quite adamant about that it only works for semisimple Lie groups. (I don't think analyticity of representations is required however, just smoothness.)
Generally, the trick works by establishing a 1:1-correspondence between the (finite-dimensional) representations of a complex Lie group G with that of a compact real form G₀. And I think this can be done if the group G is reductive (hence a compact real form exists) provided we consider only rational representations.
The critical step, I believe, is that a complex representation of (\mathfrak{g}=\mathfrak{g}_0^\mathbb{C}) is determined by its restriction to (\mathfrak{g}_0). This fails for our counterexample, but seems plausibly true for rational (or even analytic?) representations. We just need the infinitesimal representation (d\rho: \mathfrak{g}\to\mathfrak{gl}(n,\mathbb{C})) to be complex-linear, right?
@pschwahn I'll check out the Fulton-Harris book then. It would be great if we could also track down a reference for the rational-representation case.
I'm pretty sure the confusion in the Wikipedia article and in the math stackexchange discussions all stem from conflating these two cases.
I have just one question right now: G is a complex Lie group (i.e. a complex manifold with holomorphic group operations) while G_0 is a real Lie group (with smooth operations), correct? It somehow feels wrong to consider smooth non-holomorphic representations of a complex Lie group, but maybe that's just me.
@AxelBoldt If I find anything about the rational case, I'll get back to you!
Right, the operations of G are holomorphic - but representations of Lie groups are usually taken as being just smooth (the representation doesn't necessarily "see" the complex structure of G). The problem (existence of non-rational representation) probably also applies to other (noncompact) real forms of G.
@pschwahn I find in Fulton-Harris right after exercise 7.7 the statement "A representation of a complex Lie group
G is a map of complex Lie groups from G to GL(V) for an n-dimensional complex vector space V; note that such a map is required to be
complex analytic."
I am starting to believe that Weyl's trick relates the complex analytic representations of a complex (reductive?) Lie group to the smooth representations of a real compact Lie group.
@AxelBoldt you're right, that does make sense. I just realized that (z\mapsto z^k|z|^w) is a representation of (\mathbb{C}^\times) for all fixed (k\in\mathbb{Z}, w\in\mathbb{C}) but only analytic when (w=0).
So we should really be talking about complex analytic representations.
However for a group such as GL(𝑛,ℝ) it looks like the issue persists - a representation such as 𝑔↦|det(𝑔)|ʷ for non-integer w will be real-analytic, but it will not correspond to a representation of U(n).
This is turning out more subtle than I had imagined...
"However for a group such as GL(𝑛,ℝ) it looks like the issue persists - a representation such as 𝑔↦|det(𝑔)|ʷ for non-integer w will be real-analytic, but it will not correspond to a representation of U(n)."
U(n) is not a maximal compact subgroup of GL(n,ℝ) - it's not even contained in GL(n,ℝ). So there's no way to restrict a representation of GL(n,ℝ) to U(n), and you shouldn't expect an equivalence (or even a functor) from the category of representations of GL(n,ℝ) to those of U(n).
The maximal compact subgroup of GL(n,ℝ) is O(n), so you can restrict representations of GL(n,ℝ) is O(n). But a bunch of different real-analytic representations of GL(n,ℝ) restrict to the same representation of O(n), like all the representations 𝑔↦|det(𝑔)|ʷ. If I remember correctly this particular example is the "only problem". Of course it has spinoffs: you can tensor any representation of GL(n,ℝ) by a representation 𝑔↦|det(𝑔)|ʷ and get a new one which is the same on O(n).
I hope I'm remembering this correctly: every finite-dimensional smooth representation of GL(n,ℝ) is completely reducible, and every irreducible smooth representation comes from one described by a by Young diagram, possibly tensored by a representation 𝑔↦det(𝑔)ʷ where w is some real number, possibly also tensored by a representation 𝑔↦|det(𝑔)|ʷ where w is some real number.
It's a lot easier to find treatments of the 'algebraic' representations of GL(n,ℝ), and it's even easier to find them for SL(n,ℝ).
@johncarlosbaez@AxelBoldt oh yeah, that does make sense! I was trying to compare representations of GL(n,ℝ) and U(n) because they have the same complexification, but of course, the maximal compact of GL(n,ℝ) is O(n).
This description of irreducible smooth representations does sound plausible! However I don't think every finite-dimensional smooth representation of GL(n,ℝ) is completely reducible - take the example from the beginning,
[\mathrm{GL}(1,\mathbb{R})\to\mathrm{GL}(2,\mathbb{R}),\quad t\mapsto\begin{pmatrix}1&\log|t|\0&1\end{pmatrix}.]
But this counterexample of course goes away if we only care about rational representations.
@pschwahn@AxelBoldt - Ugh! I should have stuck with rational representations, which is what people usually talk about when studying representations of linear algebraic groups.
I'm pretty sure that every finite-dimensional rational representation of GL(n,ℝ) is completely reducible, and every irreducible rational representation comes from one described by a by Young diagram, possibly tensored by a representation 𝑔↦det(𝑔)ⁿ where n is some integer.
(There is some overlap here since the nth exterior power of the tautologous representation, described by a Young diagram, is also the representation 𝑔↦det(𝑔).)
It's annoying that the basic facts about finite-dimensional representations of GL(n,ℝ) aren't on Wikipedia! Someday I'll have to put them on there... once I get enough references to make sure I'm not screwing up!
@johncarlosbaez@AxelBoldt Yep, I think rational representations of GL(n,ℝ) can without problems be complexified to rational representations of GL(n,ℂ), which are then complex analytic (so in particular they correspond to the smooth finreps of U(n)). And complex analytic representations of GL(n,ℂ) are discussed in Fulton-Harris Section 15.5 - they are exactly as you said!
@pschwahn@AxelBoldt - right, that's one way to proceed. I've been doing a lot of work lately with representations of GL(n,𝔽) for 𝔽 an arbitrary field of characteristic zero. For subfields of ℂ this trick of complexifying and reducing to the case 𝔽 = ℂ works fine. But in fact the representation theory works exactly the same way even for fields of characteristic zero that aren't subfields of ℂ!
It's not that I really care about such fields. I just find it esthetically annoying to work only with subfields of ℂ when dealing with something that's purely algebraic and shouldn't really involve the complex numbers. So I had to learn a bit about how we can develop the representation theory of GL(n,𝔽) for an arbitrary field of characteristic zero. Milne's book 𝐴𝑙𝑔𝑒𝑏𝑟𝑎𝑖𝑐 𝐺𝑟𝑜𝑢𝑝𝑠 does this, and a preliminary version is free:
but unfortunately it's quite elaborate if all you want is the basics of the representation theory of GL(n,𝔽).
(For 𝔽 not of characteristic zero everything changes dramatically, since you can't symmetrize by dividing by n!. Nobody even knows all the irreps of the symmetric groups.)
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