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Notes on SL(2,C): representation theory and harmonic analysis

These are preliminary notes for the article Representations of the Lorentz group. The finite-dimensional representation theory of SU(2) will be assumed as known. All its irreducible representations—the representation V_k of half-integer spin k ≥ 0 defined for example on the space of homogeneous polynomials in two variables of degree 2k—extend analytically to holomorphic representations of its complexification SL(2,C). On the other hand K = SU(2) acts unitarily on the infinite-dimensional Hilbert spaces of square integrable functions on the the 2-sphere and 3-sphere. In group theoretic terms the first space is the quasi-regular representation L²(S²) = L₂(K / T), where T is the maximal torus consisting of diagonal matrices in SU(2). The decomposition into irreducible representations is given by L₂(K / T) = V₀ ⊕ V₁ ⊕ V₂ ⊕ ⋅⋅⋅ The second space is the regular representation L²(S³) = L²(K) on which K × K acts by right and left translation. Its decomposition into irreducible representations of SU(2) × SU(2) is given by

$L^{2}(K)=\bigoplus _{k}V_{k}\otimes V_{k}^{*},$

where the sum is over all half-integer spins. In addition C(K) acts on L²(K) by left and right convolution; it generates norm-closed *-algebras isomorphic to ⊕ End V_k. Moreover the quasi-regular representation is naturally a subrepresentation of the regular representation, consisting of functions invariant under right translation Νby T. All of these infinite-dimensional results extend in a non-trivial way to the non-compact group G = SL(2,C). The corresponding quasi-regular representation of G is on L²(G / K), the square integrable functions on the quaternionic upper half-plane. The regular representation of G × G is on L²(G).

The quasi-regular representations can be decomposed as a direct integral of irreducible representations from the spherical principle series. Technically this is the simplest decomposition. It can be derived using the elementary theory of spherical functions, developed by Gelfand and Godement, which reduces to the spectral theory of the commutative *-algebra generated by convolution operators corresponding to K-bivariant functions C_c(K \ G / K). This in turn deduces to the spectral theory of the Laplacian operator on L² (K \ G / K). Using the polar decomposition G =KAK, the latter space can be identified with L²(A) = L²(R), where the Laplacian essentially becomes the −d² /dt². So in the case the spectral theory is a consequences of the Fourier transform. In terms of operator algebras, the commutant of G acting on L²(G / K) is generated by the commutative algebra of biinvariant functions acting by right convolution.

The quasi-regular representation is the representation of G induced from the trivial representation of K. For each irreducible representation V_k, there is a corresponding induced representation and the regular representation L²(G) breaks up naturally as the direct sum of these representations (its decomposition according to the action of K by right translation). As various mathematicians have observed—in particular Godement, Takahashi and Kornwinder—the commutant of G on each of these spaces is also commutative, so that the representation is also a direct integral of irreducible representations, again corresponding to the spectral decomposition of a Laplacian operator (the Casimir operator). The commutant is generated by convolution operators from the algebra of k spherical functions, which generalise the biinvariant functions. If χ_k is the normalised character on K of V_k (the trace multiplied by the dimension 2k + 1), then the k-spherical functions in C_c(G) are those invariant under conjugation by elements of K and left (or right) convolution by χ_k. When k > 0, the operator corresponding to the Laplacian has the form Sturm-Liouville form −d² / dt² + q, where the potential q is matrix-valued. Unlike the case k = 0, this cannot be treated in a direct elementary way. However, the theory of intertwining operators, due to Gelfand, Graev, Kunze, Stein and Knapp, provides a method of extending the theory to the case k > 0. The intertwining operators arise from singular integral operators on C, generalising operators due to Riesz. Although this theory is non-elementary, it gives a uniform approach and plays an important role in Harish-Chandra's extension of the Plancherel theorem to semisimple Lie groups.

After establishing the Plancherel theorem on L²(G / K), the theory of intertwining operators for the spherical principal series of SL(2,C) will be explained. Much of this theory extends without difficulty to the whole principal series, but this case will be presented because it is the most elementary, is a prototype for the general theory and has been fully developed by various authors, in particular Kunze, Stein, Knapp, Schiffmann, Helgason and Wallach. It is the simplest setting to explain how Harish Chandra's c-function arises from natural operators and how it is related to the Plancherel measure.

An elementary approach to the Plancherel theorem for SL(2,C) was given by Gelfand and Naimark; it is presented in detail in Naimark's book on representations of the Lorentz group and in the 1962 book of Gelfand, Graev and Vilenkin, translated into English in 1966.

After the Plancherel theorem was proved, the structure of C*(G) the full C* algebra of SL(2,C) was determined by Michael Fell, but only as a consequence of the Plancherel theorem. Fell's results can also be established directly using more recent results. Let A denote the reduced C* algebra of G in B(L²(G)), i.e. the closure in the operator norm of the *-algebra of convolution operators λ(f) for f in C_c(G). Equivalently it is λ(C*(G)). There is a natural homomorphism of A into

$B=C_{0}([0,\infty ),K(H_{0}))\oplus \bigoplus _{k>0}C_{0}((-\infty ,\infty ),K(H_{k})),$

by taking the fields of operators corresponding to f in the principal series. The image of A is closed. By a generalisation of the Stone–Weierstrass theorem (a special case of a theorem of Glimm), the image is dense in B and therefore the whole of B. On the other hand the theory of tempered representations can be combined with theory of C* algebras to prove that the homomorphism is injective. The relevant theory is explained in Dixmier's book on C* algebras and Volume 2 of Wallach's Representations of reductive groups. In fact the representations that occur in the Plancherel theorem are those weakly contained in the regular representation. These are precisely the irreducible representations of G which vanish on the kernel of the homomorphism of C*(G) onto A. As Harish-Chandra showed, for connected semisimple Lie groups these are the tempered representations, those for which the matrix coefficients of K-finite vectors are in L^{2 +ε} for every ε > 0. Cowling, Haagerup & Howe gave a direct argument for this which avoided the Plancherel theorem. The Plancherel theorem can thus be deduced from the above isomorphism using intertwining operators and their link with the c-function (Helgason, Wallach, Knapp, etc).

Kato–Birman theory

In this section, basic results of perturbation theory due to Kato, Birman, Rosenblum and Kuroda will be described. This guarantees that if A and B are unbounded self-adjoint operators on a Hilbert space H such that A has only absolutely continuous spectrum and (A + i I )^–1 – (B + i I )^–1 is of trace-class, then the operator W(t) = e^iAt e^–iBt has unitary limits W(±∞) in the weak operator topology called wave operators and satisfy e^iAt W(±∞) = W(±∞) e^iBt. Thus A = W(±∞) B W(±∞)* (Theorem of Birman-Kuroda), so that the wave operators give an explicit unitary equivalence between A and B. This result has the following corollary, due to Kato and Kuroda. If D is self-adjoint, with only absolutely continuous spectrum and Q is a self-adjoint operator with domain containing the domain of D, such that (D + iI)⁻¹Q is compact and D + iI)⁻¹Q(D + iI)⁻¹ trace-class, then D + Q is self-adjoint and unitarily equivalent to D by the wave operator for A = D and B = D + Q.

This applies in particular when D = −d²/ dx² on L²((0,∞)) and Q is multiplication by a continuous real-valued function q on (0,∞) such that ∫ (1+ x) |q| < ∞. It extends easily to the operator D ⊗ I on L²((0,∞), Cⁿ) and Q is multiplication by a continuous function q from (0,∞) into the self-adjoint n × n matrices such that ∫ (1+ x) ||q|| < ∞. So in this case, it is automatic that the Sturm-Liouville operator D⊗I + Q is unitarily equivalent to the operator D.

Almost all abstract perturbation theory results, including those stated above, follow from the following theorem. It is Pearson's generalisation of Kato-Birman theory, explained in Volume 3 of Reed & Simon (Scattering Theory) or Simon's book on Trace Ideals. If A and B are self-adjoint operators, with B having only absolutely continuous spectrum, and T a bounded operator such that TA − BT is trace-class, then e^iBtT e^–iAt has a weak operator limit as t tends to ∞.

The scattering operator is defined by S = W(−∞)*W(∞). It is unitary and commutes with A.

Ordinary differential equation

The generalised matrix coefficients of the principal series can be expressed in terms of vector-valued functions h(t) of a real variable t which satisfy the following ordinary differential equation:

$-h^{\prime \prime }(t)+\sinh ^{-2}(t)\,Ah(t)+\cosh(t)\sinh ^{-2}(t)\,Bh(t)+\lambda ^{2}h(t)=0,$

where A and B are self-adjoint matrices and λ is real. Note that this ODE is invariant under the transformation sending t to –t. Let s = e^t and set g(s) = h(t). Then

$-\left(s{d \over ds}\right)^{2}g(s)+4(s-s^{-1})^{-2}Ag(s)+2(s+s^{-1})(s-s^{-1})^{-2}Bg(s)+\lambda ^{2}g(s)=0.$

This ODE is invariant under the transformation sending s to s^–1 so can be expressed in terms of z = (s + s^–1)/2. Set f(z) = g(s). Then

${dz \over ds}=(1-s^{-2})/2.$

So

$s{d \over ds}=(s-s^{-1})/2\,{d \over dz}$

and

$\left(s{d \over ds}\right)^{2}=s/2\,{d \over ds}(s-s^{-1}){d \over dz}=(s-s^{-1})^{2}/4\,{d^{2} \over dz^{2}}+s(1+s^{-2})/2\,{d \over dz}=(z^{2}-1){d^{2} \over dz^{2}}+z{d \over dz}.$

Hence

$-(z^{2}-1){d^{2}f \over dz^{2}}-z{df \over dz}+(z^{2}-1)^{-1}Af(z)+z(z^{2}-1)^{-1}Bf(z)+\lambda ^{2}f(z)=0.$

In the finite complex plane this has singularities only at ±1. Performing the coordinate change w = z^–1 and setting F(w) = f(z), the equation becomes:

$-{d \over dw}\left(w(w^{2}-1){dF \over dw}\right)+w^{2}{dF \over dw}+w(1-w^{2})^{-1}AF+(1-w^{2})^{-1}BF+w^{-1}\lambda ^{2}F=0.$

This has regular singular points at 0, 1 and –1. Let x(w) = F(w) and y(w) = w(w² – 1) dF/dw. Then

${d \over dw}{\begin{pmatrix}x\\y\end{pmatrix}}={\begin{pmatrix}0&w^{-1}(w^{2}-1)^{-1}I\\\lambda ^{2}w^{-1}I-(w^{2}-1)^{-1}B-w(w^{2}-1)^{-1}A&w(w^{2}-1)^{-1}I\end{pmatrix}}{\begin{pmatrix}x\\y\end{pmatrix}}.$

Thus if v = (x,y)^t,

${dv \over dw}=((w-1)^{-1}P+(w+1)^{-1}Q+w^{-1}R)v,$

where

$P={\begin{pmatrix}0&{1 \over 2}I\\&{1 \over 2}I\end{pmatrix}},\,\,Q={\begin{pmatrix}0&{1 \over 2}I\\&{1 \over 2}I\end{pmatrix}},\,\,R={\begin{pmatrix}0&-I\\\lambda ^{2}I&0\end{pmatrix}}.$