# Harmonic Analysis on the real special linear group

I would like to understand the representation theory and generalized Fourier transform of $SL(3, \mathbb{R})$ in as concrete a manner as possible. My ultimate goal is to develop an algorithm that can perform the Fourier transform of a discretely sampled function on this group (or a homogeneous space on which it acts), so I will need concrete formulas for the Haar measure and the matrix elements of irreducible representations.

Background

By "generalized Fourier transform" I mean the expansion of a suitably regular function $f : G \rightarrow \mathbb{R}$ in terms of matrix elements of irreducible representations. Let $U^\lambda$ denote a unitary irreducible representation of $G$ on a Hilbert space, then the Fourier transform is defined as: $$\hat{f}(\lambda) = \int_G f(g) U^\lambda(g^{-1}) d\mu(g)$$

Where $\mu$ is the bi-invariant Haar measure on $SL(3, \mathbb{R})$, which exists because this group is semisimple and hence unimodular.

To make this work, we need to know what all the IURs $U^\lambda$ are, and which ones we need in order to decompose our function $f$. The irreducible unitary representations of semisimple groups have been classified, and it turns out that the spectrum (indexed by $\lambda$) contains both a discerete and a continuous part. However, it is my understanding that not all of the IURs appear in the decomposition of a function space such as $L^2(G)$. Knapp & Trapa write that:

Roughly (but not exactly) the members of the unitary dual [the set of equivalence classes of IURs] that appear in $L^2(G)$ can be obtained by a process called "induction" that starts from discrete series of certain reductive subgroups of G.

Hence not all IURs are needed to decompose $L^2(G)$.

Questions

1. What is the explicit form of the Haar measure on $SL(3, \mathbb{R})$ (in terms of some convenient parameterization)?
2. Which IURs do I need in order to decompose a function in $L^2(G)$? I'm not set on this particular function space, so if another one is easier feel free to modify the question.
3. Are there any explicit formulas known for the matrix elements of the relevant IURs? What about the basis functions of irreducible representation spaces in $L^2(G)$?
4. Is anything known about the asymptotics of these functions? This is important because the IURs of $SL(3, \mathbb{R})$ are infinite-dimensional matrices, so an algorithm will have to choose some cutoff. If functions $U^\lambda_{mn}$ do not decay sufficiently fast (or at all) with $m,n$, that may be a showstopper.

Although I'm ultimately interested in concrete formulas, partial answers including references or somewhat abstract claims about the representation theory and harmonic analysis on this group are still very much appreciated.

Some relevant sources that I am aware of, but have not read in their entirety yet:

[1] Knapp, A. W., & Trapa, P. E. Representations of Semisimple Lie Groups.

[2] Gel’fand, I. M., Graev, ‎M. I., & Vilenkin, ‎N. Y. (n.d.). Generalized Functions Vol. 5: Integral Geometry and Representation Theory.

[3] Vilenkin, N. J., & Klimyk, A. U. (n.d.). Representation of Lie Groups and Special Functions - volume 1: Simplest Lie Groups, Special Functions and Integral Transforms.

[4] Ehrenpreis, L., & Mautner, F. I. (n.d.). Some Properties of the Fourier Transform on Semi-Simple Lie Groups I.

• I will answer your question as soon as I have time, today or tomorrow; it is very interesting. I think your program is far too optimistic, but there are many good things to be said. – guest Dec 29 '14 at 17:05
• Thanks you very much! Looking forward to your answer! – John von N. Dec 30 '14 at 8:42

First of all, you seem to be bundling up a lot of requests on your self :"develop an algorithm that can perform the Fourier transform of a discretely sampled function on this group (or a homogeneous space on which it acts)" - these are two very different things, if the homogeneous space has finite volume.

Secondly, you need to clarify (not here, but in your program) what you mean by 'performing the FT on sampled functions'. As you write in your post, the FT takes $L^2(G)$ functions to operators (distributions). Whether and when such distributions are even given by locally integrable functions is an issue that took Harish-Chandra decades to settle with his regularity theorem. When you form the matrix coefficents, you need to decide which space of functions you will use; fortunately there is a canonical choice (the Schwartz space, I will talk about this below).

Now suppose we sample the same function with the same accuracy on the same 'grid', but my grid is just tilted a little bit in a compact direction. I expect the distributions that result to be very different unless your space of test functions is very restricted.

And even worse: suppose we sample two different functions on the same grid and the samples have the same values (perfectly possible with $L^2$ or even smooth functions - remember partitions of unity). What then? The sampled transforms will be identical and any reconstruction via Plancherel (see below) cannot give both my function and yours. You may say: it will give the function up to some distortion that becomes small with the mesh of the grid. I doubt that, unfortunately. There are all sorts of boundedness issues for these inverse operators that cannot possibly be true in this generality.

What I do think can work and is feasible is to construct an algorithm that at least decomposes functions in $C^\infty_c(K\ G/K)$ via the spherical Fourier transform. This simplifies things very much due to the symmetries of spherical functions (cutting down on what you need to approximate) and is still a good and formidable project. Plus, in that case, the Paley-Wiener theorem can be used as a steering wheel for the algorithm, if the latter will be some iterative procedure. I will give you references for all these things in a bit.

What you are looking for is an explicit Plancherel formula for $\textrm{SL}(3,\mathbb{R})$. The best source for the Plancherel formula at the level you seem to be comfortable with is Varadarajan's 'An introduction to Harmonic Analysis on Semisimple Lie Groups'. It is much ligher than either Knapp's great treatise (Representation theory of semisimple lie groups) and Wallach's fairly complete reference (real reductive groups) but gives the representations and Plancherel in the case of $\textrm{SL}(2,\mathbb{R})$ explicitly, describes in detail the process of induction on parabolic subgroups from their discrete series and also treats explicitly the Plancherel formula for complex groups; in fact, I recommend you start there: if you cannot do the approximation on $\textrm{SL}(2,\mathbb{C})$ and then $\textrm{SL}(3,\mathbb{C})$, it will be hopeless to try for a vastly more complicated group like $\textrm{SL}(3,\mathbb{R})$. Varadarajan's book also has details on everything else I mentioned so far.

1. There are many ways to decompose the Haar measure. The most common one is to use the Iwasawa decomposition $G=KAN$ and write $dh= \delta^2(a) dk\,da\,dn$ where $\delta^2$ comes from the modular function of the Borel subgroup of upper triangular matrices and you can compute it by yourself, or find it in any book on Lie groups in the case of $\textrm{SL}(n)$, including Var. Chapter 4.4. Note that $dn$ is Lebesgue measure on a space homeomorphic to some $\mathbb{R}^n$ but $dk$, the Haar measure on the compact group, can still be opaque. Treating $K$-bi-invariant functions as I suggested above allows you to use Weyl's beautiful integration formulas on compact groups; you may be sick of seeing hyperbolic functions, but they are at least very explicit.

2. The representations that go into the spectral measure are so-called tempered representations, and these are exactly those that occur in the Plancherel formula. The latter is the means by which you invert (in principle) the Fourier transforms of a well behaved function. They are discrete series for the group (complicated) and induced representations from discrete series of Levi subgroups of parabolics (these are the principal series). You can find all this in Chapters 3 and 4 of Varadarajan's book.

3. I do not think so. Even for the spherical functions the integral representations are the best you will get, but they should suffice for approximation. Since most special functions of classical analysis already appear as matrix coefficients of irreps, it makes sense that the latter can be very complicated and not expressible in closed form.

4. Not in general. First of all you now need to restrict to zero mean vectors, things like $L^2_0(G)$ for anything to decay. In that space, no, there are no good decay properties in general (you can find examples in Howe-Tan, Non-abelian harmonic analysis). If you restrict to smooth, $K$-finite zero mean vectors, yes; your best bet is the Schwartz class, defined for instance in Chapter 8 of Varadarajan. There the matrix coefficients of tempered representations decay exponentially in any norm on the Lie algebra, i.e. as $e^{-C\|\log(g)\|}$. Better still, their decay properties are controlled by the $\Xi$ function, which is very well behaved.

Finally, let me recapitulate: try $\textrm{SL}(2,\mathbb{C})$ first. Then $\textrm{SL}(2,\mathbb{R})$. Already in the second case you will encounter huge problems. The Plancherel formula is derived explicitly in Serge Lang's book $\textrm{SL}(2,\mathbb{R})$. I urge you to look there for the spherical transform and focus on that case. For the higher rank case, your best bet is again the spherical transform, and complex $\textrm{SL}$, for which Varadarajan gives an explicit treatment.

• Thanks for writing such a detailed and thoughtful answer! Will need some time for this to sink in.. – John von N. Jan 1 '15 at 9:32
• @JohnvonN. I understand. You see, harmonic analysis on non-compact non-abelian Lie groups is EXTREMELY complicated even in the simplest cases. There are two principles you should be aware of: complex groups are vastly simpler than real groups and rank 1 groups vastly simpler than higher rank groups. This is why I am recommending you start with $\textrm{SL}(2,\mathbb{C})$ and then proceeding to $\textrm{SL}(2,\mathbb{R})$ rather than tackling your problem directly. You need experience and background knowledge before leaping into the dragon's lair (a full plate armor would not hurt either). – guest Jan 1 '15 at 9:38
• I had a feeling this was going to be challenging, but it's quite clear to me now that $\operatorname{SL}(3,\mathbb{R})$ is more than I can handle for now. Luckily the 2D affine group is almost as good for my application, and this is the semidirect product of $\mathbb{R}^2$ and $\operatorname{SL}(2, \mathbb{R})$ so that may be doable. – John von N. Jan 1 '15 at 10:14
• @JohnvonN. That is different. It is not reductive, so the usual theory does not apply. For this semidirect product, what you need is the $SL(2)$ case, which is explicitly written in Lang, and Mackey's theory for semidirect products. Search "system of imprimitivity" in Google or make a new question; I can answer that. Yes, due to Mackey's theory and the explicit formulae you have for $\textrm{SL}(2)$, this is much more doable. – guest Jan 1 '15 at 10:16
• @JohnvonN. Also, a huge amount of harmonic analysis on this semidirect product has been done explicitly in the book of Howe and Tan "Non-abelian harmonic analysis". That is where I would start. Apart from some obvious prerequisites from functional analysis and a little algebra, it is very elementary and explicit. – guest Jan 1 '15 at 10:19