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From the Peter-Weyl theorem in wikipedia, this theorem applies for compact group. I wonder whether there is a non-compact version for this theorem.

I suspect it because the proof of the Peter-Weyl theorem heavily depends on the compactness of Lie group. It is related to the spectral decomposition of compact operators.

Thanks Mariano pointing out the Peter-Weyl theorem does not hold for non-compact group. But I really wants to know is: is there any Peter-Weyl analogue decomposition for non-compact group, say decompose to integral representations but not finite dimensional representations?

Another related questions is about the definition of quantized flag variety. In the work of Lunts and Rosenberg on localization for quantum group, they tried to establish the quantum analogue of Beilinson-Bernstein localization theorem. They defined the quantized flag variety in the framework of noncommutative algebraic geometry. They used the Peter-Weyl philosophy for quantum group to define the coordinate ring of quantized base affine space as the direct sum of all simple $U_{q}(g)$-modules with highest weight $\lambda$(positive).(Denoted by $R_{+}$)

Then one can define category of quasi coherent sheaves on "quantized flag variety" as proj-category of graded $R_{+}$.

What I want to ask is there any other way to define quantized flag variety? In the classical case, It is well known that flag variety can be define as $G/B$, say $G$ is general linear group and $B$ is Borel subgroup. Is there any analogue for quantum case? Is there a definition like $G_{q}$ as "quantum linear group" and $B_{q}$ as quantum analogue of Borel subgroup?

However,I suspected, because the quantum flag variety is essentially not a real space

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I am 100% ignorant of the work done by Lunts and Rosenberg, but I am pretty sure you can get quantized flag varieties (at least, one version of them) as the coinvanriant sublgebra for an appropriate coaction of one quantum group on another. You can get the quantum Grassmannians as coinvariant subalgebras of quantum affine planes under coactions of quantum groups, for example. –  Mariano Suárez-Alvarez Apr 7 '10 at 15:56
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(And, of course, my point is that taking coinvariant subalgebras corresponds precisely to constructing quotients of the 'spaces') –  Mariano Suárez-Alvarez Apr 7 '10 at 15:56
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It's not clear in the question what kind of "quantum group" is wanted, since they come in different flavors. Apart from that, Harish-Chandra and many others have contributed to the classical theory for noncompact groups. The questions here need a more specific context. –  Jim Humphreys Apr 7 '10 at 22:20

7 Answers 7

up vote 5 down vote accepted

What I want to ask is there any other way to define quantized flag variety? In the classical case, it is well known that flag variety can be defined as $G/B$, say $G$ is general linear group and B is Borel subgroup. Is there any analogue for quantum case? Is there a definition like $G_q$ as "quantum linear group" and $B_q$ as quantum analogue of Borel subgroup?

The Borel subgroup of the quantized function algebra $G_q$ is of course certain quotient Hopf algebra $p:\mathcal{O}(G_q)\to \mathcal{O}(B_q)$ which then canonically coacts on $G_q$ from both sides (from the left by $(p\otimes id)\circ\Delta_{G_q}:\mathcal{O}(G_q)\to \mathcal{O}(B_q)\otimes \mathcal{O}(G_q)$. In the case of $SL_q(n)$ you get the lower Borel by quotienting by the Hopf ideal generate by the entries of the matrxi $T$ of the generators standing above the diagonal.

Of course, if you define $Qcoh(G_q)$ as the category ${}_{\mathcal{O}(G_q)}\mathcal{M}$ of left modules over $\mathcal{O}(G_q)$ then the category of relative left-right Hopf modules ${}_{\mathcal{O}(G_q)}\mathcal{M}^{\mathcal{O}(B_q)}$ is precisely the category of quasicoherent sheaves over the quantum flag variety $G_q/B_q$. If you look at several of my earlier articles they together show that in the case of $SL_q(n)$ at least, there is a collection of localizations of the above category of relative Hopf modules which are affine, which gives it a structure of a noncommutative scheme with a canonical atlas whose cardinality is the cardinality of the Weyl group (the generalization to other parabolics is easy). Moreover the forgetful functor from the category of Hopf modules to ${}_{\mathcal{O}(G_q)}\mathcal{M}$ is the inverse image part of the geometric morphism which is a Zariski locally trivial $B_q$-fibration; the local triviality, which is obtained with the help of the quantum Gauss decomposition, boils down to the Schneider's equivalence for the extension of the algebras of localized coinvariants into the corresponding localization of $G_q$. The sketch of the whole program is in

  • Localizations for construction of quantum coset spaces, in "Noncommutative geometry and Quantum groups", W.Pusz, P.M. Hajac, eds. Banach Center Publications vol.61, pp. 265--298, Warszawa 2003, math.QA/0301090.

For the Peter-Weyl look at the original works of S. Woronowicz or

  • A. Klimyk, K. Schmüdgen, Quantum groups and their representations, Springer 1997.
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Thank you very much! This might be exactly I am looking for –  Shizhuo Zhang Apr 7 '10 at 22:15

The conclusion of the part of the Peter-Weyl theorem stating that unitary representations of the group split as an orthogonal direct sum of finite dimensional ones is false if the group is not compact. There are even non-compact groups which simply do not admit any unitary finite dimensional representations, like the group $\mathrm{SL}(2,\mathbb R)$, which has, on the other hand, irreducible infinite dimensional representations.

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Thanks for pointing out. Of course, one can not expect finite dimensional representations. But can we expect integral representations? –  Shizhuo Zhang Apr 7 '10 at 15:47
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Since $\operatorname{SL}(2,\mathbb{R})$ was mentioned, maybe I should point out that the name of this group is also the title of a book by Serge Lang? ams.org/mathscinet-getitem?mr=430163 –  Harald Hanche-Olsen Apr 7 '10 at 16:06
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Maybe it should be added that this book of Lang's is not at all held in high esteem by the specialists, who seem to find it out of focus on the main issues. I don't have an opinion of my own on this. –  Jim Humphreys Apr 7 '10 at 22:15
    
Hmm, okay, thanks for the information. (I haven't read it myself.) –  Harald Hanche-Olsen Apr 9 '10 at 1:57

I don't know anything about quantum groups, but the Peter-Weyl theorem for compact groups generalizes nicely to Type I second-countable locally compact topological groups, a result of Segal and Mautner. For such groups, there is a decomposition of the unitary biregular representation: $$L^2(G) \cong \int_{\pi \in \hat G} E_\pi d \pi,$$ where

  • $\hat G$ is a (slightly well-chosen) set of representatives for each isomorphism class of irreducible unitary representation
  • When $\pi \in \hat G$, $E_\pi$ is the space of Hilbert-Schmidt endomorphisms of the Hilbert space of $\pi$.
  • $d \pi$ is the "Plancherel measure", uniquely determined from a choice of Haar measure on $G$.

When $G$ is compact, the direct integral -- i.e. spectral measure -- is replaced by a Hilbert space direct sum, and the above formula is the Peter-Weyl theorem.

See I. Segal, "An extension of Plancherel's formula to separable unimodular groups", Ann. of Math. 1950, for the unimodular case. Also see Mautner, "Note on the Fourier inversion formula on groups.", Trans. AMS 1955.

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Is that an end or an integral? –  Harry Gindi Apr 7 '10 at 16:01
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THe $E_\pi$ is a space of endomorphisms. The integral $\int$ is a Hilbert space integral -- the space of $L^2$-sections of a Hilbert space bundle over a measure space. Try looking around for "direct integral of Hilbert spaces" or "spectral measure" to understand better. –  Marty Apr 7 '10 at 16:09
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One could add that if $G$ is a reductive Lie group then Harish-Chandra computed the Plancherel measure explicitly (and that this was a very major piece of mathematics!). –  Emerton Apr 7 '10 at 16:22

Compact groups are pretty much the same as complex reductive groups, but one must replace finite dimensional unitary representations with finite dimensional holomorphic represententations. The Weierstrass approximation theorem says that polynomials are dense in $L^2$ for compact spaces. Putting this together, this yields an algebraic Peter-Weyl theorem that polynomial functions on, say, $GL_n(C)$ are the direct sum of endomorphisms of the irreducible holomorphic representations. One can also prove this directly, by essentially the same methods as the Peter-Weyl theorem, but rather easier.

If you are interested in non-compact groups, then this may not be the answer you're looking for, but if you're interested in quantum groups, it is a reasonable starting point.

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If you are happy to work with the Operator Algebra approach to quantum groups, then Woronowicz's definition of a Compact Quantum Group admits a perfect analogue of the Peter-Weyl theorem (looking at unitary corepresentations). The Wikipedia page doesn't give any references, but a good (and freely available) survey is: arXiv:math/9803122v1 [math.FA].

But maybe this isn't what you're after...

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I am not really sure what you are after. On the face of it, the question of "how much of the Peter-Weyl theorem survives for arbitrary locally compact groups?" admits the bald answer "Not very much" - there is a standard example of the free group on two generators admitting a unitary representation which admits a direct integral decomposition in two essentially different ways. (I first saw this on page 182 of Robert's book "Introduction to the representation theory of compact and locally compact groups" but unfortunately don't know of other, more easily accessible sources.)

But perhaps you are more interested in algebraic groups, or Lie groups? Well, my very limited understanding is that people have been bashing their brains against this for over 60 years. Supposedly the case of semisimple Lie groups is the easiest one to get a handle on. (It is relevant that semisimple Lie groups are Type I, so that one doesn't get the same "pathology" of non-uniqueness of decomposition that afflicts the case of the free group.)

Perhaps this overview/advert may have pointers to some literature or background that answer your question better.

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very helpful, thanks –  Shizhuo Zhang Apr 7 '10 at 22:23

Marty's answer discusses the Plancherel formula, and in a comment on his answer, I mentioned Harish-Chandra's work on the Plancherel formula in the case of reductive Lie groups. Yemon Choi's answer also mentions the case of semisimple Lie groups as being easier than the general case. The point of this answer is to elaborate slightly on my comment, and to point out that, while the semi-simple case might be easier, it is a very substantial piece of mathematics; indeed, it is essentially Harish-Chandra's life's work.

When Harish-Chandra began his work, it was known (thanks to Mautner?) that semisimple Lie groups were type I, and hence that for such a group $G$, the space $L^2(G)$ admits a well-defined direct integral decomposition into irreducibles (typically infinite dimensional, if $G$ is not compact). However, this is a far cry from knowing the precise form of the decomposition.

The most fundamental, and difficult, question, turns out to be whether there are any atoms in the Plancherel measure, i.e. whether $L^2(G)$ contains any non-zero irreducible subspaces, i.e. whether the group $G$ admits discrete series. This was solved by Harish-Chandra, who established his famous criterion: $G$ admits discrete series if and only if it contains a compact Cartan subgroup. He also gave a complete enumeration of the discrete series representations up to isomorphism, and described their characters via formulas analogous to the Weyl character formula.

Harish-Chandra then want on to describe the Plancherel measure on $L^2(G)$ inductively in terms of direct integrals of parabolic inductions of discrete series represenations of Levi subgroups of $G$. (It is the appearance of Levi subgroups, which are always reductive but typically never semisimple, that also forces one to generalize from the semisimple to the reductive case.)

After completing the theory of the Plancherel measure for reductive Lie groups, he then went on to develop the analogous theory for $p$-adic reductive groups. However, in this case, one still doesn't have a complete enumeration of the discrete series representations in general: there are certain "atomic" discrete series representations, called "supercuspidal", which have no analog for Lie groups, and which aren't yet classified in general (i.e. for all $p$-adic reductive groups).

Harish-Chandra's work, as well as standing on its own as an amazing edifice, was a central inspiration for Langlands in his development of the Langlands program, and remains at the core of the Langlands proram today.

For a very nice introduction to Harish-Chandra's work, and the surrounding circle of ideas, one can read this article by Rebecca Herb.

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One name I've seen mentioned in connection with "semisimple Lie implies Type I" is Godement -- there are certain shortcuts one can take using operator-algebraic machinery or trickery. I'm not sure if this was before or after Mautner. In any case, as you say, the analysis of the situation by Harish-Chandra, at the level of the Plancherel measure and discrete series, is much deeper (at least according to my novice-level reading of the literature). –  Yemon Choi Apr 8 '10 at 8:58

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