SL 2(R)

Let G be a locally compact group, always assumed Hausdorff. Let H be a Banach space (which in most of our applications will be a Hubert space). A representation of G in H is a homomorphism of G into the group of continuous linear automorphisms of H, such that for each vector v ∈ H the map of G into H given by is continuous. One may say that the homomorphism is strongly continuous, the strong topology being the norm topology on the Banach space. [We recall here that the weak topology on H is that topology having the smallest family of open sets for which all functionals on H are continuous.]

In this section we essentially work out a special case of representation theory over compact groups, but in the context of SL ₂(R), providing a good introduction for what follows. We bring out immediately the important role of a maximal compact subgroup, the circle group K, i.e. the group of matrices A character of k is by definition a continuous homomorphism of k into the unit circle, and the characters are indexed by integers

We shall study SL ₂(R) by decomposing it as a product of certain closed subgroups (not normal). Here we recall the general foundations for integration on coset spaces.

In this chapter and the next we study the algebra of functions on G which are invariant on the left and on the right by K, and relate the characters of this algebra to representation theory, This amounts to studying those representations which contain a K-fixed vector. We cover §3, §4 of the last chapter of Helgason's book [He 2]. We work with the abstract nonsense of Haar measure and convolution, without differential operators. This point of view was emphasized by Godement [Go 6]; see also Tamagawa [Tarn], which we follow in part. To prove that all spherical functions are those which we exhibit explicitly, we need the differential equations, and the proof is postponed to Chapter X, §3.

In this chapter we study the spectral decomposition of the algebra C _c ^∞ (G//K), consisting of those C^∞ functions with compact support, bi-invariant under K. We shall also determine the bounded spherical functions. We let G = SL ₂(R) throughout.

In this chapter for the first time we begin to deal with the C ^∞ or (real) analytic structure of G, rather than with just measure theory. We shall see how a representation of G gives rise to an algebraic representation of the Lie algebra on a dense subspace, for an arbitrary Lie group G. In the case of SL ₂(R), this representation has an especially simple form, as shown in §2.

In this chapter we deal systematically with the trace in infinite dimensional representations, especially those we have explicitly constructed. We prove that in the induced representations, the trace expressed as the integral of a kernel over the diagonal can be identified with the usual sum of diagonal matrix coefficients. We then compute the trace in various representations.

We shall put together the facts we have learned about traces in order to prove the Plancherel formula, giving an expansion of a function in terms of its characters. The proof is due to Harish-Chandra [H-C 6]. It consists in expanding out the Fourier series of the Harish transform H ^Kψ(θ) using the relation between the trace of the discrete series and the Harish transform on A given in Theorem 6 of the preceding chapter, and then performing a Fourier transform on some of the terms to get the final formula. It turns out that H ^Kψ(k_θ) is not continuous, the discontinuity occurring at those elements of k which also lie in A, i.e. at ± 1. The first calculus lemma serves to determine the jumps, and also shows that the derivative (H ^Kψ)’(k_θ) is continuous at those points, and that the Fourier series converges for the derivative at those points. This gives us the value ψ(1), in terms of a series involving traces in the discrete and principal series.

In this chapter we give various unitary realizations of the discrete series, i.e. those irreducible representations which admit a lowest weight vector of weight ≥ 2 and highest weight vector of weight ≤ -2. It turns out that in each case, one is the complex conjugate of the other, so essentially we need only look at those with a lowest weight vector. We shall see that they admit an infinitesimal embedding in L ²(G) with the action of left translation, and also that they can be represented as operations on certain function spaces in the upper half plane.

So far we have avoided to a large extent the more refined behavior of functions with respect to Lie derivatives. For the theory of spherical functions, we dealt with eigenvectors of convolution operators. The time has come to relate some invariants we have found in the representation theory with some of the invariant differential operators on G. Bargmann [Ba] saw how coefficient functions are eigenfunctions of such operators, Harish-Chandra got a complete insight into the situation by determining the center of the algebra of invariant differential operators, the centralizer of K in this algebra. Gelfand characterized spherical functions as eigenfunctions of this centralizer. In this chapter, we give Harish-Chandra’s result that there are no other spherical functions, besides those described in Chapter IV, on SL ₂(R) where the proofs are short and easy.

There is a whole aspect of SL ₂(R) into which we shall not go, namely the various models which can be found in an infinitesimal equivalence class of representations, and the possibility of finding canonical models, e.g. the Whittaker model in such a class. We refer the reader to Jacquet-Langlands [Ja, La], Knapp-Stein [Kn, St], and Stein [St 2] for more information in this direction, and a discussion of intertwining operators among various models. Helgason [He 3] gives a particularly interesting model of representations in eigenspaces of the Laplacian. I include here just the special model of the Weil representation because of its particular interest in number theoretic applications, and the possibility of constructing automorphic forms with it, as in Shalika-Tanaka [Sh, Ta]. Besides, since Weil’s Acta paper [We] is written in an extremely general context, it may be useful to have a naive treatment of the special case as an introduction. Finally, the way the Weil representation is constructed provides an excuse for giving generators and relations for SL ₂, and for mentioning the Brahat decomposition. I did not want to get very much involved in the matters discussed here, and so the chapter is somewhat arbitrary.

The algebraic and arithmetic properties of SL ₂ begin to be felt when we consider the representation on Г\G for some discrete subgroup Г. In this chapter, after a general discussion of the nature of the factor space Г\G or Г\G/K = Г\ℌ, which is essentially classical, we prove that on a certain subspace ⁰L²(Г\G) of L²(Г\G) the representation is completely reducible when Г = SL ₂(Z). The method works just as well for any “arithmetic” subgroup, i.e. a subgroup of finite index in SL ₂(Z). It uses the Poisson summation formula, in addition to some estimates. It has the advantage of being very rapid and of using a minimum of analysis.

We now look at the orthogonal complement of ⁰ L ²(Г\G) and prove a spectral decomposition theorem following Godemenťs paper [Go 2], using the Poisson summation formula. The method works for arithmetic subgroups Г, and has the advantage of being rapid and easy. It fails for more general discrete subgroups, and the question is reconsidered by other methods in the next and last chapter. The spectral decomposition is achieved by the Eisenstein transform, which maps the orthogonal complement of ⁰ L ²(Г\G) and the constant functions on the L ² space of a positive real line—with our normalization, the upper half of the imaginary line Re s = 1/2.

This chapter reproduces, with a number of details added, a paper of Faddeev [Fa 1].