PLANCHEREL THEOREM FOR THE UNIVERSAL COVER OF THE CONFORMAL GROUP Rebecca A. Herb* Joseph A. Wolff Department of Mathematics Department of Mathematics University of Maryland University of California College Park, Maryland 20742 Berkeley, California 94720 §0. INTRODUCTION Ever since the discovery of conformal invariance of Maxwell's equations, conformal groups and conformal structures have had an important role in mathematical physics. For the most part, this role has been confined to consequences of the geometric action of the conformal group and to the use of certain unitary represen- tations. Now there is a fairly explicit theory of harmonic analysis on the conformal group. It seems likely that this theory will be of some physical importance, e.g. in partial wave analysis. When a group G acts by geometric symmetries on a space X, it also acts on various spaces of functions on X. These functions are better understood by taking the symmetries into account. That is, of course, the basic idea in Fourier analysis. It has also been exploited in the use of spherical harmonics, where G=SO(3) and X=S 2, and in the application of other sorts of special functions. Now the machinery is available for the case where G is the conformal group or one of its coverings, and X is either the space G itself or is a symmetric homogen- eous space of G. In this article we describe some of those developments, first sketching the general theory and then describing the case of the simply connected covering group S'U(2,2) of the conformal group. *Research partially supported by National Science Foundation grant DMS-84-01374. tResearch partially supported by National Science Foundation grant DMS-85-13467. 228 §I. Tempered Representations of ~emisimple Groups §2. Tempered Representations of SU(2,2) §3. Plancherel Theorem for Semisimple Groups §4. Explicit Plancherel Theorem for S'U(2,2) We assume that the reader is well acquainted with Mackey theory but not so well acquainted with Harish-Chandra theory. The results in §§I and 2 have been known for some time. The Plancherel formula in §3 was worked out by Harish-Chandra in the 1960's (published somewhat later in [I], [2] and [3]) for semisimple groups with finite center. We recently developed another approach ([4],[5]) which allows infinite center, as in the group ~(2,2). In §4 we work out the constants to obtain an explicit formula for ~U(2,2). §I. TEMPEREDREPRESENTATIONS OF SEMISIMPLE GROUPS We describe the representations involved in the Plancherel formula for a semi- simple group. To do this for a class of semisimple groups, one must ensure that certain subgroups belong to the same class. Our class consists of the reductive Lie groups G that have a closed normal abelian subgroup Z such that (l .l) Z centralizes the identity component Go and G/ZG° is finite and (I .2) If x E G then Ad(x) is an inner automorphism of g~ . Here "reductive" means that the Lie algebra B of G is (commutative)~(semisimple). If ~ E G, the set of equivalence classes of irreducible unitary representations of G, then ~ has three types of characters. The central character ~ is the scalar valued function on the center ZG that is given by ~(z) = ~(z).l where I is the identity on the representation space Jf(~). The infinitesimal character ×~ is the map on the center 3(g) of the enveloping algebra ~(g) given by d~(D) : ×~(D).I for every Casimir operator D. We view it as a homomorphism ~(g)--->~ of associa- tive algebras. The character or distribution character 0(~) is the Schwartz distribution on G given by co (1.3) E)(~:f) = trace ~(f) for f E Cc(G) where ~(f) = ~f(x)z(x)dx. The equivalence class of ~ is specified by e(~). The differential equations (1.4) De(~) = ×(D)-e(~) for D e 3(~) 229 lead to the information that e(~) is integration against a locally L z function T(~), (1.5) ~(~:f) = IGT(~:x)f(x)dx g which is real analytic on a certain dense open subset ("regular set") G' in G. We may suppose Z n Go = ZG0. Then ~ c G belongs to the relative discrete series if its coefficients (1.6) @u,v(X) = <u, ~(x)v)j/(~) , u,v E~(~) , are L 2 modulo Z. The representations we will use will be constructed from relative discrete series representations. Choose a Cartan involution e of G. In other words, e is an automorphism of G, 8 2=I, e is the identity on ZG(G°), and the fixed point set K of B satisfies: K/ZG(G°) is a maximal compact subgroup of G/ZG(G°). Let h be a Cartan subalgebra of B, i.e. a maximal diagonalizable (over C) subalgebra. Then H = {x E G: Ad(x)~ = ~ for all ~ E h} is the corresponding Cartan subgroup. One can find x E Go such that Ad(x)h and xHx-I are e-stable. If G has relative discrete series representations, then K contains a Cartan subgroup of G. Conversely, let T c K be a Cartan subgroup of G, ~ its Lie algebra, and ~+ = @+(~,t) a system of positive roots. Let p : ½~@+a and set <~,~> ~ 0 for all ~ E @+ and (I.7) A' = ~ E i~*: I e~-p is well defined on To . If ~ e A' then there is a relative discrete series representation ~# of Go such that (1.8) T(~T~: X) = (constant)A(x) -~ ~] det(w) eW~(x) w We where x E T O n G' A(x) = -IF (e ~/2-e-~/2)(x), and W° is the Weyl group of (G o ,TO). , ~,@+ The central character of ~ is eh-PlzGo. If X E ZG(G°)^ agrees with ~T~ on ZGO, then (1.9) X ® ~ ~ (G~)^ , Gt = ZG(G°)G° is well defined and is a relative discrete series representation of Gt. Finally, the relative discrete series of G consists of the (I .I0) ~X,}, = indG.(x{~- ® ~>)o , ~E A' , XE ZG(G0)^ 230 where ~ and X agree on ZG0. Now let H be any e-stable Cartan subgroup of G. Then h = t H ~ a H, ±I eigen- spaces of elh, and H = TH×A H where TH = HnK and AH = exPG(aH). Then the centralizer ZG(AH) = MHxA H where eMH=M H and TH is a Cartan subgroup of MH. Let @+aH = @+(g,aH) be a system of positive a--roots, n-= ~ g , N- the corre- H H ~@~+ ~ H sponding analytic subgroup of G, and PH =MHAHNH the associate~aH"cuspidal parabolic" subgroup of G. If w c AMHr , qw0 is the corresponding relative discrete series representatlon• of MH,0 and for X E ZMH(M~)^ consistent with h~0 then qX, ~ = IndM~(x ® q~) is the corresponding relative discrete series representation MH of MH. If aE a~ now n×,w ® e i~ E (MHAH)^ extends to PH' trivial on NH, and we have the unitary representation (I.II) ~×,w,~ = Ind~H(q×,v ® ei~) + It does not depend on choice of @aH. The representations (I.II) of G constitute the H-series. If AH = {I} that is the relative discrete series. If AH is maximal it is the principal series. Given v and X, ~X,~,a is irreducible for almost all ~. The irreducible constituents of representations ~X,v,o' H variable, are the tempered representations of G. When we are dealing with several Cartan subgroups we will write (l .12a) ~(H: ×:w: ~) for the H-series representation ~X,w,a ' (l .12b) O(H:X:w: ~) for the distribution character of ~X,v,~ ' and (l .12c) e(H :X:v:~: f) for the trace of ~×,v,~(f), f e C~(G) . §2. TEMPERED REPRESENTATIONS OF SU(2,2) The conformal group is usually realized as the identity component S0(2,4) of the orthogonal group of the real bilinear form -xly I - x2y2 + x3y3 +... +x6y6 on ~2,4. The space of light-like lines in ~2,4 is the conformal completion of Minkowsky space ~1,3, and that gives the action of S0(2,4) there. We will find it much more convenient to use the complex form (2.1) SU(2,2) = {x E GL(4; C): xJx*=J and det x = l} -I -I ) of the double cover of S0(2,4), J = 1 1 ' for the linear group. The two to one correspondence is 231 (2.2) SU(2,2) --> S0(2,4) by A2(vector representation) antisymmetrization of the usual representation of SU(2,2) on C4, Write G for the universal covering group SU(2,2) of SU(2,2), 4: G--> SU(2,2) for the covering, and ZI =Ker(~). Since SU(2,2) has center of order 4, ZI has index 4 in Z=Z G. 0 The maximal compactly embedded subgroup K=[K,K]×Z K where [K,K]- SU(2)×SU(2) maps one-to-one to SU(2,2), and where ZK0 - IR. Let t denote the compactly embedded Cartan subalgebra of B given by (2.3) ~ : {diag(i81,i62,i83,i84): 8j real, ~8j=O} Then ~ : (~n[k,k]) • ~K with tn[k,k] : {diag(i81,-i01,i82,-i82): 8j real} (2.4) ~K = {diag(iB,ie,-ie,-ie): 8 real} This gives us a parameterization of the corresponding Cartan subgroup T of G: (2.5a) T = {t(O1,02)Zu: 0 <81<2~, 0__<02<2~, -~<u<~} where (2.5b) t(81,82) = exPG diag(iel,-iel,iB2,-ie2) and (2.5c) z u = exPG diag(iu,iu,-iu,-iu) Notice that ~ sends t(Ol,O2)z u to diag(e i(81+u), ei('Oz+u),e i(02-u), ei('82"u)), so Z = ZG = ~'I({±I, ±il}) is given by (2.6) {t(O,O)Zk~ , t(~,~)Zk~, t(O,~)Z(k+½)~, t(~,O)Z(k+½) : ke #} Let ~j: diag(a l,a 2,a~,a 4) ~-> aj as usual.