Acta Applicandae Mathematicae 81: 191–214, 2004. © 2004 Kluwer Academic Publishers. Printed in the Netherlands.
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Canonical and Boundary Representations on a Hyperboloid of One Sheet V. F. MOLCHANOV Tambov State University, Internatsionalnaya 33, 392622 Tambov, Russia. e-mail:
[email protected] Abstract. For the hyperboloid of one sheet X = G/H , G = SO0 (1, 2), H = SO0 (1, 1), canonical representations Rλ,ν , λ ∈ C, ν = 0, 1, are defined as the restrictions to G of representations of the = SO0 (2, 2) associated with a cone. They act on the torus containing two copies of X overgroup G as open G-orbits. We study boundary representations generated by Rλ,ν . For some λ, they contain Jordan blocks. The decomposition of Rλ,ν into irreducible constituents includes a finite number (depending on λ) of irreducible parts of the boundary representations. Mathematics Subject Classification (2000): 43A85. Key words: symmetric spaces, canonical representations, Berezin form, Berezin transform, Poisson and Fourier transforms, spherical functions, boundary representations.
Canonical representations on Hermitian symmetric spaces G/K were introduced by Berezin [1] and Vershik, Gelfand and Graev [7]. They are unitary with respect to some invariant nonlocal inner product (the Berezin form). As it seems to us, it is more natural to give up the condition of unitarity and to consider canonical representations on symmetric spaces G/H in a wider sense. Moreover, sometimes it is natural to consider at once several spaces G/Hi , possibly with different Hi , embedded as open G-orbits into a compact manifold , where G acts. We obtain that could be called the ‘mixed’ harmonic analysis. One of sourses of getting canonical representations consists of the following. (overgroup) containing G, take a series of representations We take some group G associated with G/H , R of G induced by characters of some parabolic subgroup P and restrict these representations to G. In this paper we carry out this program for the one sheeted hyperboloid X = G/H, G = SO0 (1, 2), H = SO0 (1, 1). This space is a para-Hermitian symmetric = SO0 (2, 2), the representations space (pseudo-Riemannian). The overgroup is G R are representations associated with a cone, the manifold is the torus S 1 × S 1 . It contains two open G-orbits ± diffeomorphic to X.
Supported by grants of the Netherlands Organization for Scientific Research (NWO): 047-008-
009, the Russian Foundation for Basic Research (RFBR): 01-01-00100-a, the Minobr RF: E00-1.0156, the NTP ‘Univ. Rossii’: UR.04.01.037.
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Canonical representations Rλ,ν , λ ∈ C, ν = 0, 1, generate boundary representations which act on distributions concentrated at the boundary 0 of ± and on jets transversal to 0 . The decomposition of them is based on the meromorphic structure of Poisson and Fourier transforms associated with the canonical representations: simple poles give the diagonalization, poles of the second order give Jordan blocks. The decomposition of Rλ,ν into irreducible constituents includes a finite number (depending on λ) of irreducible parts of the boundary representations. Similar results were obtained in [6] for the Lobachevsky plane G/K, G = SU(1, 1), K = U(1) (a Riemannian space).
1. Hyperboloid of One Sheet Let G be the group SO0 (1, 2), it is a connected group of linear transformations of R3 , preserving the form [x, y] = −x1 y1 + x2 y2 + x3 y3 . We consider that G acts on R3 from the right. In accordance with this we write vectors in the row form. Let g be twice the Casimir element in the universal enveloping algebra of the Lie algebra g of G. Let X denote the hyperboloid in R3 defined by equation [x, x] = 1. It is a homogeneous space of the group G with respect to translations x → xg. The stabilizer H of the point x 0 = (0, 0, 1) is isomorphic to the group SO0 (1, 1). The hyperboloid X has a G-invariant metric. It gives rise to the measure dx and the Laplace–Beltrami operator X , both G-invariant. In polar coordinates t, α: x = (sinh t, cosh t · sin α, cosh t · cos α) we have: dx = cosh t · dt dα. If M is a manifold, then D(M) denotes the Schwartz space of compactly supported infinitely differentiable C-valued functions on M, with the usual topology, and D (M) denotes the space of distributions on M – of antilinear continuous functionals on D(M). For a differentiable representation of a Lie group, we retain the same symbol for the corresponding representations of its Lie algebra and of the universal enveloping algebra. Let us denote by U the representation of our group G by translations on functions on X (quasiregular representation): (U (g)f )(x) = f (xg). A space where U acts will be specially indicated, if necessary. In particular, the representation U on the space L2 (X, dx) is unitary with respect to the inner product F, f X =
F (x)f (x) dx.
(1.1)
X
We have U (g ) = X .
(1.2)
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2. Representations of the Group SO0 (1, 2) Recall [8] some material about representations of the group G = SO0 (1, 2). Let S be the circle consisting of points s = (1, sin α, cos α). The Euclidean measure on S is ds = dα. For a function ϕ on S, sometimes we write ϕ(α) instead of ϕ(s). The representation Tσ , σ ∈ C, of G acts on the space D(S) as follows: (Tσ (g)ϕ)(s) = ϕ(sg/(sg)1)(sg)σ1 . The element g goes to a scalar operator: Tσ (g ) = σ (σ + 1)E.
(2.1)
The Hermitian form ψ, ϕS = ψ(s)ϕ(s) ds
(2.2)
S
is invariant with respect to the pair (Tσ , T−σ −1 ), i.e. Tσ (g)ψ, ϕS = ψ, T−σ −1 (g −1 )ϕS .
(2.3)
Define an operator Aσ in D(S): (Aσ ϕ)(s) = (−[s, u])−σ −1 ϕ(u) du. S
The integral converges absolutely for Re σ < −1/2 and can be continued meromorphically to the whole σ -plane. It has simple poles at σ ∈ −(1/2) + N. Here µ at a pole µ = −(1/2) + r, r ∈ N, is and further N = {0, 1, 2, . . .}. The residue A a differential operator of order 2r. For Aσ we have Aσ ψ, ϕS = ψ, Aσ ϕS .
(2.4)
The operator Aσ intertwines Tσ T−σ −1 . In general, if an operator C intertwines some representations S and T , i.e. T (g)C = CS(g), then we write: C intertwines S T . Take a basis ψm (α) = eimα , m ∈ Z, in D(S). It consists of eigenfunctions of Aσ : Aσ ψm = a(σ, m)ψm , where a(σ, m) = 2σ +2 π(−1)m
(−2σ − 1) . (−σ + m) (−σ − m)
The composition Aσ A−σ −1 is a scalar operator – the multiplication by (8π ω(σ ))−1, where ω(σ ) is a ‘Plancherel measure’ (3.10).
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The representation Tσ can be extended to the space D (S) by formula (2.3) where ψ, ϕS means the value of a distribution ψ at a test function ϕ. Similarly Aσ can be extended to D (S) by means of formula (2.4). If σ is not integer, then Tσ is irreducible and Tσ is equivalent to T−σ −1 (by Aσ σ ). or A Let σ ∈ Z, n ∈ N. Subspaces Vσ,+ and Vσ,+ spanned by ψm for which m −σ and m σ respectively are invariant. For σ < 0 they are irreducible and orthogonal to each other. For σ 0 their intersection Eσ is irreducible and has dimension 2σ + 1. d = V−n−1,+ + V−n−1,− . Let Tσd , σ ∈ Z, be the Let Vnd = D(S)/En and V−n−1 representation on Vσd generated by Tσ . The operator An vanishes on En and gives d . rise to the equivalence Tnd ∼ T−n−1 There are four series of unitarizable irreducible representations: the continuous series consisting of Tσ with σ = −(1/2) + iρ, ρ ∈ R, the inner product is (2.2); the complementary series consisting of Tσ with −1 < σ < 1, the inner product is Aσ ψ, ϕS with a factor; the holomorphic and antiholomorphic series. We need only their sum Tσd . We shall call Tσd the representations of discrete series. For ϕ ∈ D(S), denote by ϕ the coset of ϕ modulo En . Then the invariant inner product d (· , ·)n for Tn is , (ψ ϕ )n = cn An ψ, ϕS ,
cn = a(n, n + 1)−1 .
(2.5)
3. Harmonic Analysis on the Hyperboloid Here we adduce briefly some results from [4] in a modified form. First we determine distributions θ in D (S) invariant with respect to the subgroup H under the representations Tσ and their subquotiens. We shall use the following notation for a character of the group R∗ : t λ,m = |t|λ (sgn t)m , where t ∈ R∗ = R \ {0}, λ ∈ C, m ∈ Z. In fact this character depends only on m modulo 2. Here and further the sign ‘≡’ means the congruence modulo 2. It is easy to check that the distribution θσ,ε = s3σ,ε = [x 0 , s]σ,ε , where σ ∈ C, ε = 0, 1, is H -invariant. As a function of σ , it is a meromorphic function – with simple poles at points σ ∈ −1 − ε − 2N. Its residue at σ = −n − 1, n ≡ ε, is the distribution const · δ (n) (s3 ) concentrated at two points s = (1, ±1, 0). Here δ(t) is the Dirac delta function. The space of H -invariants has dimension 2 for σ = −n − 1, n ∈ N, and dimension 3 for σ = −n − 1. Every irreducible subfactor for Tσ , σ ∈ Z, contains, up to a factor, precisely one H -invariant. In particular, θ−n−1,n+1 and θn,n+1 have and D (S)/Vn,∓ respectively. nonzero projections into V−n−1,± The operator Aσ carries θσ,ε to θ−σ −1,ε with a factor: Aσ θσ,ε = j (σ, ε)θ−σ −1,ε ,
(3.1)
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where 1 −σ −1/2 −σ − (σ + 1) 1 − (−1)ε cos σ π . j (σ, ε) = 2 π 2 It is easy to check that j (σ, ε)j (−σ − 1, ε) = (8π ω(σ ))−1,
(3.2)
where ω(σ ) is given by (3.10). The factor j (σ, ε) has a simple pole at µ = −(1/2) + r, r ∈ N, its residue is independing of ε, so we denote it by j (µ). By a general scheme [5], the H -invariant θσ,ε gives rise to the Poisson kernel Pσ,ε (x, s) = [x, s]σ,ε , x ∈ X, s ∈ S. This kernel gives rise to two transforms. The first of them, the Poisson transform, is a linear continuous operator Pσ,ε : D(S) → C ∞ (X) defined as follows: (Pσ,ε ϕ)(x) = [x, s]σ,ε ϕ(s) ds. (3.3) S
It intertwines T−σ −1 U , therefore, its image consists of eigenfunctions of the Laplace–Beltrami operator: X ◦ Pσ,ε = σ (σ + 1)Pσ,ε of parity ε (see (2.1) and (1.2)). As a function in σ , the Poisson transform behaves like θσ,ε : it depends on σ meromorphically and has simple poles at σ ∈ −1 − ε − 2N. Formula (3.1) gives Pσ,ε Aσ = j (σ, ε)P−σ −1,ε .
(3.4)
Introduce differential operators Wσ,k , σ ∈ C, k ∈ N, by a generating function: ∞ 3 2 σ + 1 − i∂ σ + 1 + i∂ 2k , ;σ + ;z , Wσ,k z = F (3.5) 2 2 2 k=0 where F is the Gauss hypergeometric function, ∂ = ∂/∂α. Here is an explicit expression: Wσ,k =
k−1 1 (σ + 1 + 2r)2 + ∂ 2 2k [k] 2 k!(σ + (3/2)) r=0
(W0 = 1). We use the following notation for ‘generalized powers’: a [m] = a(a + 1) · · · (a + m − 1). The operator Wσ,k depends on σ meromorphically – with simple poles at σ half integer such that −k − (1/2) σ −3/2. THEOREM 3.1. Let σ be not half integer. For any K-finite function ϕ in D(S), its Poisson transform has for t > 0 the following expansion in powers of cosh t (x has polar coordinates t, α): ∞ (Cσ,k ϕ)(α)(cosh t)−2k + (Pσ,ε ϕ)(x) = (cosh t)σ (−1)ε k=0
+ (cosh t)
−σ −1
∞ (−1) j (σ, ε) (Wσ,k ϕ)(α)(cosh t)−2k , (3.6) ε
k=0
where Cσ,k = A−σ −1 W−σ −1,k .
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The operators Cσ,k , Wσ,k and Aσ are diagonal in the basis ψm , therefore they commute each with other. The operators Cσ,k are linear combinations of operators A−σ −1+m , in particular, Cσ,0 = A−σ −1 . They depend on σ meromorphically – with simple poles at σ half integer such that σ k − 1/2. The operators Dσ,ε,k = j (σ, ε)Wσ,k depend on σ meromorphically – with simple poles at σ half integer such that σ −k − 1/2 and also at poles of θσ,ε . Consider σ ∈ Z. The transform P−n−1,n+1 vanishes on En , it generates an operator on D(S)/En which intertwines Tnd U . The Poisson transform Pn,n+1 d d considered on V−n−1 intertwines T−n−1 U . By (3.4), Pn,n+1 has the same image as P−n−1,n+1 . The second transform generated by the Poisson kernel is the Fourier transform Fσ,ε defined by (Fσ,ε f )(s) = [x, s]σ,ε f (x) dx. X
It is meromorphic in σ with simple poles at points σ ∈ −1 − ε − 2N. It is a continuous map D(X) → D(S) and intertwines U Tσ . It follows from (3.1) that Aσ Fσ,ε = j (σ, ε)F−σ −1,ε .
(3.7)
For a function f ∈ D(X), let us call two functions Fσ,ε f , ε = 0, 1, the Fourier components of f corresponding to the representation Tσ . The Fourier and Poisson transforms are conjugate to each other with respect to forms (1.1) and (2.2): Pσ,ε ϕ, f X = ϕ, Fσ ,ε f S . This relation allows to extend the Poisson transform to distributions on S. Consider the reducible case. The Fourier transform Fn corresponding to Tnd is defined as the map of D(X) to D(S)/En which assigns to f ∈ D(X) the corresponding coset of the function Fn,n+1 f . By (2.5) and (3.1) we have (Fn f, Fn h)n = dn F−n−1,n+1 f, Fn,n+1 hS ,
dn = 2n!2 /π(2n + 1)!
(3.8)
d is F−n−1,n+1 . The Fourier transform corresponding to T−n−1 For σ ∈ C, ε = 0, 1, define a distribution σ,ε on the hyperboloid X whose value at f ∈ D(X) is (σ,ε , f ) = θ−σ −1,ε , Fσ ,ε f S . In other words, σ,ε = Pσ,ε θ−σ −1,ε . Let us call σ,ε the spherical function corresponding to the representation Tσ . For each σ we have two functions. The spherical function σ,ε is invariant with respect to H , is an eigenfunction of the Laplace–Beltrami operator: X σ,ε = σ (σ + 1)σ,ε , is meromorphic in σ with simple poles at σ = n, σ = −n − 1, ε ≡ n (n ∈ N), has the following symmetry property with respect to σ : −σ −1,ε = σ,ε , and is regular, i.e. is a locally integrable function σ,ε (x). This function depends on x3 only and has parity ε.
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Define the Legendre functions Pσ (c), Qσ (c) at the cut −∞ < c < 1 as half the sum of limit values from above and below of corresponding functions. For −1 < c < 1 it coinsides with [2]. For a function f (c) on the real line, we denote f(c) = f (−c). The spherical function σ,ε (x) has the following expression in terms of the Legendre functions: σ,ε (x) = −
2π Pσ (x3 ) + (−1)ε Pσ (x3 ) . sin σ π
Consider the reducible case. Let us call the distribution n,n+1 the spherical function corresponding to the representation Tnd and denote by nd . It is the Legendre function of the second kind times 4: nd (x) = n,n+1 (x) = −n−1,n+1 (x) = 4Qn (x3 ). Let be an arbitrary distribution on X invariant with respect to H . Assign to it a map D(X) → C ∞ (X) called the convolution with : ( f )(x) = , U (g)f X ,
x = x 0 g,
and also a sesqui-linear functional K(|f, h): f (x), U (g)hX dx, K(|f, h) = f, hX =
x = x 0 g.
(3.9)
X
For the spherical function σ,ε , the convolution is the composition of Fourier and Poisson transforms: σ,ε f = P−σ −1,ε Fσ,ε f , and the functional is the inner product of Fourier components: K(σ,ε |f, h) = F−σ −1,ε f, Fσ ,ε hS . In particular, if σ = −(1/2) + iρ (the continuous series), then −σ − 1 = σ and K(σ,ε |f, h) = Fσ,ε f, Fσ,ε hS . Furthermore, for the spherical function nd , we have by (3.8) K(nd |f, h) = F−n−1,n+1 f, Fn,n+1 hS = dn−1 (Fn f, Fn h)n . THEOREM 3.2. The quasiregular representation U of the group G on the space L2 (X, dx) decomposes into irreducible unitary representations of the continuous series with multiplicity 2 and the discrete series with multiplicity 1. Namely, let us assign to a function f ∈ D(X) the set of its Fourier components Fσ,ε f of the continuous series (σ = −(1/2) + iρ, ε = 0, 1) and Fourier components Fn f of the discrete series Tnd (n ∈ N). This correspondence is G-equivariant. There is the inversion formula: ∞ ∞
ω(σ ) P−σ −1,ε Fσ,ε f
dρ + ωn Pn,n+1 F−n−1,n+1 f, f = −∞
ε
σ =−1/2+iρ
n=0
where ω(σ ) =
1 (2σ + 1) cot σ π, 8π 2
ωn = 2−3 π −2 (2n + 1),
(3.10)
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and the Plancherel formula: ∞
f, hX = ω(σ ) Fσ,ε f, Fσ,ε hS
−∞
+
∞
σ =−1/2+iρ
ε
dρ +
ωn dn−1 (Fn f, Fn h)n .
(3.11)
n=0
Therefore, the correspondence can be extended from D(X) to L2 (X, dx). The integrands in the theorem are even in ρ, it reflects the equivalence of Tσ and T−σ −1 . It is more convenient for us to take integrals over R. Notice that ω(−1/2 + iρ) = 2−4 π −2 ρ tanh ρπ . Theorem 3.2 is equivalent to the decomposition of the delta function δ concentrated at x 0 in terms of spherical functions: δ=
∞
ω(σ ) −∞
ε
σ,ε
σ =−1/2+iρ
dρ +
∞
ωn nd .
(3.12)
n=0
In its turn, (3.12) is deduced from the decomposition of the distribution Nλ,ν = (x3 − 1)λ,ν , λ ∈ C, ν = 0, 1, which is in fact the decomposition of the Berezin form, see below, so we do not write it here. The distribution Nλ,0 has at λ = −1 a pole of the second order, the first Laurent coefficient is 4δ. It gives (3.12). The Berezin kernel on the hyperboloid X is the following function of two variables x, y ∈ X: Eλ,ν (x, y) = c(λ, ν)([x, y] + 1)λ,ν ,
(3.13)
where λ ∈ C, ν = 0, 1, and c(λ, ν) = 2−λ−2 π −1 (λ + 1) cot((λ − ν)π/2). The Berezin kernel gives rise to the sesqui-linear form Bλ,ν (f, h) on D(X), let us call it the Berezin form. This form is K(|f, h), see (3.9), where is the distribution c(λ, ν)(x3 + 1)λ,ν . This distribution is regular for Re λ > −1 and can be meromorphically extended to the λ-plane – with simple poles at points λ ∈ (−2 − N) ∪ (ν + 2N) and simple zeros at points λ ∈ 1 − ν + 2N. The same is true for the Berezin form. THEOREM 3.3. Let λ ∈ / Z. Let f, h ∈ D(X). Then for Re λ < −1/2 and for λ = −1/2 we have ∞
ω(σ ) (λ, ν; σ, ε)Fσ,ε f, F−σ −1,ε hS
dρ + Bλ,ν (f, h) = −∞
+
∞ n=0
ε
ωn dn (λ; n)(Fn f, Fn h)n ,
σ =−1/2+iρ
(3.14)
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for k − (1/2) < Re λ < k + (1/2), k ∈ N, and for λ = k + (1/2) we have ∞ ∞ k + + M(λ, m)F−λ−1+m,ν−m f, Fλ−m,ν−m hS , (3.15) Bλ,ν (f, h) = −∞
n=0
m=0
and for Re λ = k − (1/2), λ = k − (1/2), we have ∞ ∞ k−1 1 + + + M(λ, k)F−λ−1+k,ν−k f, Fλ−k,ν−k hS . (3.16) Bλ,ν (f, h) = 2 −∞ n=0 m=0 The integral and the series in (3.15) and (3.16) denote the same integral and the series as in (3.14), the sum in (3.16) contains the same summands as the sum in (3.15), and (−λ − σ − 1) (−λ + σ ) sin λπ + (−1)ν+ε sin σ π · , (−λ − 1) (−λ) sin λπ (−λ − n − 1) (−λ + n) (λ; n) = , (3.17) (−λ − 1) (−λ) (λ + 1) (λ + 2) 1 . M(λ; m) = − 2 (2λ − 2m + 1) 8π m! (2λ − m + 2) (λ, ν; σ, ε) =
Notice that the Berezin form is not positive or negative defined on D(X) even for λ real. THEOREM 3.4. Under the condition −1 Re λ < −1/2 the Berezin form is bounded in L2 (X, dx). Therefore, the Berezin form and the decomposition (3.14) can be extended to L2 (X, dx). Proof. It is sufficient to show that the coefficients in (3.14) as functions of ρ and n are bounded. It follows from their asymptotic behaviour at infinity: (λ, ν; −(1/2) + iρ, ε) ∼ const · |ρ|−2λ−2 ,
(λ; n) ∼ const · n−2λ−2 , 2
see [2], 1.18(6), (4). 4. Representations of the Group SO0 (2, 2) Associated with a Cone
= In this section we recall some material about representations of the group G SO0 (2, 2) associated with a cone, see [3]. We adapt the notation to our goals is the connected component of the identity in the group of here. The group G linear transformations of R4 preserving the following bilinear form: [x, y] = acts from the right. −x0 y0 − x1 y1 + x2 y2 + x3 y3 . As before, we consider that G Let C denote a cone in R4 defined by [x, x] = 0, x = 0. For λ ∈ C, ν = 0, 1, let Dλ,ν (C) denote the space of functions f of class C ∞ on C homogeneous ‘of degree λ, ν’, i.e. f (tx) = t λ,ν f (x),
x ∈ C, t ∈ R∗ .
(4.1)
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λ,ν of G acts on D−λ−2,ν (C) by translations: The representation R . λ,ν (g)f )(x) = f (xg), x ∈ C, g ∈ G (4.2) (R For x ∈ R4 , denote |x| = x02 + x12 . Let be the set consisting of points u ∈ C such that |u| = 1. It is a torus: u = (cos α, sin α, sin β, cos β).
(4.3)
The Euclidean measure on is du = dα dβ. Functions in Dλ,ν (C) are completely determined by their restrictions to . These restrictions form the space Dν () of functions in D() of parity ν. λ,ν can be realized on Dν (): The representation R λ,ν (g)f )(u) = f (ug/|ug|)|ug|−λ−2 . (R The Hermitian form f (u)h(u) du f, h =
(4.4)
(4.5)
−λ−2,ν λ,ν , R ): is invariant with respect to the pair (R ¯ −λ−2,ν λ,ν (g)f, h = f, R (g −1 )h . R ¯
(4.6)
An operator Bλ,ν on Dν () defined by (Bλ,ν f )(u) = [u, v]λ,ν f (v) dv
−λ−2,ν . It interacts with form (4.5) as follows: λ,ν R intertwines R Bλ,ν f, h = f, Bλ,ν h .
(4.7)
The composition Bλ,ν and B−λ−2,ν is a scalar operator: Bλ,ν B−λ−2,ν = γ (λ, ν) · E, −2 where γ (λ, ν) = (λ + 1)/(4π ) · cot((λ − ν)π/2) .
(4.8)
5. Canonical Representations = SO0 (2, 2) as a Let us embed the group G = SO0 (1, 2) into the group G subgroup which preserves the coordinate x0 . The canonical representation Rλ,ν , λ ∈ C, ν = 0, 1, of the group G is defined as the restriction to G of the represen associated with a cone. It acts on D−λ−2,ν (C) by translations, see λ,ν of G tation R (4.2) with g ∈ G. of the cone C by the plane x0 = 1. We can idenConsider the section X tify it with the hyperboloid X: to a point x = (x1 , x2 , x3 ) of X we assign the
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= X. point x = (1, x1 , x2 , x3 ). Let us restrict functions in D−λ−2,ν (C) to X We obtain a space D−λ−2,ν (X) of functions on X. It contains D(X) and is contained in C ∞ (X). On the space D−λ−2,ν (X) the representation Rλ,ν becomes the representation U by translations. On the space Dν () the representation Rλ,ν acts according to (4.4), g ∈ G. The Hermitian form (4.5) is invariant with respect to the pair (Rλ,ν , R−λ−2,ν ). The group G has four orbits on with respect to the action u → ug/|ug|, namely, two open orbits (of dimension 2): + = {u0 > 0} and − = {u0 < 0}, and two orbits of dimension 1 (two circles): S + = {u0 = 0, u1 = +1} and S − = {u0 = 0, u1 = −1}. The orbit S + can be identified with the circle S from Section ◦ 2: to a point s = (1, sin t, cos t) of S we assign the point s= (0, 1, sin t, cos t) in S + . Denote = + ∪ − and 0 = S + ∪ S − . The latter is the boundary and therefore of ± . Each of orbits ± can be identified with the section X with the hyperboloid X. The map is constructed by means of generating lines of the cone C: to a point u given by (4.3) we assign the point x = u/u0 = (1, tan α, sin β/ cos α, cos β/ cos α). In these coordinates α, β the measure on X is dx = u−2 0 du. Let a function f belong to D−λ−2,ν (C). Its restrictions f (u) and f (x) to and = X are connected as follows: to X u f (x), x ←→ x= . (5.1) f (u) = u−λ−2,ν 0 u0 Therefore, 1 2 |f (x)| dx = |f (u)|2 |u0 |2Re λ+2 du, 2 X the factor 1/2 appears because covers X twice. Thus, if Re λ > −3/2, then the restriction of f ∈ D−λ−2,ν (C) to X belongs to L2 (X,dx). Let f ∈ D−λ−2,ν (C), h ∈ D λ,ν (C). Then by (5.1) we have f, hX = (1/2)f, h . Introduce the operator Qλ,ν on Dν (): 1 (Qλ,ν f )(u) = c(λ, ν) [u, vI ]λ,ν f (v) dv, 2 where c(λ, ν) is the factor in (3.13), I = diag{−1, 1, 1, 1}. Since elements g in G commute with I , this operator intertwines Rλ,ν R−λ−2,ν . In the realization on X its kernel is exactly the Berezin kernel (3.13). So we call Qλ,ν the Berezin transform. By (4.8), the inverse operator is Q−λ−2,ν : Q−λ−2,ν Qλ,ν = E.
(5.2)
Let Dν () be the space of distributions on of parity ν. We extend Rλ,ν and Qλ,ν to Dν () by (4.6) and (4.7) and retain their names and the notation. For λ ∈ C, let us define a sesqui-linear form on Dν (): (f, h)λ,ν = Qλ,ν f, h .
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It is invariant with respect to the pair (Rλ,ν , Rλ,ν ¯ ). In the realization on X it is twice the Berezin form: (f, h)λ,ν = 2Bλ,ν (f, h).
(5.3)
6. Boundary Representations The canonical representation Rλ,ν gives rise to two representations Lλ,ν and Mλ,ν associated with the boundary 0 of the manifolds ± (boundary representations). The first one acts on distributions concentrated at 0 , the second one acts on jets orthogonal to 0 . Denote by Dν (0 ) the space of functions ϕ in D(0 ) of parity ν = 0, 1. This space can be identified with the space D(S): to a function ϕ in D(S) we assign the function ϕ ν in Dν (0 ) such that ◦
ϕ ν (s) = ϕ(s),
◦
ϕ ν (− s) = (−1)ν ϕ(s).
(6.1)
For a function f in Dν (), consider its Taylor series a0 + a1 u0 + a2 u20 + · · · in powers of u0 , corresponding to the domain u1 > 0. Here ak = ak (f ) are functions in D(S). Denote by ak∗ (f ) Taylor coefficients of the function (1 − u20 )−1/2 f (u). We have (1 − u20 )−1/2
∞ k=0
ak (f )uk0 =
∞
ak∗ (f )uk0 .
(6.2)
k=0
It allows to express ak∗ (f ) in terms of ak (f ) and inversely – by means of triangular matrices with the unit diagonal. Define the map u → s(u) = (0, sgn u1 , u2 , u3 ) of points u ∈ with u1 = 0 onto 0 . Let us denote by m ν (), m ∈ N, ν = 0, 1, the space of distributions in Dν (0 ) having the form: ϕ(s(u))δ (m) (u0 ),
(6.3)
where ϕ ∈ Dν−m (0 ). Distribution (6.3) acts on functions f ∈ Dν () as follows. Instead of variables α, β, see (4.3), let us take for local coordinates on the variables u0 , β, where u0 = cos α. In these coordinates we have: du = (1 − u20 )−1/2 du0 dβ. Therefore, ϕδ (m) (u0 ), f = 2(−1)m m!ϕ, am∗ (f )S .
(6.4)
Let us set νk () = 0ν () + 1ν () + · · · + kν () and ν () = νk (). The representation Rλ,ν preserves ν () and the filtration 0ν () = ν0 () ⊂ ν1 () ⊂ · · ·
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(but not each kν (), k 1). Denote by Lλ,ν the restriction of Rλ,ν to ν (). Any distribution ζ in νk () has the form ϕ0 δ + ϕ1 δ + · · · + ϕk δ (k) . Let us assign to this ζ the column (ϕ0 , ϕ1 , . . . , ϕk , 0, 0, . . .). Then Lλ,ν is a upper triangular matrix with the diagonal T−λ−1 , T−λ , T−λ+1 , . . . . Distributions in νk () can be extended in a natural way to a space wider than Dν (). Namely, let Tνk () denote the space of functions f on of parity ν and of class C ∞ on each G-orbit on and having the Taylor expansion of order k in a neighbourhood of S + : f (u) = a0 + a1 u0 + · · · + ak uk0 + o(uk0 )
(6.5)
+
uniformly with respect to s ∈ S , here am = am (f ) belong to D(S), in (6.5) they depend on s(u). Then (6.4) preserves for f ∈ Tνk () for m k. Now let us define the representation Mλ,ν . Let a(f ) denote the column of Taylor coefficients am (f ). Then Mλ,ν acts on these columns: Mλ,ν (g)a(f ) = a(Rλ,ν (g)f ). It is a lower triangular matrix with the diagonal T−λ−2 , T−λ−3 , T−λ−4 , . . . . 7. Poisson Transform Associated with a Canonical Representation For a function ϕ ∈ D(S), let us consider the Poisson transform Pσ,ε ϕ as the = X of a function on C of homogeneity −λ − 2, ν, see (4.1). restriction to X ◦ to . For that, in (3.3) we write [ Now let us pass from X x , s] instead of [x, s] and use (5.1). Then we obtain the map Pλ,ν;σ,ε : D(S) → C ∞ ( ) given by ◦ −λ−σ −2,ν−ε [u, s]σ,ε ϕ(s) ds. (Pλ,ν;σ,ε ϕ)(u) = u0 S
It intertwines T−σ −1 Rλ,ν . Let us call it the Poisson transform associated with the canonical representation Rλ,ν . It has parity ν. Here we consider Rλ,ν as the restriction to C ∞ ( ) of the representation Rλ,ν acting on distributions in D (). The Poisson transform interacts with the intertwining operators Aσ and Qλ,ν as follows: Pλ,ν;σ,ε Aσ = j (σ, ε)P−λ−2,ν;σ,ε , Qλ,ν Pλ,ν;σ,ε = (λ, ν; σ, ε)P−λ−2,ν;σ,ε .
(7.1)
The first of these relations follows at once from (3.4), the second one follows from formula (8.3) below with the help of duality (8.2). Let us expand the Poisson transform in powers of u0 . It is sufficient to do it for the domain u1 > 0, i.e. for 0 < α < π , see (4.3). For 0 < α < π/2 we use (3.6), and for π/2 < α < π we use parity ε of Pσ,ε . We obtain: −2,ν−ε (−1)ε (Pλ,ν;σ,ε ϕ)(u) = u−λ−σ 0
∞ (Cσ,k ϕ)(s)u2k 0 + k=0
−1,ν−ε (−1)ε j (σ, ε) + u−λ+σ 0
∞ (Wσ,k ϕ)(s)u2k 0 . k=0
(7.2)
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Here s = (1, sin β, cos β) for u given by (4.3). Expansion (7.2) is true for K-finite −2,ν−ε −1,ν−ε functions ϕ and for σ ∈ 1/2 + Z. Let us call factors u−λ−σ and u−λ+σ 0 0 the leading factors. They give poles in σ depending on λ: σ = λ − k,
σ = −λ − 1 + l,
(7.3)
where k, l ∈ N and k ≡ ν − ε, l ≡ ν − ε. Let us write the Laurent expansion of the Poisson transform at a pole µ of the first and the second order in the following form respectively: λ,ν,µ P + ···, σ −µ Pλ,ν,µ Pλ,ν,µ + ···. = + 2 (σ − µ) σ −µ
Pλ,ν;σ,ε =
(7.4)
Pλ,ν;σ,ε
(7.5)
The first Laurent coefficient intertwines T−µ−1 Rλ,ν . If µ is of the second order, then Rλ,ν (g)Pλ,ν,µ = Pλ,ν,µT−µ−1 (g),
(g), Rλ,ν (g)Pλ,ν,µ = Pλ,ν,µ T−µ−1 (g) −Pλ,ν,µ T−µ−1
(7.6) (7.7)
where Tσ = (d/dσ )Tσ . Let us write the first Laurent coefficients in (7.4) and (7.5) explicitly. Residues P for poles of the second order have rather complicated expressions, we omit them, cf. [6]. If a pole µ belongs only to one of series (7.3), then it is simple and 1 Pλ,ν,λ−k = 2(−1)ν j (λ − k, ν − k)ξλ,ν,k , k! 1 Pλ,ν,−λ−1+l = −2(−1)ν ξλ,ν,l ◦ Aλ−l , l!
(7.8)
where ξλ,ν,k is the following operator D(S) → νk (): ξλ,ν,k (ϕ) =
k! (Wλ−k,r ϕ)ν−k δ (k−2r)(u0 ), (k − 2r)! 0rk/2
recall the notation ϕ ν for ϕ ∈ D(S), see (6.1). It depends on λ meromorphically with simple poles at points λ ∈ 1/2 + N such that λ + 3/2 k 2λ + 1. It intertwines T−λ−1+k Lλ,ν . Let a pole µ belongs to both series (7.3). Then 2λ + 1 ∈ N, so that either λ ∈ N or λ ∈ −(1/2) + N. By (7.3) we have µ = λ − k = −λ − 1 + l, where k, l ∈ N, and 0 k, l 2λ + 1, k + l = 2λ + 1, l − k = 2µ + 1. If λ ∈ N, then µ is of the first or second order. The second order appears because of the factor j (σ, ε), the residues include summands which are not belong to ν (). We can avoide this discomfort by a normalization of the H -invariant
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taking off its poles. But the poles µ ∈ Z are not important for us, so we omit details. Now let λ ∈ −(1/2) + N, then µ is of the second order and k ≡ l. We have: ν
λ−l , k l, λ,ν,µ = −4 (−1) ξλ,ν,µ ◦ A P l!ν (7.9) λ,ν,µ = 4 (−1) j (µ)ξλ,ν,µ, k l. P k! Let us take the residue of both sides of (7.1) at σ = λ − k with ε ≡ ν − k. By (7.8), (3.2) and using the notation (8.12) below, we obtain − Qλ,ν ξλ,ν;k = (−1)ν k!j (−λ − 1 + k, ν − k)M(λ, k) · P−λ−2,ν,k ,
(7.10)
where M(λ, k) is given by (3.17). 8. Fourier Transforms Associated with the Canonical Representation Let us apply to the Fourier transform Fσ,ε , see Section 3, the same treatments as in the beginning of Section 7. Throwing off the factor 1/2, we define the map Fλ,ν;σ,ε : Dν () → D(S) as follows: ◦ ν−ε f (u) du. (8.1) (Fλ,ν;σ,ε f )(s) = [u, s]σ,ε uλ−σ, 0
It intertwines Rλ,ν Tσ . Let us call this map the Fourier transform associated with the canonical representation Rλ,ν . Integral (8.1) converges absolutely for Re σ > −1/2, Re(λ ± σ ) > −1 and can be continued meromorphically in σ and λ. If the restriction of f ∈ D−λ−2,ν (C) to X belongs to D(X), then the Fourier transforms differ by the factor 1/2: Fσ,ε f = (1/2)Fλ,ν;σ,ε f . The Fourier and Poisson transforms are conjugate to each other: Fλ,ν;σ,ε f, ϕS = f, P−λ−2,ν; ¯ σ¯ ,ε ϕ .
(8.2)
It allows to transfer statements for one of them to the other. The Fourier transform interacts with the intertwining operators Aσ and Qλ,ν as follows: Aσ Fλ,ν;σ,ε = j (σ, ε)Fλ,ν;−σ −1,ε , F−λ−2,ν;σ,ε Qλ,ν = (λ, ν; σ, ε)Fλ,ν;σ,ε .
(8.3)
The first of these equalities follows just from (3.7), the second one will be proved in Section 10. Besides of poles indicated in Section 3, the Fourier transform has poles in σ depending on λ. They are situated at points σ = −λ − 2 − k,
σ = λ + 1 + l,
where k, l ∈ N satisfy conditions: ε ≡ ν − k, ε ≡ ν − l.
(8.4)
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For Laurent coefficients we use the same notation as for the Poisson transform. The first Laurent coefficient intertwines Rλ,ν Tµ . For a pole µ of the second order there are relations dual to (7.6), (7.7). Let us write first Laurent coefficients explicitly. If a pole µ belongs only to one of series (8.4), then it is simple and λ,ν,−λ−2−k = 4(−1)ν−k j (−λ − 2 − k, ν − k)bλ,k , F λ,ν,λ+1+l = −4(−1)ν−l A−λ−2−l ◦ bλ,l , F
(8.5) (8.6)
where bλ,k is a ‘boundary’ operator Dν () → D(S) which is defined in terms of Taylor coefficients am (f ) taken at S + as follows: ∗ W−λ−2−k,r ak−2r (f ). (8.7) bλ,k (f ) = 0rk/2
Now let a pole µ belongs to both series (8.4), so that −2λ − 3 ∈ N. Consider λ ∈ −3/2 − N only, then the pole µ is of the second order and −λ−2−l ◦ bλ,l , k l, λ,ν,µ = 8A F k l. F λ,ν,µ = 8j (µ)bλ,k , Boundary operators b and ξ are conjugate (up to a factor): ξ−λ−2,ν,k (ϕ), f = 2(−1)k k!ϕ, bλ,k (f )S ,
(8.8)
where f ∈ Dν (), ϕ ∈ D(S). The operator bλ,k is a meromorphic function of λ, with simple poles at points λ ∈ −5/2−N for which −λ−1/2 k −2λ−3. It intertwines Rλ,ν T−λ−2−k . The operators bλ,m with m k can be in the natural way extended to the space Tνk (), see Section 6. We see from formula (8.7) that bλ,k (f ) are expressed in terms of Taylor coefficients am (f ) by means of a triangular matrix with the unit diagonal. Therefore, coefficients am (f ) can be expressed in terms of bλ,k (f ): Wλ+1+m−2r,r bλ,m−2r (f ), (8.9) am (f ) = 0rm/2
this formula will be proved in Section 10. There is a similar formula for am∗ (f ): ∗ Wλ+1+m−2r,r bλ,m−2r (f ). (8.10) am∗ (f ) = 0rm/2 ∗ is the operator on D(S) defined similarly to Wσ,r – by means of a Here Wσ,r generating function: ∞ 3 σ + 2 − i∂ σ + 2 + i∂ ∗ 2k , ; σ + ; z2 . Wσ,k z =F (8.11) 2 2 2 k=0
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By [2], 2.1(23), the hypergeometric function (8.11) is the hypergeometric function (3.5) multiplied by (1 − z2 )−1/2 . Together with (6.2) it deduces (8.10) from (8.9). In its turn, formula (8.10) with (6.4) and (8.8) gives an expression of the ‘old basis’ ϕδ (m) (u0 ) in ν () in terms of the ‘new basis’ ξλ,ν,k : ϕδ (m) (u0 ) =
0rm/2
m! ∗ (ϕ)). ξλ,ν,m−2r (W−λ−1+m−2r,r (m − 2r)!
Let us consider the Poisson and Fourier transforms at poles of each other. Then these transforms have some special properties. Since the notation for these transforms become too long, let us rename them: + Pλ,ν;λ+1+m,ν−m = Pλ,ν,m ,
Fλ,ν;−λ−1+m,ν−m =
+ Fλ,ν,m ,
− Pλ,ν;−λ−2−m,ν−m = Pλ,ν,m ,
Fλ,ν;λ−m,ν−m =
(8.12)
− Fλ,ν,m .
Let us return to formula (8.3). Both sides of it have no singularities at points (8.4) since the function has zero at these points. Evaluating the indeterminacy in the right-hand side by means of (8.5), (8.6), we obtain + Qλ,ν = 2(−1)ν T (λ, m)A−λ−2−m ◦ bλ,m , F−λ−2,ν,m
− Qλ,ν = 2(−1)ν T (λ, m)j (−λ − 2 − m, ν − m)bλ,m , F−λ−2,ν,m
where T (λ, m) =
m! (λ + 1) (λ + 2) . (2λ + 3 + m)
(8.13)
−2λ−3−m,m ± the leading factors are um . For the Poisson transformation Pλ,ν,m 0 and u0 The first one is a polynomial in u0 . Let us fix k ∈ N. Let Re λ < −k − 3/2. Then for any m = 0, 1, . . . , k the second leading factor is o(uk0 ), so that, in particular, the ± lies in Tνk () and we can apply to it the boundary image of the transform Pλ,ν,m operators bλ,r , r k. These boundary operators turn out to be the inverse operators for the Poisson transforms – up to a factor or the operator Aσ . Namely, we have
THEOREM 8.1. Let k ∈ N. Let Re λ < −k − 3/2 and λ ∈ −3/2 − N. Then for m k we have: − = (−1)ν−m Aλ+1+m , bλ,m ◦ Pλ,ν,m
bλ,m ◦
+ Pλ,ν,m
= (−1)
ν−m
j (λ + 1 + m, ν − m) · E,
(8.14) (8.15)
and for r, m k, r = m, we have ± = 0. bλ,r ◦ Pλ,ν,m ± is an operator D(S) → D(S) intertwining Proof. The composition bλ,r ◦ Pλ,ν,m Tλ+1+m T−λ−2−r . Therefore, if r = m, then this operator is equal to 0. If r = m,
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V. F. MOLCHANOV
− then it is the operator Aλ+1+m up to a factor. By (7.2) the coefficient for Pλ,ν,m ϕ m ν−m ν−m in front of u0 is (−1) C−λ−2−m,0 ϕ which is equal to (−1) Aλ+1+m ϕ. On the other hand, the dominant term in bλ,m is the Taylor coefficient am . Therefore, the factor before Aλ+1+m mentioned above is (−1)ν−m . It proves (8.14). Similarly we prove (8.15). 2
Under the same conditions as in Theorem 8.1 we can apply distributions from νk () to the images of the Poisson transforms occuring in Theorem 8.1. Therefore, we can take as f in (8.8) a function from these images. Then, using Theorem 8.1, we obtain the following action of basic distributions from νk (). THEOREM 8.2. Let k ∈ N. Let Re λ < −k − 3/2 and λ ∈ −3/2 − N. Then for r, m k we have:
2(−1)ν m!Aλ+1+m ψ, ϕS , r = m, − ξ−λ−2,ν;r (ψ), Pλ,ν,m ϕ = 0, r = m, + ϕ ξ−λ−2,ν;r (ψ), Pλ,ν,m
ν 2(−1) m!j (λ + 1 + m, ν − m)ψ, ϕS , = 0, r = m.
r = m,
± to distribNow, using duality (8.2), we can extend the Fourier transforms Fλ,ν,m k utions ζ ∈ ν (), m k. Namely, for Re λ > k − 1/2 (we replaced λ by −λ − 2) and m k, we set: ± ζ, ϕS = ζ, P−±λ−2,ν,m ϕ . Fλ,ν,m ¯
Then Theorem 8.2 implies (here r, m k):
2(−1)ν m!A−λ−1+m , r = m, − Fλ,ν,m ◦ ξλ,ν,r = 0, r = m,
2(−1)ν m! · j (−λ − 1 + m, ν − m) · E, + ◦ ξλ,ν,r = Fλ,ν,m 0, r = m.
r = m,
(8.16)
± defined originally as maps from νk () These formulae show that the maps Fλ,ν,m to Dν (S) are actually maps νk () → Dν (S).
9. Decomposition of Boundary Representations Let Vλ,ν,k be the image of the map ξλ,ν,k . This space is contained in νk () and its projection to kν () is the whole kν (). THEOREM 9.1. Let λ ∈ 1/2 + N. Then the boundary representation Lλ,ν is diagonalizable which means that ν () decomposes into the direct sum of Vλ,ν,k , k ∈ N, and the resriction of Lλ,ν to Vλ,ν,k is equivalent to T−λ−1+k (by ξλ,ν,k ).
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Let λ ∈ 1/2 + N. If λ + 3/2 k 2λ + 1, then ξλ,ν,k is not defined, so that Vλ,ν,k is absent. Then denote by Vλ,ν,k the image of the operator Pλ,ν,λ−k , see (7.5). k This space is included in ν () and has the full projection to kν (). Therefore, the whole space ν () is the direct sum of subspaces Vλ,ν,k with k < λ + 3/2 and . For k such that λ+3/2 k 2λ+1, let l ∈ N be k > 2λ+1 and subspaces Vλ,ν,k such that l + k = 2λ + 1. Then k > l, and by (7.9) the subspace Vλ,ν,l is the image of Pλ,ν;λ−k . By (7.6) and (7.7) the subspace Vλ,ν,l + Vλ,ν,k is invariant with respect λ,ν,µϕ − Pλ,ν,µ ψ to Lλ,ν . Assign to a pair of functions ϕ, ψ in D(S) the element P in Vλ,l + Vλ,k , where µ = λ − k. By (7.6) and (7.7) the restriction of Lλ,ν to Vλ,ν,l + Vλ,ν,k is equivalent to T−µ−1 T−µ−1 , µ = λ − k. (9.1) 0 T−µ−1 The block (9.1) can not be diagonalizable. It is sufficient to check it for g by (2.2). THEOREM 9.2. Let λ ∈ 1/2 + N. Then the space ν () is the direct sum of the with subspaces Vλ,ν,k with k λ + 1/2 and k 2λ + 2 and the subspaces Vλ,ν,k λ + 3/2 k 2λ + 1. The representation Lλ,ν is equivalent to the direct sum of , k + l = 2λ + 1, λ + 1/2 Jordan blocks (9.1) acting on subspaces Vλ,ν,l + Vλ,ν,k and the representations T−1/2 and Tλ+1 , Tλ+2 , . . . . Now let us decompose the second boundary representation Mλ,ν . Let λ ∈ −5/2 − N. Then the boundary operators bλ,k are defined for all k ∈ N. Denote by τλ the map which to any sequence a = (a0 , a1 , a2 , . . .) of functions ak ∈ D(S) assigns the sequence bλ = (bλ,0 , bλ,1 , bλ,2 , . . .) by (8.7) – without f . This map is given by a lower triangular matrix with the unit diagonal. The inverse map τλ−1 is given by (8.9). THEOREM 9.3. Let λ ∈ −5/2 − N. Then the boundary representation Mλ,ν is diagonalizable which means that τλ Mλ,ν τλ−1 is a diagonal matrix with the diagonal T−λ−2 , T−λ−3 , . . . . Let λ ∈ −5/2 − N. Then the boundary operators bλ,k are not defined for k satisfying −λ − 1/2 k −2λ − 3. For these k, the Fourier transform Fλ,ν;σ,ε has poles of the second order in σ at µ = −λ − 2 − k. In this case define the following : operators bλ,k λ,ν,µ , = (4j (µ, ν))−1 F bλ,k
µ = −λ − 2 − k.
These operators are linear combinations of operators am with m = 0, 1, . . . , k, bλ a sequence which where the coefficient in front of ak is equal to 1. Denote by is obtained from the above-mentioned sequence bλ when bλ,k with k satisfying . Denote by τλ the map which to any −λ − 1/2 k −2λ − 3 are replaced by bλ,k bλ . This map is given by a lower sequence a = (a0 , a1 , . . .) assigns the sequence triangular matrix wich the unit diagonal. We have
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V. F. MOLCHANOV
THEOREM 9.4. Let λ ∈ −5/2 − N. Then the representation Mλ,ν has −λ − 3/2 Jordan blocks of the second order. Namely, the matrix τλ Mλ,ν τλ−1 is a lower triangular matrix with the diagonal T−λ−2 , T−λ−3 , . . . and with −λ − 3/2 Jordan blocks, these Jordan blocks are situated at the intersections of kth rows and lth columns with k satisfying −λ − 1/2 k −2λ − 3 and l + k = −2λ − 3 (so that k > l), these blocks are: Tµ 0 , µ = −λ − 2 − k. Tµ Tµ
10. Decomposition of Canonical Representations We consider λ lying in strips Ik : − 3/2 + k < Re λ < −1/2 + k, k ∈ Z. For λ lying at lines Re λ ∈ 1/2 + Z (boundaries of the strips Ik ), decompositions can be done similarly – in the spirit of the decomposition of the Berezin form. Case (A): λ ∈ I0 . Let f ∈ D−λ−2,ν (C), h ∈ Dλ,ν (C). For λ ∈ I0 , the restric = X belong to L2 (X, dx), see Section 5. Let us take tions of both functions to X the Plancherel formula (3.11) for these restrictions and pass to restrictions to by (5.1) and to Fourier transforms Fλ,ν;σ,ε . We obtain ∞
1
ω(σ ) Fλ,ν;σ,ε f, F−λ,−2,ν;−σ −1,ε hS
dρ + f, h = σ =−1/2+iρ −∞ 2 ε ∞ 1
+
n=0
2
ωn Fλ,ν;−n−1,n+1 f, F−λ−2,ν;n,n+1 hS .
(10.1)
Now, using conjugacy (8.2), we transfer the Fourier transform from h to f as the Poisson transform. We obtain a formula which is the decomposition of the function f considered as a distribution in Dν (): ∞
1
ω(σ ) Pλ,ν;−σ −1,ε Fλ,ν;σ,ε f
dρ + f = σ =−1/2+iρ −∞ 2 ε +
∞ 1 n=0
2
ωn Pλ,ν;n,n+1 Fλ,ν;−n−1,n+1 f.
(10.2)
Let now f ∈ D−λ−2,ν (C), h ∈ D−λ−2,ν (C). Then the restrictions of both functions = X belong to L2 (X, dx) again. Similarly as before, by Theorem 3.4 and to X formula (5.3) we obtain the decomposition of the sesqui-linear form ( , )λ,ν : ∞
1
ω(σ ) (λ, ν; σ, ε)Fλ,ν;σ,ε f, Fλ,ν;−σ −1,ε hS
dρ + (f, h)λ,ν = σ =−1/2+iρ 2 −∞ ε +
∞ 1 n=0
2
ωn (λ, ν)Fλ,ν;−n−1,n+1 f, Fλ,ν;n,n+1 hS
(10.3)
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first for −1 Re λ < −1/2 by Theorem 3.4 and then, using (5.2), for −3/2 < Re λ −1. THEOREM 10.1. Let λ ∈ I0 . Then the canonical representation Rλ,ν decomposes as the quasiregular representation from Section 7 into representations of the continuous series with multiplicity 2 and representations of the discrete series with multiplicity 1. Namely, let us assign a function the family of its Fourier components {Fλ,ν;σ,ε f, Fλ,,ν;−n−1,n+1 f }, σ = −1/2 + iρ, n ∈ N. This correspondence is Gequivariant. There is an inversion formula, see (10.2), and a ‘Plancherel formula’ for the form ( , )λ,ν , see (10.3). Now we are able to prove equality (8.3) and hence equality (7.1) – by conjugacy (8.2). Let us take equality (10.1) with −λ − 2 instead of λ and with Qλ,ν f instead of f and compare it with (10.3). It gives (8.3) for λ ∈ I0 . By analyticity we extend it to all possible λ. Case (B): λ ∈ Ik+1 , k ∈ N. Let us continue the decomposition (10.2) analytically in λ from I0 to Ik+1 , k ∈ N. Some poles in σ of the integrand intersect the integrating line – the line Re σ = −1/2. They are poles σ = λ − m and σ = −λ − 1 + m of the Poisson transform with ε ≡ ν − m, 0 m k. They give additional summands to the right-hand side. So after the continuation we obtain: f =
∞ −∞
+
∞
+
n=0
k
(10.4)
πλ,ν,m(f ),
m=0
where the integral and the series mean the same as in (10.2) and πλ,ν,m (f ) = (−1)ν
1 1 + · · ξλ,ν,m(Fλ,ν,m f ). 2m! j (−λ − 1 + m, ν − m)
Similarly, the continuation of (10.3) gives (f, h)λ,ν =
∞ −∞
+
∞ n=0
+
k 1
2 m=0
+ − M(λ, m)Fλ,ν,m f, Fλ,ν,m h , S
(10.5)
where the integral and the series mean the same as in (10.3) and M(λ, m) is given by (3.17). The operators πλ,ν,m , m k, can be extended from Dν () to the space νk (), see Section 6, and therefore to the sum Dνk () = Dν () + νk (). Then, as it follows from (8.16), these operators πλ,ν,m are projection operators onto Vλ,ν,m . Decomposition (10.4) can be also extended to the space Dνk (). In particular, for f ∈ νk () the integral and the series in (10.4) disappear. Indeed, for h ∈ Dν (), the function P−λ−2,ν;σ ,ε (F−λ−2,ν;−σ −1,ε h) has leading factors
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V. F. MOLCHANOV
,ν−ε uλ−σ and u0λ+σ +1,ν−ε which are o(uk+1 0 0 ) for λ ∈ Ik+1 . Therefore, this decomposition becomes the decomposition of the distribution f into its projections in Vλ,ν,m, m k. Using (7.10), we obtain that Dνk () is included in the domain of the form ( , )λ,ν . In particular, we have orthogonality relations:
1 + − f, Fλ,ν,m h , M(λ, m)Fλ,ν,m S 2 = 0, r = m,
(πλ,ν;m (f ), πλ,ν,m(h))λ,ν = (πλ,ν;m(f ), πλ,ν,r (h))λ,ν
so that (10.5) is a ‘Pythagorean theorem’ for decomposition (10.4). THEOREM 10.2. Let λ ∈ Ik+1 , k ∈ N. Then the space Dν () has to be completed to Dνk (). On this space the representation Rλ,ν splits into the sum of two terms: the first one decomposes as Rλ,ν does in Case (A), the second one decomposes into the sum of k + 1 irreducible representations T−λ−1+m , m = 0, 1, . . . , k (there are no Jordan blocks). Namely, let us assign to any f ∈ Dνk () the family {Fλ,ν;σ,ε f, Fλ,ν;n,n+1 f, πλ,ν,m(f )} where σ = −(1/2) + iρ, n ∈ N, m = 0, 1, . . . , k. This correspondence is G-equivariant. There is an inverse formula, see (10.4), and a ‘Plancherel formula’, see (14.5). Case (C): λ ∈ I−k−1 , k ∈ N. Let us continue decomposition (10.2) analytically in σ from I0 to I−k−1 . Here poles σ = −λ − 2 − m and σ = λ + 1 + m, ε ≡ ν − m, m k, of the integrand (they are poles of the Fourier transform) give additional terms. We obtain ∞ ∞ k + + λ,ν,m(f ), (10.6) f = −∞
n=0
m=0
where λ,ν,m =
(−1)ν−m P + ◦ bλ,m . j (λ + 1 + m, ν − m) λ,ν,m
+ . They can be extended The operators λ,ν,m map Dν () in the image of Pλ,ν,m k to the space Tν (), m k, since the operators bλ,m , m k, are defined on this + , space. In particular, the operators λ,ν,m can be applied to the images of Pλ,ν,r r k. Hence we can consider the products λ,ν,mλ,ν,r . It follows from (8.14) + . Moreover, and (8.15) that λ,ν,m are projection operators onto images of Pλ,ν,m they are orthogonal in the sense of the form ( , )λ,ν :
(λ,ν,m (f ), λ,ν,m(h))λ,ν = N(λ, ν, m)A−λ−2−m bλ,m (f ), bλ,m (h)S , (λ,ν,m(f ), λ,ν,r (h))λ,ν = 0, r = m, where 2(−1)m T (λ, m), j (λ + 1 + m, ν − m) T (λ, m) is given by (8.13). N(λ, ν, m) =
CANONICAL AND BOUNDARY REPRESENTATIONS ON A HYPERBOLOID OF ONE SHEET
213
Now we continue decomposition (10.3) from I0 to I−k−1 . Poles of the integrand which intersect the integrating line Re σ = −1/2 and give additional terms (they are poles of the Fourier transform) turn out fortunately to be of the first order, since at these points the function λ,ν;σ,ε has zero of the first order. After the continuation we obtain: ∞ ∞ k (f, h)λ,ν = + + N(λ, ν, m)A−λ−2−m bλ,m (f ), bλ,m (h)S . (10.7) −∞
n=0
m=0
It is a ‘Pythagorean theorem’ for decomposition (10.6). Now we are able to prove (8.9). It follows from (10.6) and (10.7) that a function f and the last sum in (10.6) have the same Fourier coefficients am with m k. By (7.2) we have q k Wλ+1+q,r (bλ,q (f ))u2r λ,ν,q (f ) = u0 0 + o(u0 ). 0r(k−q)/2
Summarizing it on q = 0, 1, . . . , k and collecting terms with um 0 together, we obtain formula (8.9) – first for λ ∈ I−k−1 and then by analyticity for all possible λ. THEOREM 10.3. Let λ ∈ I−k−1 , k ∈ N. Then the representation Rλ,ν considered on the space Tνk () splits into the sum of two terms. The first one acts on the subspace of functions f such that their Taylor coefficients am (f ) are equal to 0 for m k and decomposes as Rλ,ν does in Case (A), the second one decomposes into the direct sum of k + 1 irreducible representations T−λ−2−m , m k, acting on the + (there are no Jordan blocks). sum of the images of the Poisson transforms Pλ,ν,m There is an inverse formula, see (10.6), and a ‘Plancherel formula’ for the form ( , )λ,ν , see (10.7).
References 1. 2.
3. 4.
5.
6.
Berezin, F. A.: Quantization in complex symmetric spaces, Izv. Akad. Nauk SSSR, Ser. Mat. 39(2) (1975), 363–402. Engl. transl.: Math. USSR Izv. 9 (1975), 341–379. Berezin, F. A.: Connection between co- and contravariant symbols of operators on the classical complex symmetric spaces, Dokl. Akad. Nauk SSSR 241(1) (1978), 15–17. Engl. transl.: Soviet Math. Dokl. 19(4) (1978), 786–789. Molchanov, V. F.: Representations of the pseudo-orthogonal group associated with a cone, Math. Sb. 81(3) (1970), 358–375. Engl. transl.: Math. USSR Sb. 10 (1970), 333–347. Molchanov, V. F.: Decomposition of the tensor square of a representation of a complementary series of the unimodular group of real second order matrices, Sibirsk. Math. Zh. 18(1) (1977), 174–188. Engl. transl.: Siberian Math. J. 18 (1977), 128–138. Molchanov, V. F.: Harmonic analysis on homogeneous spaces, Itogi nauki i tekhn., Sovr. probl. mat. Fund. napr., tom 59, 1990, 5–144. Engl. transl.: Encycl. Math. 59, Springer-Verlag, Berlin, 1995, pp. 1–135. Molchanov, V. F. and Grosheva, L. I.: Canonical and boundary representations on the Lobachevsky plane, Acta Appl. Math. 73(1&2) (2002), 59–77.
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Vershik, A. M., Gelfand, I. M. and Graev, M. I.: Representations of the group SL(2, R) where R is a ring of functions, Uspekhi Mat. Nauk 28(5) (1973), 83–128. Engl. transl.: London Math. Soc. Lect. Note Ser. 69, Cambridge Univ. Press, 1982, pp. 15–110. Vilenkin, N. Ya.: Special Functions and the Theory of Group Representations, Nauka, Moscow, 1965. Engl. transl.: Transl. Math. Monographs 22, Amer. Math. Soc., Providence, RI, 1968.
