The inversion formula and holomorphic extension of the minimal representation of the conformal group

Toshiyuki Kobayashi^{∗} and Gen Mano

RIMS, Kyoto University, Sakyo-ku, Kyoto, 606-8502, Japan

  toshi@kurims.kyoto-u.ac.jp; gmano@kurims.kyoto-u.ac.jp

Dedicated to Roger Howe on the occasion of his 60th birthday

Abstract

The minimal representation π of the indefinite orthogonal group O(m + 1, 2) is realized on the Hilbert space of square integrable functions on ℝ^{m} with respect to the measure |x|^{−1}dx_{1} ⋯ dx_{m}. This article gives an explicit integral formula for the holomorphic extension of π to a holomorphic semigroup of O(m + 3, ℂ) by means of the Bessel function. Taking its ‘boundary value’, we also find the integral kernel of the ‘inversion operator’ corresponding to the inversion element on the Minkowski space ℝ^{m,1}.

Mathematics Subject Classifications (2000) : Primary 22E30; Secondary 22E45, 33C10 35J10, 43A80, 43A85, 47D05, 51B20.

Key words and phrases : minimal representation, holomorphic semigroup, Hermite operator, highest weight module, conformal group, Bessel function, Hankel transform, Schrödinger model.

Contents

1 Introduction 3 1.1 Semigroup generated by a differential operator D . . . . . . 3 1.2 Comparison with the Hermite operator 𝒟 . . . . . . . . . . . 5 1.3 The action of SL(2, ℝ) × O(m) . . . . . . . . . . . . . . . . . 6 1.4 Minimal representation as hidden symmetry . . . . . . . . . . 7 2 Preliminary results on the minimal representation of O(m +

∗Partially supported by Grant-in-Aid for Scientific Research 18340037, Japan Society for the Promotion of Science.

1, 2) 10 2.1 Maximal parabolic subgroup of the conformal group . . . . . 11 2.2 L^{2}-model of the minimal representation . . . . . . . . . . . . 12 2.3 K-type decomposition . . . . . . . . . . . . . . . . . . . . . . 13 2.4 Infinitesimal action of the minimal representation . . . . . . . 14

3 Branching law of π_{+} 15 3.1 Schrödinger model of the minimal representation . . . . . . . 15 3.2 K-finite functions on the forward light cone C_{+} . . . . . . . . 16 3.3 Description of infinitesimal generators of 𝔰𝔩(2, ℝ) . . . . . . . 18 3.4 Central element Z of 𝔨_{ℂ} . . . . . . . . . . . . . . . . . . . . . 21 3.5 Proof of Proposition 3.2.1 . . . . . . . . . . . . . . . . . . . . 22 3.6 One parameter holomorphic semigroup π_{+}(e^{tZ}) . . . . . . . . 23

4 Radial part of the semigroup 24 4.1 Result of the section . . . . . . . . . . . . . . . . . . . . . . . 25 4.2 Upper estimate of the kernel function . . . . . . . . . . . . . 27 4.3 Proof of Theorem 4.1.1 (Case Re t > 0) . . . . . . . . . . . . 28 4.4 Proof of Theorem 4.1.1 (Case Re t = 0) . . . . . . . . . . . . 31 4.5 Weber’s second exponential integral formula . . . . . . . . . . 32 4.6 Dirac sequence operators . . . . . . . . . . . . . . . . . . . . . 32

5 Integral formula for the semigroup 33 5.1 Result of the section . . . . . . . . . . . . . . . . . . . . . . . 34 5.2 Upper estimates of the kernel function . . . . . . . . . . . . . 35 5.3 Proof of Theorem 5.1.1 (Case Re t > 0) . . . . . . . . . . . . 36 5.4 Proof of Theorem 5.1.1 (Case Re t = 0) . . . . . . . . . . . . 37 5.5 Spectra of an O(m)-invariant operator . . . . . . . . . . . . . 38 5.6 Proof of Lemma 5.3.1 . . . . . . . . . . . . . . . . . . . . . . 40 5.7 Expansion formulas . . . . . . . . . . . . . . . . . . . . . . . . 40

6 The unitary inversion operator 42 6.1 Result of the section . . . . . . . . . . . . . . . . . . . . . . . 42 6.2 Inversion and Plancherel formula . . . . . . . . . . . . . . . . 44 6.3 The Hankel transform . . . . . . . . . . . . . . . . . . . . . . 45 6.4 Forward and backward light cones . . . . . . . . . . . . . . . 45 7 Explicit actions of the whole group on L^{2}(C) 46

7.1 Bruhat decomposition of O(m + 1, 2) . . . . . . . . . . . . . . 46 7.2 Explicit action of the whole group . . . . . . . . . . . . . . . 48

8 Appendix: special functions 49 8.1 Laguerre polynomials . . . . . . . . . . . . . . . . . . . . . . . 49 8.2 Hermite polynomials . . . . . . . . . . . . . . . . . . . . . . . 50 8.3 Gegenbauer polynomials . . . . . . . . . . . . . . . . . . . . . 50 8.4 Spherical harmonics and Gegenbauer polynomials . . . . . . . 51 8.5 Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . . 52

1 Introduction

1.1 Semigroup generated by a differential operator D

Consider the differential operator

D := |x|(\frac{Δ}{4} − 1)
= \frac{1}{4}(∑_{j=1}^{m} x_{j}^{2})^{\frac{1}{2}} (∑_{j=1}^{m} \frac{∂^{2}}{∂x_{j}^{2}} − 4)  (1.1.1)

on ℝ^{m}. A distinguishing feature here is that D has the following properties (see Remark 3.4.4):

1) D extends to a self-adjoint operator on L^{2}(ℝ^{m}, \frac{dx}{|x|}).

2) D has only discrete spectra \{−(j + \frac{m−1}{2}) : j = 0, 1, ⋯\} in L^{2}(ℝ^{m}, \frac{dx}{|x|}). Therefore, one can define a continuous operator

e^{tD} = ∑_{k=0}^{∞} \frac{t^{k}}{k!}D^{k}

for Re t ≥ 0 with the operator norm e^{− \frac{m−1}{2} Re t} satisfying the composition law

e^{t_{1}D} ∘ e^{t_{2}D} = e^{(t_{1}+t_{2})D}.

Thus, e^{tD} (Re t > 0) forms a holomorphic one-parameter semigroup. Besides, the operator e^{tD} is self-adjoint if t is real, and is unitary if t is purely imaginary.

We ask:

Question. Find an explicit formula of e^{tD}.

In this context, our main results are stated as follows:

Theorem A (see Theorem 5.1.1). The holomorphic semigroup e^{tD} (Re t > 0) is given by

(e^{tD}u)(x) = ∫_{ℝ^{m}} K^{+}(x, x′; t)u(x′)\frac{dx′}{|x′|} for u ∈ L^{2}(ℝ^{m}, \frac{dx}{|x|}),

where the integral kernel K^{+}(x, x′; t) is defined by

K^{+}(x, x′; t) := \frac{2e^{−2(|x|+|x′|) coth \frac{t}{2}}}{π^{\frac{m−1}{2}} sinh^{m−1} \frac{t}{2}} Ĩ_{\frac{m−3}{2}} (\frac{ψ(x, x′)}{sinh \frac{t}{2}}).

Here, we set Ĩ_{ν}(z) := (\frac{z}{2})^{−ν} I_{ν}(z) (I_{ν} is the I-Bessel function (see (8.5.2)) and

ψ(x, x′) := 2\sqrt{2(|x||x′| + ⟨x, x′⟩)} = 4|x|^{\frac{1}{2}} |x′|^{\frac{1}{2}} cos \frac{θ}{2},

where θ ≡ θ(x, x′) is the Euclidean angle between x and x′ in ℝ^{m}.

Particularly important is the special value at t = π\sqrt{−1}. We set

K^{+}(x, x′) := lim_{ε↓0} K^{+}(x, x′; π\sqrt{−1} + ε)
= \frac{2^{\frac{m−1}{2}} ψ(x, x′)^{− \frac{m−3}{2}}}{\sqrt{−1}^{m−1}π^{\frac{m−1}{2}}} J_{\frac{m−3}{2}} (ψ(x, x′))
= \frac{2}{\sqrt{−1}^{m−1}π^{\frac{m−1}{2}}} J̃_{\frac{m−3}{2}} (ψ(x, x′)).

Corollary B (see Theorem 6.1.1 and Corollary 6.2.1). The unitary operator e^{π\sqrt{−1}D} on L^{2}(ℝ^{m}, \frac{dx}{|x|}) is given by the Hankel-type transform:

T : L^{2}(ℝ^{m}, \frac{dx}{|x|}) → L^{2}(ℝ^{m}, \frac{dx}{|x|}), u ↦ ∫_{ℝ^{m}} K^{+}(x, x′)u(x′)\frac{dx′}{|x′|}.

This transform has the following properties:

(Inversion formula) T^{−1} = (−1)^{m+1}T.
(Plancherel formula) ∥T u∥_{L^{2}(ℝ^{m}, \frac{dx}{|x|})} = ∥u∥_{L^{2}(ℝ^{m}, \frac{dx}{|x|})}.

Let us explain the backgrounds and motivation of our question from three different viewpoints:

(1) Hermite semigroup and its variants (see Subsection 1.2).

(2) 𝔰𝔩_{2}-triple of differential operators on ℝ^{m} (see Subsection 1.3).

(3) Minimal representation of the reductive group O(m + 1, 2) (see Subsection 1.4).

1.2 Comparison with the Hermite operator 𝒟

Let us compare our operator D on L^{2}(ℝ^{m}, \frac{dx}{|x|}) with the well-known operator 𝒟 on L^{2}(ℝ^{m}, dx) defined by

𝒟 := \frac{1}{4}(Δ − |x|^{2}).  (1.2.1)

We call this operator the Hermite operator following the terminology of R. Howe and E.-C. Tan [17]. Analogously to D, the Hermite operator 𝒟 satisfies the following properties:

1) 𝒟 extends to a self-adjoint operator on L^{2}(ℝ^{m}, dx).

2) 𝒟 has only discrete spectra \{−\frac{1}{2}(j + \frac{m}{2}) : j = 0, 1, ⋯\} in L^{2}(ℝ^{m}, dx). We recall from [16, §5.3 and §6] that 𝒟 gives rise to a holomorphic

semigroup e^{t𝒟} (Re t > 0) (Hermite semigroup). Then the following results may be regarded as a prototype of Theorem A and Corollary B.

Fact C (see [16, §5] [31, §4.1]). The holomorphic semigroup e^{t𝒟} (Re t > 0) is given by

(e^{t𝒟} u)(x) = ∫_{ℝ^{m}} 𝒦(x, x′; t)u(x′)dx′,

where 𝒦 is the Mehler kernel defined by

𝒦(x, x′; t) := \frac{1}{(2π sinh \frac{t}{2})^{\frac{m}{2}}} exp(−\frac{1}{2} ^{t}(\begin{matrix}x \\ x′ \end{matrix}) A(t) (\begin{matrix}x \\ x′ \end{matrix})).  (1.2.2)

Here, we set

A(t) := \frac{1}{sinh \frac{t}{2}} (\begin{matrix}cosh \frac{t}{2}I_{m} & −I_{m} \\ −I_{m} & cosh \frac{t}{2}I_{m} \end{matrix}) ∈ GL(2m, ℝ).

In light of the limit formula:

lim_{ε↓0} 𝒦(x, x′; π\sqrt{−1} + ε) = \frac{1}{(2π\sqrt{−1})^{\frac{m}{2}}} e^{−\sqrt{−1}⟨x,x′⟩},

the special value of the operator e^{t𝒟} at t = π\sqrt{−1} reduces to the (ordinary) Fourier transform:

Fact D. The unitary operator e^{π\sqrt{−1}𝒟} on L^{2}(ℝ^{m}, dx) is nothing other than the Fourier transform ℱ:

(ℱf)(x) = \frac{1}{(2π\sqrt{−1})^{\frac{m}{2}}} ∫_{ℝ^{m}} e^{−\sqrt{−1}⟨x,x′⟩}f(x′)dx′.

We shall see in Section 6 a group theoretic interpretation of the fact that T is unitary and T^{4} = id as well as the fact that ℱ is unitary and ℱ^{4} = id.

1.3 The action of SL(2, ℝ) × O(m)

The self-adjoint operator D defined by (1.1.1) arises in the context of the 𝔰𝔩_{2}-triple of differential operators on ℝ^{m} as follows. We define

h̃ = 2 ∑_{j=1}^{m} x_{j} \frac{∂}{∂x_{j}} + m − 1, ẽ = 2\sqrt{−1}|x|, f̃ = \frac{\sqrt{−1}}{2}|x|Δ.  (1.3.1)

These operators h̃, ẽ and f̃ are skew self-adjoint operators on L^{2}(ℝ^{m}, \frac{dx}{|x|}), and satisfy the 𝔰𝔩_{2}-relation:

[h̃, ẽ] = 2ẽ, [h̃, f̃] = −2f̃, [ẽ, f̃] = h̃.

The operator D has the following expression

D = \frac{1}{2\sqrt{−1}}(−ẽ + f̃),  (1.3.2)

which means that \sqrt{−1}D corresponds to a generator of 𝔰𝔬(2) in 𝔰𝔩(2, ℝ). The 𝔰𝔩_{2}-module C_{0}^{∞}(ℝ^{m} \ {0}) exponentiates to a unitary representation of SL(2, ℝ) on L^{2}(ℝ^{m}, \frac{dx}{|x|}) (see Subsection 2.3 and Lemma 3.4.1).

On the other hand, there is a natural unitary representation of the orthogonal group O(m) on the same space L^{2}(ℝ^{m}, \frac{dx}{|x|}), and the actions of SL(2, ℝ) and O(m) mutually commute. Then we have the following discrete and multiplicity-free decomposition into irreducible representations of SL(2, ℝ) × O(m) (see [22, Theorem A]):

L^{2}(ℝ^{m}, \frac{dx}{|x|}) ≃ \begin{matrix}∞ \\ ∑ & ⊕ \\ j=0 \end{matrix} π_{2j+m−1}^{SL(2,ℝ)} ⊗ ℋ^{j}(ℝ^{m}).

Here, ℋ^{j}(ℝ^{m}) denotes the space of harmonic polynomials on ℝ^{m} of degree j, and π_{b}^{SL(2,ℝ)} stands for the irreducible unitary lowest weight representation of SL(2, ℝ) with minimal K-type ℂ_{b} for b ∈ ℕ_{+} = {1, 2, ⋯ }. It is the limit of discrete series if b = 1, and holomorphic discrete series if b ≥ 2.

In contrast, the Hermite operator 𝒟 (see (1.2.1)) arises from the following 𝔰𝔩_{2}-triple:

h̃′ := ∑_{j=1}^{m} x_{j} \frac{∂}{∂x_{j}} + \frac{m}{2}, ẽ′ := \frac{\sqrt{−1}}{2}|x|^{2}, f̃′ := \frac{\sqrt{−1}}{2}Δ,  (1.3.3)

where the Hermite operator 𝒟 is given by

𝒟 = \frac{1}{2\sqrt{−1}}(−ẽ′ + f̃′).  (1.3.4)

This 𝔰𝔩_{2}-triple also gives rise to the commutative actions of the double covering group SL(2, ℝ) of SL(2, ℝ) and O(m) on L^{2}(ℝ^{m}), whose irreducible decomposition amounts to (see [17, Chapter III, Theorem 2.4.4])

L^{2}(ℝ^{m}, dx) ≃ \begin{matrix}∞ \\ ∑ & ⊕ \\ j=0 \end{matrix} π_{j+ \frac{m}{2}}^{SL(2,ℝ)}⊗ ℋ^{j}(ℝ^{m}).

Here, π_{b}^{SL(2,ℝ)}stands for the irreducible unitary lowest weight representation of SL(2, ℝ) with minimal K-type ℂ_{b} for b ∈ \frac{1}{2}ℕ_{+} = \{\frac{1}{2}, 1, \frac{3}{2}, ⋯ \}. It is the Weil representation if b = \frac{1}{2}, and is obtained by the representation of SL(2, ℝ) if b ∈ ℕ_{+}.

1.4 Minimal representation as hidden symmetry

The representation of SL(2, ℝ) × O(m) on L^{2}(ℝ^{m}, \frac{dx}{|x|}) in Subsection 1.3 extends to the irreducible unitary representation π_{+} of the double covering group G := SO_{0}(m + 1, 2) of the indefinite orthogonal group (see Subsections 2.3 and 3.1). If m is odd, this representation is well-defined also as a representation of SO_{0}(m + 1, 2).

Similarly, the representation of SL(2, ℝ) × O(m) on L^{2}(ℝ^{m}, dx) extends to the unitary representation ϖ of the metaplectic group G′ = Sp(m, ℝ).

These groups G and G′ may be interpreted as hidden symmetry of SL(2, ℝ) × O(m). Conversely, the group SL(2, ℝ) × O(m) forms a ‘dual pair’ in each of the groups G and G′.

The unitary representations π_{+} and ϖ are typical examples of ‘minimal representations’ of reductive Lie groups in the sense that the Gelfand– Kirillov dimension attains its minimum among infinite dimensional unitary representations or in the sense that its annihilator is the Joseph ideal in the enveloping algebra.

The unitary representation π_{+} may be interpreted as the mass-zero spinzero wave equation, or as the bound states of the Hydrogen atom (in m space dimensions), while the representation ϖ is sometimes referred to as the oscillator representation or as the (Segal–Shale–)Weil representation.

We shall review the L^{2}-realization of the minimal representation π_{+} of SO_{0}(m + 1, 2), that is, the analog of the Schrödinger model on L^{2}(ℝ^{m}, \frac{dx}{|x|}) in Subsection 3.1. See also [8, 16] for a nice introduction to the original Schrödinger model of the Weil representation of Sp(m, ℝ) on L^{2}(ℝ^{m}).

To be more precise, we take

e = (\begin{matrix}0 & 1 \\ 0 & 0 \end{matrix}) , f = (\begin{matrix}0 & 0 \\ 1 & 0 \end{matrix}) , h := [e, f] = (\begin{matrix}1 & 0 \\ 0 & −1 \end{matrix})

to be a basis of 𝔰𝔩(2, ℝ) and define injective Lie algebra homomorphisms

ϕ : 𝔰𝔩(2, ℝ) → 𝔰𝔬(m + 1, 2),
φ : 𝔰𝔩(2, ℝ) → 𝔰𝔭(m, ℝ)

(see Subsection 3.3) such that the differential operators (1.3.1) and (1.3.3) are obtained via ϕ and φ, respectively, that is,

dπ_{+}(ϕ(X)) = X̃, dϖ(φ(X)) = X̃′

holds for X = e, f, h. Next we set

z := \frac{1}{2\sqrt{−1}}(−e + f) = \frac{1}{2\sqrt{−1}} (\begin{matrix}0 & −1 \\ 1 & 0 \end{matrix}) ∈ \sqrt{−1}𝔰𝔩(2, ℝ).

Then e^{\sqrt{−1}ℝϕ(z)} is the center of the maximal compact subgroup of G for m > 1, while e^{\sqrt{−1}ℝφ(z)} is that of G′ (we use the same notations ϕ and φ for their complex linear extensions).

In this context, we shall see in Lemma 3.4.3 and Remark 3.4.5 that the differential operators D and 𝒟 are given by

D = dπ_{+}(ϕ(z)),  𝒟 = dϖ(φ(z)).  (1.4.1)

Thanks to these formulas, Theorem A and Corollary B are also useful in the analysis on minimal representations of SO_{0}(m+1, 2) as well as Sp(m, ℝ) in the following contexts:

1) The Gelfand–Gindikin program — Theorem A.

The Gelfand–Gindikin program asks for extending a given unitary representation of a real semisimple Lie group G to a holomorphic object of some complex submanifold in its complexification G_{ℂ}. Stanton and Olshanskiǐ [29, 27] independently gave a general framework of the Gelfand– Gindikin program for holomorphic discrete series. Their abstract results are enriched, for example for G′ = Sp(m, ℝ), by the explicit formula of the Hermite semigroup e^{t𝒟} = ϖ(e^{tφ(z)}) for the Weil representation ϖ on L^{2}(ℝ^{m}) by Howe [16]. Likewise, Theorem A gives an explicit formula of the semigroup e^{tD} = π_{+}(e^{tϕ(z)}) for the minimal representation of G = SO_{0}(m+1, 2). Since e^{tϕ(z)} ∈ G_{ℂ} \ G for Re t > 0, Theorem A can be interpreted as a descendent of the Gelfand–Gindikin program.

2) The unitary inversion operator — Corollary B.

In the Schrödinger model of the minimal representation, G acts only on the function space L^{2}(ℝ^{m}, \frac{dx}{|x|}) but does not act on the underlying geometry ℝ^{m} itself. One may observe this fact by the aforementioned formula f̃ = dπ_{+}(ϕ(f)), which does not act on functions on ℝ^{m} as a vector field but acts as a differential operator of second order on ℝ^{m} (see (1.3.1)). To see how G acts on L^{2}(ℝ^{m}, \frac{dx}{|x|}), we use the facts that

1) G is generated by \overline{P^{max}} and w_{0}.

2) The \overline{P^{max}} action on L^{2}(ℝ^{m}, \frac{dx}{|x|}) is easily described (see Subsection 2.2).

Here, \overline{P^{max}} is a maximal parabolic subgroup of G (see Subsection 2.1) and w_{0} := e^{π\sqrt{−1}Z} ∈ G sends \overline{P^{max}} to its opposite parabolic subgroup P^{max}. Geometrically, \overline{P^{max}} is essentially the conformal affine transformation group on the flat standard Lorentz manifold ℝ^{m,1} (the Minkowski space), and w_{0} := e^{π\sqrt{−1}Z} ∈ G acts on ℝ^{m,1} as the ‘inversion’ element (see Subsection 6.1).

Thus the representation π_{+} of G on L^{2}(ℝ^{m}, \frac{dx}{|x|}) would be understood if

we find an explicit formula for π_{+}(w_{0}). But since the formula w_{0} = e^{π\sqrt{−1}ϕ(z)} implies π_{+}(w_{0}) = e^{π\sqrt{−1}D}, Corollary B answers this question. This parallels the fact that the Weil representation is generated by the (natural) action of the Siegel parabolic subgroup P_{Siegel} and the Fourier transform ℱ = e^{π\sqrt{−1}𝒟} (see Fact D).

Briefly, we pin down the analogy in the table below. Howe [16] established the left-hand side of the table for the oscillator representation ϖ of Sp(m, ℝ), while Theorem A and Corollary B supply the right-hand side of the table for the minimal representation π_{+} of SO_{0}(m + 1, 2).

𝔰𝔭(m, ℝ)  𝔰𝔬(m + 1, 2)

minimal representation (dϖ, L^{2}(ℝ^{m})) (dπ_{+}, L^{2}(ℝ^{m}, \frac{dx}{|x|}))

e  \frac{\sqrt{−1}}{2}|x|^{2}  2\sqrt{−1}|x|
f  \frac{\sqrt{−1}}{2}Δ  \frac{\sqrt{−1}}{2}|x|Δ
h  E_{x} + \frac{m}{2}  2E_{x} + m − 1
z = \frac{−e+f}{2\sqrt{−1}}  𝒟 := \frac{1}{4}(Δ − |x|^{2}) D := \frac{1}{4}|x|Δ − |x|
holomorphic semigroup e^{tz} 𝒦(x, x′; t) K(x, x′; t)
inversion e^{π\sqrt{−1}z}  Fourier transform Hankel transform

maximal parabolic subgroup P_{Siegel}

\overline{P^{max}}

Analogous results to Corollary B were previously known for some singular unitary highest weight representations. For example, see a paper [4] by Ding, Gross, Kunze and Richards for those of U(n, n). Since SU(2, 2) is a double covering group of SO_{0}(4, 2), Corollary B in the case m = 3 essentially corresponds to [4, Corollary 7.5] in the case (n, k) = (2, 1) in their notation. However, our proof based on an analytic continuation (see Theorem A) is different from theirs. Also in [20], we shall find the inversion operator π(w_{0}) for the minimal representation of O(p, q) for p + q even, and in particular give yet another proof of Corollary B for odd m.

We also present explicit integral formulas of π(e^{tZ}) and π(w_{0}) when restricted to radial functions and alike (see Theorem 4.1.1). This yields a group theoretic interpretation of some classic formulas of special functions of one variable including Weber’s second exponential integral formula on Bessel function and the reciprocal and the Parseval-Plancherel formula for the Hankel transform.

This article is organized as follows. After summarizing the preliminary results on the L^{2}-model of the minimal representation π, we find explicitly which function arises for describing K-types in L^{2}(ℝ^{m}, \frac{dx}{|x|}), and define a holomorphic extension π_{+}(e^{tZ}) in Section 3. The integral formula of the ‘radial’ part of π_{+}(e^{tZ}) is given in Theorem 4.1.1. Theorem A is proved in Section 5 by using the result of Section 4. Taking the special value at t = π\sqrt{−1}, we obtain the integral formula of π(w_{0}) corresponding to the inversion element w_{0}. This corresponds to Corollary B, and is proved in Section 6. Our integral formula for π(w_{0}) enables us to write explicitly the action of the whole group G = SO_{0}(m + 1, 2) on L^{2}(ℝ^{m}, \frac{dx}{|x|}). This is given in Section 7. For the convenience of the reader, we collect basic formulas of special functions in a way that we use in this article.

The main results of the paper were announced in [19] with a sketch of the proof.

Notation: ℕ = {0, 1, 2, … }, ℕ_{+} = {1, 2, 3, …}, ℝ_{≤0} = {x ∈ ℝ : x ≤ 0}, ℝ_{+} = {x ∈ ℝ : x > 0}.

2 Preliminary results on the minimal representation of O(m + 1, 2)

This section gives a brief review on the known results of the L^{2}-model of the minimal representation of O(m + 1, 2) in a way that we shall use later. We shall give an explicit action of the maximal parabolic subgroup \overline{P^{max}} and the Lie algebra 𝔤. Furthermore, we state an explicit K-type decomposition of L^{2}(C_{+}) even though the action of K itself is not given explicitly here (see Section 7 for this).

2.1 Maximal parabolic subgroup of the conformal group

Let O(m + 1, 2) be the indefinite orthogonal group which preserves the quadratic form x_{0}^{2}+⋯+x_{m}^{2}−x_{m+1}^{2}−x_{m+2}^{2} of signature (m+1, 2). We denote by e_{0}, ⋯, e_{m+2} the standard basis of ℝ^{m+3}, and by E_{ij} (0 ≤ i, j ≤ m + 2) the matrix unit. We set

ε_{j} := \begin{cases}1 & (1 ≤ j ≤ m), \\ −1 & (j = m + 1). \end{cases}

We take the following elements of the Lie algebra 𝔬(m + 1, 2):

\overline{N}_{j} := E_{j,0} + E_{j,m+2} − ε_{j}E_{0,j} + ε_{j}E_{m+2,j} (1 ≤ j ≤ m + 1),
N_{j} := E_{j,0} − E_{j,m+2} − ε_{j}E_{0,j} − ε_{j}E_{m+2,j} (1 ≤ j ≤ m + 1), (2.1.1)
E := E_{0,m+2} + E_{m+2,0},

and define subalgebras of 𝔬(m + 1, 2) by

\overline{𝔫^{max}} := ∑_{j=1}^{m+1} ℝ\overline{N}_{j}, 𝔫^{max} := ∑_{j=1}^{m+1} ℝN_{j}, 𝔞 := ℝE.

Then we define the following subgroups of O(m + 1, 2):

M_{+}^{max} := {g ∈ O(m + 1, 2) : g · e_{0} = e_{0}, g · e_{m+2} = e_{m+2}}
≃ O(m, 1),
M^{max} := M_{+}^{max} ∪ {−I_{m+3}} · M_{+}^{max}
≃ O(m, 1) × ℤ_{2},
\overline{N^{max}} := exp(\overline{𝔫^{max}}),
N^{max} := exp(𝔫^{max}),
A := exp(𝔞).

For b = (b_{1}, …, b_{m+1}) ∈ ℝ^{m+1}, we set

\overline{n}_{b} := exp(∑_{j=1}^{m+1} b_{j}\overline{N}_{j})  (2.1.2)

= I_{m+3} + ∑_{j=1}^{m+1} b_{j}\overline{N}_{j} + \frac{Q(b)}{2}(−E_{0,0} − E_{0,m+1} + E_{m+1,0} + E_{m+1,m+1}),

where Q(b) is the quadratic form of signature (m, 1) given by

Q(b) := b_{1}^{2} + ⋯ + b_{m}^{2} − b_{m+1}^{2}.

The Lie group \overline{N^{max}} is abelian, and we have an isomorphism of Lie groups:

ℝ^{m+1} →^{∼} \overline{N^{max}}, b ↦ \overline{n}_{b}.

It is readily seen from (2.1.2) that

\overline{n}_{b}(e_{0} − e_{m+2}) = e_{0} − e_{m+2},  (2.1.3)
\overline{n}_{b}(e_{0} + e_{m+2}) = ^{t}(1 − Q(b), 2b_{1}, …, 2b_{m+1}, 1 + Q(b)).  (2.1.4)

We also note

e^{tE}(e_{0} + e_{m+2}) = e^{t}(e_{0} + e_{m+2}).  (2.1.5)

The subgroup M_{+}^{max}\overline{N^{max}} is isomorphic to the semidirect product group O(m, 1) ⋉ ℝ^{m+1} via the bijection ℝ^{m+1} →^{∼} \overline{N^{max}}, b ↦ \overline{n}_{b}. In this context, M_{+}^{max}\overline{N^{max}} is regarded as the group of isometries of the Minkowski space ℝ^{m,1}, while O(m + 1, 2) is the group of Möbius transformations on ℝ^{m,1} preserving its conformal structure.

Next, we define a maximal parabolic subgroup

\overline{P^{max}} := M^{max}A\overline{N^{max}}.

In our analysis of the minimal representation of O(m + 1, 2), \overline{P^{max}} plays an analogous role to the Siegel parabolic subgroup of the metaplectic group Sp(m, ℝ) for the Weil representation.

2.2 L^{2}-model of the minimal representation

We shall briefly review the L^{2}-model of the minimal representation of O(m+ 1, 2). Let C_{±} be the forward and the backward light cone respectively:

C_{±} := {(ζ_{1}, ⋯, ζ_{m+1}) ∈ ℝ^{m+1} : ±ζ_{m+1} > 0, ζ_{1}^{2} + ⋯ + ζ_{m}^{2} = ζ_{m+1}^{2}},

and C be its disjoint union C_{+} ∪ C_{+}, that is, C is the conical subvariety with respect to the quadratic form of signature (m, 1):

C = {ζ ∈ ℝ^{m+1} \ {0} : Q(ζ) = 0}.  (2.2.1)

Note that M_{+}^{max} acts on C transitively.

The measure dμ on C is naturally defined to be δ(Q), the generalized function associated to the quadratic form Q (see [11, Chapter III, §2]).

Then we form a unitary representation π of \overline{P^{max}} on the Hilbert space L^{2}(C, dμ) ≡ L^{2}(C) as follows: for ψ ∈ L^{2}(C),

(π(e^{tE})ψ)(ζ) := e^{− \frac{m−1}{2} t}ψ(e^{−t}ζ)  (t ∈ ℝ),  (2.2.2)
(π(m)ψ)(ζ) := ψ(^{t}mζ)  (m ∈ M_{+}^{max}), (2.2.3)
(π(−I_{m+3})ψ)(ζ) := (−1)^{\frac{m−1}{2}} ψ(ζ),  (2.2.4)
(π(\overline{n}_{b})ψ)(ζ) := e^{2\sqrt{−1}(b_{1}ζ_{1}+···+b_{m+1}ζ_{m+1})}ψ(ζ)  (b ∈ ℝ^{m+1}). (2.2.5)

Then π is irreducible and unitary as a \overline{P^{max}}-module, and it is proved in [23, Theorem 4.9] that the \overline{P^{max}}-module (π, L^{2}(C)) extends to an irreducible unitary representation of O(m + 1, 2) if m is odd. We shall denote this representation of O(m + 1, 2) by the same letter π. The direct sum decomposition

L^{2}(C) = L^{2}(C_{+}) ⊕ L^{2}(C_{−})  (2.2.6)

yields a branching law π = π_{+} ⊕ π_{−} with respect to the restriction O(m + 1, 2) ↓ SO_{0}(m + 1, 2), where SO_{0}(m + 1, 2) is the identity component of O(m + 1, 2). The irreducible representations π_{+} and π_{−} of SO_{0}(m + 1, 2) are contragredient to each other, one is a highest weight module, and the other is a lowest weight module.

2.3 K-type decomposition

Let SO(2) be the double covering of SO(2). We write η for the unique element of SO(2) of order two. Then, we have an exact sequence:

1 → {1, η} → SO(2) → SO(2) → 1.

Let

G = SO_{0}(m + 1, 2)  (2.3.1)

be the double covering group of SO_{0}(m + 1, 2) characterized as follows: a maximal compact subgroup K is of the form K_{1} × K_{2} ≃ SO(m + 1) × SO(2) and the kernel of the covering map G → SO_{0}(m + 1, 2) is given by {(1, 1), (1, η)}. Likewise, the double covering group O(m+1, 2) of O(m+1, 2) is defined.

If m is odd, the irreducible representation (π_{±}, L^{2}(C_{±})) defined in Subsection 2.2 extends to that of SO_{0}(m + 1, 2), and therefore, also that of G = SO_{0}(m + 1, 2) as we proved it more generally for O(p, q) (p + q : even) in [23]. We shall use the same letter π_{+} to denote the extension to SO_{0}(m + 1, 2) or G. If m is even, by [28], the irreducible unitary representation (π_{±}, L^{2}(C_{±})) is still well-defined as a representation of G, whose Lie algebra 𝔤 = 𝔰𝔬(m + 1, 2) of G acts in the same manner as in the case of odd m (see Subsection 2.4).

The K-type formula of (π_{±}, L^{2}(C_{±})) is given as follows:

L^{2}(C_{±})_{K} ≃ ⨁_{a=0}^{∞} ℋ^{a}(ℝ^{m+1}) ⊠ ℂe^{±(a+ \frac{m−1}{2} )\sqrt{−1}θ}.  (2.3.2)

(For example, this formula can be read from [28, §1.3] by substituting d = m − 1, p = 1 and q = 0.) Here, ℋ^{a}(ℝ^{m+1}) stands for the representation of SO(m + 1) on the space of the spherical harmonics which is irreducible if m > 1 (see Subsection 8.4).

Likewise, the representation (π, L^{2}(C)) of O(m + 1, 2) decomposes when restricted to its maximal compact subgroup as follows (see [21, Theorem 3.6.1]):

L^{2}(C)_{K} ≃ ⨁_{a=0}^{∞} ℋ^{a}(ℝ^{m+1}) ⊠ ℋ^{a+ \frac{m−1}{2}} (ℝ^{2}).  (2.3.3)

ℋ^{a+ \frac{m−1}{2}} (ℝ^{2}) decomposes into ℂe^{(a+ \frac{m−1}{2} )\sqrt{−1}θ} ⊕ ℂe^{−(a+ \frac{m−1}{2} )\sqrt{−1}θ} as SO(2)- modules (see Subsection 8.4). This corresponds to the decomposition

L^{2}(C) = L^{2}(C_{+}) ⊕ L^{2}(C_{−}),

for which the K-type formula is given by (2.3.2).

2.4 Infinitesimal action of the minimal representation

For \overline{N}_{j}, N_{j}(1 ≤ j ≤ m + 1) and E, we define linear transformations on the space 𝒮′(ℝ^{m+1}) of tempered distributions by

dϖ̂(\overline{N}_{j}) := 2\sqrt{−1}ζ_{j},  (2.4.1)
dϖ̂(N_{j}) := \sqrt{−1} (−\frac{m + 3}{2}ε_{j} \frac{∂}{∂ζ_{j}} − E_{ζ}ε_{j} \frac{∂}{∂ζ_{j}} + \frac{1}{2}ζ_{j}□_{ζ}) ,  (2.4.2)
dϖ̂(E) := −\frac{m + 3}{2} − E_{ζ},  (2.4.3)

where we set

□_{ζ} := \frac{∂^{2}}{∂ζ_{1}^{2}} + \frac{∂^{2}}{∂ζ_{2}^{2}} + ⋯ + \frac{∂^{2}}{∂ζ_{m}^{2}} − \frac{∂^{2}}{∂ζ_{m+1}^{2}} , E_{ζ} := ∑_{j=1}^{m+1} ζ_{j} \frac{∂}{∂ζ_{j}} .

Then, we recall from [23] that this generates the infinitesimal action dπ of the Lie algebra 𝔰𝔬(m + 1, 2), and we have the following commutative diagram for any X ∈ 𝔰𝔬(m + 1, 2):

\begin{matrix}L^{2}(C)_{K} & ⟶^{ı} & 𝒮′(ℝ^{m+1}) \\ dπ(X) ↓ & & ↓ dϖ̂(X) \\ L^{2}(C)_{K} & ⟶^{ı} & 𝒮′(ℝ^{m+1}). \end{matrix}  (2.4.4)

Here, ı : L^{2}(C) → 𝒮′(ℝ^{m+1}) is given by u(ζ) ↦ u(ζ)δ(Q). This is welldefined and injective if m > 1 (see [23, §3.4]).

3 Branching law of π_{+}

The main goal of this section is Proposition 3.2.1, which explicitly describes special functions that arise as K-types in the ‘Schrödinger model’ on L^{2}(ℝ^{m}, |x|^{−1}dx) of the minimal representation of the double covering group G of SO_{0}(m + 1, 2).

3.1 Schrödinger model of the minimal representation

We have used the variables ζ = (ζ_{1}, ⋯, ζ_{m+1}) for the positive cone C_{+} ⊂ ℝ^{m+1} in Section 2, and will use the letter x = (x_{1}, ⋯, x_{m}) for the coordinate of ℝ^{m}. The projection

p : ℝ^{m+1} → ℝ^{m}, (ζ_{1}, ⋯, ζ_{m}, ζ_{m+1}) ↦ (ζ_{1}, ⋯, ζ_{m}).  (3.1.1)

induces a diffeomorphism from C_{+} onto ℝ^{m}\{0}, and the measure dμ on C_{+} is given by δ(Q), and therefore is pushed forward to \frac{1}{2|x|}dx = \frac{1}{2|x|}dx_{1} ⋯ dx_{m}. Thus, we have a unitary isomorphism:

\sqrt{2}p^{∗} : L^{2}(ℝ^{m}, \frac{dx}{|x|}) →^{∼} L^{2}(C_{+}).  (3.1.2)

Through this isomorphism, we can realize the minimal representation of G on L^{2}(ℝ^{m}, \frac{dx}{|x|}) as well. Named after the Schrödinger model of the Weil representation, we say this model is the Schrödinger model of the minimal representation of G. We shall work with this model from now on.

3.2 K-finite functions on the forward light cone C_{+}

This section refines the K-type decomposition (2.3.2) by providing an explicit irreducible decomposition:

L^{2}(ℝ^{m}, \frac{dx}{|x|})_{K} = ⨁_{a=0}^{∞} W_{a} = ⨁_{a=0}^{∞} ⨁_{l=0}^{a} W_{a,l}  (3.2.1)

according to the following chain of subgroups:

G  ⊃  K  ⊃ R := K ∩ (M_{+}^{max})_{0}
∥  ∥  ∥
SO_{0}(m + 1, 2) ⊃ SO(m + 1) × SO(2) ⊃  SO(m).

Here, the K-irreducible subspace W_{a} of L^{2}(ℝ^{m}, \frac{dx}{|x|}) and the R-irreducible supspace W_{a,l} of W_{a} is characterized by

W_{a} ≃ ℋ^{a}(ℝ^{m+1}) ⊠ ℂe^{(a+ \frac{m−1}{2} )θ} (see (2.3.2)),  (3.2.2)
W_{a,l} ≃ ℋ^{l}(ℝ^{m}),  (3.2.3)

as K-modules and as R-modules, respectively. Here, we note that the SO(m)-module ℋ^{l}(ℝ^{m}) occurs exactly once in the O(m+1)-module ℋ^{a}(ℝ^{m+1}) if 0 ≤ l ≤ a (see Subsection 8.4 (4)).

Proposition 3.2.1 will describe the subspace W_{a,l} of L^{2}(ℝ^{m}, \frac{dx}{|x|}) by means of Laguerre polynomials L_{n}^{α}(x) (see (8.1.1) for the definition). For this, we set

f_{a,l}(r) := L_{a−l}^{m−2+2l}(4r)r^{l}e^{−2r} (0 ≤ l ≤ a),  (3.2.4)

and define injective linear maps by

j_{a,l} : ℋ^{l}(ℝ^{m}) → C^{∞}(ℝ^{m}), ϕ(ω) ↦ (j_{a,l}ϕ)(rω) := f_{a,l}(r)ϕ(ω).

Here, we have identified ℝ^{m} with ℝ_{+} × S^{m−1} by the polar coordinate

ℝ × S^{m−1} → ℝ^{m}, (r, ω) ↦ rω.  (3.2.5)

Then, we have:

Proposition 3.2.1. 1)

j_{a,l}(ℋ^{l}(ℝ^{m})) ⊂ L^{1}(ℝ^{m}, \frac{dx}{|x|}) ∩ L^{2}(ℝ^{m}, \frac{dx}{|x|}).  (3.2.6)

2) Furthermore, the image of j_{a,l} coincides with the R-type W_{a,l}:

j_{a,l}(ℋ^{l}(ℝ^{m})) = W_{a,l}.

Remark 3.2.2. Proposition 3.2.1 (2) asserts in particular that j_{a,l}(ℋ^{l}(ℝ^{m})) and j_{a′,l}(ℋ^{l}(ℝ^{m})) are orthogonal to each other if a ≠ a′ (a, a′ ≥ l). More than this, it gives a representation theoretic proof of the fact that {f_{a,l}(r) : a = l, l + 1, ⋯} forms a complete orthogonal basis of L^{2}(ℝ_{+}, r^{m−2}dr) (see Lemma 4.3.1 for the normalization).

Remark 3.2.3. The indefinite orthogonal group O(p, q) (p+q : even, p, q ≥ 2, and (p, q) ≠ (2, 2)) has a minimal representation π whose minimal Ktype is of the form ℋ^{0}(ℝ^{p}) ⊗ ℋ^{\frac{p−q}{2}} (ℝ^{q}) if p ≥ q. In the L^{2}-model of π, we have proved that any M^{max}-fixed vector in ℋ^{0}(ℝ^{p}) ⊗ ℋ^{\frac{p−q}{2}} (ℝ^{q}) is a scalar multiple of the function r^{\frac{3−q}{2}} K_{\frac{q−3}{2}} (2r) (see [23, Theorem 5.5] for a precise statement), where K_{ν}(z) denotes the K-Bessel function. Since K_{− \frac{1}{2}} (2r) =

\frac{\sqrt{π}}{2} r^{− \frac{1}{2}} e^{−2r} and L_{0}^{α}(x) = 1, we have r^{\frac{3−q}{2}} K_{\frac{q−3}{2}} (2r) = \frac{\sqrt{π}}{2} e^{−2r} = \frac{\sqrt{π}}{2} f_{0,0}(r) if q = 2. This vector is a generator of the one dimensional vector space W_{0,0}.

Remark 3.2.4 (Weil representation). Let us compare our representation on L^{2}(ℝ^{m}, |x|^{−1}dx) with the (original) Schrödinger model on L^{2}(ℝ^{m}) of the Weil representation of G′ = Sp(m, ℝ). The counterpart to Proposition 3.2.1 can be stated as follows: we set for 0 ≤ l ≤ a

f′_{a,l}(r) := L_{a−l}^{\frac{m−2}{2} +l}(r^{2})r^{l}e^{− \frac{r^{2}}{2}} ,  (3.2.7)

and define linear maps by

j′_{a,l} : ℋ^{l}(ℝ^{m}) → C^{∞}(ℝ^{m}), ϕ(ω) ↦ (j′_{a,l}ϕ)(rω) = f′_{a,l}(r)ϕ(ω).

Then for any ϕ ∈ ℋ^{l}(ℝ^{m}), j′_{a,l}(ϕ) is square integrable on ℝ^{m}, and its image j′_{a,l}(ℋ^{l}(ℝ^{m})) is characterized by the following properties: Let (K′, R′) = (U(m) , O(m) ).

1) j′_{a,l}(ℋ^{l}(ℝ^{m})) is isomorphic to ℋ^{l}(ℝ^{m}) as R′-modules,

2) it is contained in the K′-type isomorphic to S^{a}(ℂ) ⊗ det^{\frac{1}{4}} .

The remaining part of this section is organized as follows. In Subsection 3.3, we shall define a central element Z of (the complexification of ) the Lie algebra 𝔨 of K, and compute its differential action dπ_{+}(Z) (see Lemma 3.4.3). By using this explicit form of dπ_{+}(Z), we prove Proposition 3.2.1 in Subsection 3.5. Finally, in Subsection 3.6, by looking at the eigenvalues of dπ_{+}(Z) (see Lemma 3.5.1), we shall see that {π_{+}(e^{tZ}) := exp(tdπ_{+}(Z)) : Re t > 0} forms a holomorphic semigroup of contraction operators.

In Section 5, we shall find an explicit integral kernel of this semigroup by using Proposition 3.2.1.

3.3 Description of infinitesimal generators of 𝔰𝔩(2, ℝ)

Let

e := (\begin{matrix}0 & 1 \\ 0 & 0 \end{matrix}) , f := (\begin{matrix}0 & 0 \\ 1 & 0 \end{matrix}) , h := [e, f] = (\begin{matrix}1 & 0 \\ 0 & −1 \end{matrix}) .

be the standard basis of 𝔰𝔩(2, ℝ). With the notation (2.1.1), we define a Lie algebra homomorphism

ϕ : 𝔰𝔩(2, ℝ) → 𝔰𝔬(m + 1, 2)  (3.3.1)

by

ϕ(e) = \overline{N}_{m+1}, ϕ(f) = N_{m+1}, ϕ(h) = −2E.  (3.3.2)

In this subsection, we shall explicitly describe dπ_{+}(ϕ(e)), dπ_{+}(ϕ(f)), and dπ_{+}(ϕ(h)) as differential operators on ℝ^{m}.

Lemma 3.3.1. Let E_{x} = ∑_{j=1}^{m} x_{j} \frac{∂}{∂x_{j}} and Δ = ∑_{j=1}^{m} \frac{∂^{2}}{∂x_{j}^{2}} . Then we have:

dπ_{+}(ϕ(h)) =2E_{x} + m − 1,  (3.3.3)
dπ_{+}(ϕ(e)) =2\sqrt{−1}|x|,  (3.3.4)
dπ_{+}(ϕ(f)) =\frac{\sqrt{−1}}{2}|x|Δ.  (3.3.5)

Remark 3.3.2. Lemma 3.3.1 corresponds to an analogous result for the Schrödinger model of the Weil representation (ϖ, L^{2}(ℝ^{m})) of Sp(m, ℝ) as follows: by the matrix realization of the real symplectic Lie algebra 𝔰𝔭(m, ℝ), we define a Lie algebra homomorphism φ : 𝔰𝔩(2, ℝ) → 𝔰𝔭(m, ℝ) by

φ(e) = (\begin{matrix}0 & I_{m} \\ 0 & 0 \end{matrix}) , φ(f) = (\begin{matrix}0 & 0 \\ I_{m} & 0 \end{matrix}) , φ(h) = (\begin{matrix}I_{m} & 0 \\ 0 & −I_{m} \end{matrix}) . (3.3.6)

Then, {dϖ(φ(h)), dϖ(φ(f)), dϖ(φ(f))} is no other than the 𝔰𝔩_{2}-triple of differential operators {h̃′, ẽ′, f̃′} on ℝ^{m} given in (1.3.3).

Proof of Lemma 3.3.1. First we compute dπ_{+}(ϕ(h)) = dπ_{+}(−2E). For this, we use the formula of dϖ̂ on ℝ^{m+1} in Subsection 2.4, and then compute the formula of dπ_{+} on the positive cone C_{+} (or on the coordinate space ℝ^{m}) through the embedding ı : L^{2}(C) → 𝒮′(ℝ^{m+1}), u(ζ) ↦ u(ζ)δ(Q). Since the distribution δ(Q) is homogeneous of degree −2, we note that E_{ζ}δ(Q) = −2δ(Q). Therefore,

(dπ_{+}(−2E)u)δ(Q) = − 2dϖ̂(E)(uδ(Q)) by (2.4.4)
=(2E_{ζ} + m + 3)(uδ(Q)) by (2.4.3)
=(2E_{ζ} + m − 1)u · δ(Q).

Now by identifying L^{2}(C_{+}) with L^{2}(ℝ^{m}, \frac{dx}{|x|}) by (3.1.1), we obtain (3.3.3). The second formula (3.3.4) follows immediately from (2.4.1). We shall show the third formula (3.3.5). In light of ϕ(f) = N_{m+1} (see

(3.3.2)), by (2.4.2), we have

dϖ̂(ϕ(f)) = \sqrt{−1}(\frac{m + 3}{2} \frac{∂}{∂ζ_{m+1}} + E_{ζ} \frac{∂}{∂ζ_{m+1}} + \frac{ζ_{m+1}}{2} □_{ζ}).  (3.3.7)

In order to compute the action dϖ̂(ϕ(f)) along the cone C_{+}, we use the following coordinate on ℝ^{m+1}:

ℝ × ℝ_{+} × S^{m−1} → ℝ^{m+1}, (Q, r, ω) ↦ (rω, \sqrt{r^{2} − Q}).  (3.3.8)

Claim 3.3.3. With the above coordinate, the differential operator dϖ̂(ϕ(f)) on 𝒮′(ℝ^{m+1}) takes the form:

dϖ̂(ϕ(f)) = \sqrt{−1}\sqrt{r^{2} − Q}(\frac{1}{2} \frac{∂^{2}}{∂r^{2}} + \frac{m − 1}{2r} \frac{∂}{∂r} + \frac{Δ_{S^{m−1}}}{2r^{2}} − 2Q\frac{∂^{2}}{∂Q^{2}} − 4\frac{∂}{∂Q}).

Proof of Claim 3.3.3. We start with a new coordinate ℝ^{m+1} = ℝ^{m} ⊕ ℝ by

ℝ_{+} × S^{m−1} × ℝ → ℝ^{m+1}, (R, ω, ζ_{m+1}) ↦ (Rω, ζ_{m+1}).  (3.3.9)

Then, clearly,

E_{ζ} =R\frac{∂}{∂R} + ζ_{m+1} \frac{∂}{∂ζ_{m+1}}
□_{ζ} =Δ_{ℝ^{m}} − \frac{∂^{2}}{∂ζ_{m+1}^{2}} = (\frac{∂^{2}}{∂R^{2}} + \frac{m − 1}{R} \frac{∂}{∂R} + \frac{1}{R^{2}} Δ_{S^{m−1}}) − \frac{∂^{2}}{∂ζ_{m+1}^{2}} .

The coordinate (3.3.8) is obtained by the composition of (3.3.9) and

r = R, Q = R^{2} − ζ_{m+1}^{2}.

In light of

\frac{∂}{∂R} = \frac{∂}{∂r} + 2r \frac{∂}{∂Q}, \frac{∂}{∂ζ_{m+1}} = −2\sqrt{r^{2} − Q}\frac{∂}{∂Q},  (3.3.10)

we get

E_{ζ} =r \frac{∂}{∂r} + 2Q\frac{∂}{∂Q},  (3.3.11)
□_{ζ} =\frac{∂^{2}}{∂r^{2}} + 4Q\frac{∂^{2}}{∂Q^{2}} + 4r \frac{∂^{2}}{∂r∂Q} + 2(m + 1)\frac{∂}{∂Q} + \frac{m − 1}{r} \frac{∂}{∂r} + \frac{Δ_{S^{m−1}}}{r^{2}} . (3.3.12)

Now substituting (3.3.10), (3.3.11) and (3.3.12) into (3.3.7), we get the claim.

Given u ∈ L^{2}(C_{+})_{K}, we extend it to a distribution ũ ≡ ũ(Q, r, ω) ∈ 𝒮′(ℝ^{m+1}) ∩ C^{∞}(ℝ^{m} \ {0}) such that

uδ(Q) = ũδ(Q).

We set

S := 2\sqrt{r^{2} − Q}(Q\frac{∂^{2}}{∂Q^{2}} + 2\frac{∂}{∂Q})(ũδ(Q)).

Then, it follows from (2.4.4) and Claim 3.3.3 that

dϖ̂(ϕ(f))(ũδ(Q))

= \sqrt{−1}(\sqrt{r^{2} − Q}(\frac{1}{2} \frac{∂^{2}}{∂r^{2}} + \frac{m − 1}{2r} \frac{∂}{∂r} + \frac{Δ_{S^{m−1}}}{2r^{2}} )ũ)δ(Q) − S

= \sqrt{−1}(1 + O((\frac{Q}{r^{2}} )))(\frac{r}{2} \frac{∂^{2}}{∂r^{2}} + \frac{m − 1}{2} \frac{∂}{∂r} + \frac{Δ_{S^{m−1}}}{2r} )u · δ(Q) − S

= \sqrt{−1}(\frac{r}{2} \frac{∂^{2}}{∂r^{2}} + \frac{m − 1}{2} \frac{∂}{∂r} + \frac{Δ_{S^{m−1}}}{2r} )u · δ(Q) − S

= \frac{\sqrt{−1}}{2}r(Δ_{ℝ^{m}}u)δ(Q) − S.

At the second last equality, we used the formula Qδ(Q) = 0. In the following claim, we shall show S = 0. Now, the proof of (3.3.5) is complete. Hence, we have shown Lemma 3.3.1.

Claim 3.3.4. For any ũ ∈ C_{0}^{∞}(ℝ^{m+1} \ {0}), we have

(Q\frac{∂^{2}}{∂Q^{2}} + 2\frac{∂}{∂Q})(ũδ(Q)) = 0.

Proof of Claim 3.3.4. By the Leibniz rule, the left-hand side amounts to

\frac{∂^{2}ũ}{∂Q^{2}} Qδ(Q) + 2\frac{∂ũ}{∂Q}(Q\frac{∂}{∂Q}δ(Q) + δ(Q)) + ũ(Q\frac{∂^{2}}{∂Q^{2}} δ(Q) + 2\frac{∂}{∂Q}δ(Q)).

Hence we see that this equals 0 in light of the formulas:

Qδ(Q) = 0, Q\frac{∂}{∂Q}δ(Q) = −δ(Q), Q\frac{∂^{2}}{∂Q^{2}} δ(Q) = −2\frac{∂}{∂Q}δ(Q).

3.4 Central element Z of 𝔨_{ℂ}

We extend the Lie algebra homomorphism ϕ : 𝔰𝔩(2, ℝ) → 𝔰𝔬(m + 1, 2) (see (3.3.2)) to the complex Lie algebra homomorphism ϕ : 𝔰𝔩(2, ℂ) → 𝔰𝔬(m + 1, ℂ). Consider a generator of 𝔰𝔬(2, ℂ) given by

z := \frac{\sqrt{−1}}{2}(e − f) = \frac{\sqrt{−1}}{2} (\begin{matrix}0 & 1 \\ −1 & 0 \end{matrix}) .  (3.4.1)

We set

Z := ϕ(z) ∈ \sqrt{−1}𝔤.  (3.4.2)

In light of (3.3.2) and (2.1.1), we have

Z = \frac{\sqrt{−1}}{2}(\overline{N}_{m+1} − N_{m+1})  (3.4.3)
= \sqrt{−1}(E_{m+1,m+2} − E_{m+2,m+1}).  (3.4.4)

Hence, \sqrt{−1}Z is contained in the center 𝔠(𝔨) of 𝔨 ≃ 𝔰𝔬(m + 1) ⊕ 𝔰𝔬(2). If m > 1, then 𝔠(𝔨) is of one dimension, and \sqrt{−1}Z generates 𝔠(𝔨). By (3.4.1), we have e^{\sqrt{−1}tZ} = I_{m+3} in SO_{0}(m + 1, 2) if and only if t ∈ 2πℤ. Hence, e^{\sqrt{−1}tZ} = 1 in G = SO_{0}(m + 1, 2) if and only if t ∈ 4πℤ (see (2.3.1)). Therefore, we have the following lemma:

Lemma 3.4.1. The Lie algebra homomorphism ϕ : 𝔰𝔩(2, ℝ) → 𝔰𝔬(m + 1, 2) (see (3.3.1)) lifts to an injective Lie group homomorphism SL(2, ℝ) → G.

By the expression (3.4.4) of Z and by the K-isomorphism (3.2.2), we have:

Lemma 3.4.2. dπ_{+}(Z) acts on W_{a} as a scalar multiplication of −(a+\frac{m−1}{2}).

Combining (3.4.1), (3.4.2), and Lemma 3.3.1, we readily get the following lemma:

Lemma 3.4.3. The differential operator dπ_{+}(Z) takes the form:

dπ_{+}(Z) = |x|(\frac{Δ}{4} − 1).

Remark 3.4.4. dπ_{+}(Z) coincides with the operator D in Introduction. In particular, D is a self-adjoint operator on L^{2}(ℝ^{m}, \frac{dx}{|x|}) because \sqrt{−1}Z ∈ 𝔤 and π_{+} is a unitary representation.

Remark 3.4.5 (Weil representation). For the Weil representation of G′, the Lie algebra homomorphism φ : 𝔰𝔩(2, ℝ) → 𝔰𝔭(m, ℝ) (see Remark 3.3.2) sends z to

Z′ := φ(z) = \frac{\sqrt{−1}}{2} (\begin{matrix}0 & I_{m} \\ −I_{m} & 0 \end{matrix}) ∈ \sqrt{−1}𝔤′.

Hence \sqrt{−1}Z′ is a central element of 𝔨′ ≃ 𝔲(m). The differential operator dϖ(Z′) amounts to the Hermite operator (see [16, §6 (d)])

𝒟 = \frac{1}{4}(Δ − |x|^{2}).

See the table in Subsection 1.4 for the differential operators corresponding to e, f and h.

3.5 Proof of Proposition 3.2.1

This subsection gives a proof of Proposition 3.2.1.

1) Let us show j_{a,l}(ϕ) = f_{a,l}ϕ ∈ L^{1}(ℝ^{m}, \frac{dx}{|x|}) ∩ L^{2}(ℝ^{m}, \frac{dx}{|x|}) for any ϕ ∈ ℋ^{l}(ℝ^{m}). By the definition (3.2.4) of f_{a,l}, f_{a,l}(r) is regular at r = 0, and f_{a,l}(r) decays exponentially as r tends to infinity. Therefore, f_{a,l} ∈ L^{1}(ℝ_{+}, r^{m−2}dr) ∩ L^{2}(ℝ_{+}, r^{m−2}dr). Since our measure \frac{dx}{|x|} has the form

\frac{dx}{|x|} = r^{m−2}drdω,  (3.5.1)

with respect to the polar coordinate (3.2.5), we have shown j_{a,l}(ϕ) ∈ L^{1}(ℝ^{m}, \frac{dx}{|x|})∩ L^{2}(ℝ^{m}, \frac{dx}{|x|}).

2) We set H_{a,l} := j_{a,l}(ℋ^{l}(ℝ^{m})). Obviously, H_{a,l} is isomorphic to ℋ^{l}(ℝ^{m}) as R-modules. To see W_{a,l} = H_{a,l}, it is sufficient to show the following inclusion:

H_{a,l} ⊂ W_{a},  (3.5.2)

because W_{a,l} is characterized as the unique subspace of W_{a} such that W_{a,l} ≃ ℋ^{l}(ℝ^{m}) as R-modules.

To see (3.5.2), we recall from (3.2.2) that W_{a} is characterized as the unique subspace of L^{2}(ℝ^{m}, \frac{dx}{|x|}) on which dπ_{+}(Z) acts with eigenvalue −(a+ \frac{m−1}{2}). Thus the inclusive relation (3.5.2) will be proved if we show the following lemma:

Lemma 3.5.1. The operator dπ_{+}(Z) acts on H_{a,l} as a scalar multiplication −(a + \frac{m−1}{2}). In other words, we have

dπ_{+}(Z)(f_{a,l}ϕ) = −(a + \frac{m − 1}{2})f_{a,l}ϕ.  (3.5.3)

Proof of Lemma 3.5.1. Writing the differential operator dπ_{+}(Z) (see Lemma 3.4.3) in terms of the polar coordinate and using the definition of f_{a,l} (see (3.2.4)), we see that the equation (3.5.3) is equivalent to

(\frac{r}{4} \frac{∂^{2}}{∂r^{2}} + \frac{m − 1}{4} \frac{∂}{∂r} + \frac{Δ_{S^{m−1}}}{4r} − r + (a + \frac{m − 1}{2}))(ψ(4r)r^{l}e^{−2r}ϕ(ω)) = 0, (3.5.4)

for ψ(r) := L_{a−l}^{m−2+2l}(r). The equation (3.5.4) amounts to

(4rψ′′(4r) + (m − 1 + 2l − 4r)ψ′(4r) + (a − l)ψ(4r))r^{l}e^{−2r}ϕ(ω) = 0.

This is nothing but Laguerre’s differential equation (8.1.2) with n = a − l, α = m − 2 + 2l. Now the lemma follows.

3.6 One parameter holomorphic semigroup π_{+}(e^{tZ})

It follows from Lemma 3.4.2 that for t ∈ ℂ the operator

π_{+}(e^{tZ}) := exp dπ_{+}(tZ) = ∑_{j=0}^{∞} \frac{1}{j!}dπ_{+}(tZ)^{j}  (3.6.1)

acts on W_{a,l} as a scalar multiplication of e^{−(a+ \frac{m−1}{2} )t} for any 0 ≤ l ≤ a, of which the absolute value does not exceed 1 if Re t ≥ 0. In light of the direct sum decomposition (3.2.1), if Re t ≥ 0 then the linear map π_{+}(e^{tZ}) : L^{2}(ℝ^{m}, \frac{dx}{|x|})_{K} → L^{2}(ℝ^{m}, \frac{dx}{|x|})_{K} extends to a continuous operator (we use the same notation π_{+}(e^{tZ})) on L^{2}(ℝ^{m}, \frac{dx}{|x|}). Furthermore, it is a contraction operator if Re t > 0. We summarize some of basic properties of π_{+}(e^{tZ}):

Proposition 3.6.1. 1) The map

\{t ∈ ℂ : Re t ≥ 0\} × L^{2}(ℝ^{m}, \frac{dx}{|x|}) → L^{2}(ℝ^{m}, \frac{dx}{|x|}), (t, f) ↦ π_{+}(e^{tZ})f

(3.6.2)

is continuous.

2) For a fixed t such that Re t ≥ 0, π_{+}(e^{tZ}) is characterized as the continuous operator from L^{2}(ℝ^{m}, \frac{dx}{|x|}) → L^{2}(ℝ^{m}, \frac{dx}{|x|}) satisfying

π_{+}(e^{tZ})u = e^{−(a+ \frac{m−1}{2} )t}u,  (3.6.3)

for any u ∈ W_{a,l} = {f_{a,l}ϕ : ϕ ∈ ℋ^{l}(ℝ^{m})} (see Proposition 3.2.1) and for any l, a ∈ ℕ such that 0 ≤ l ≤ a.

3) The operator norm ∥π_{+}(e^{tZ})∥ of π_{+}(e^{tZ}) is e^{− \frac{m−1}{2} Re t}.

4) If Re t > 0, π_{+}(e^{tZ}) is a Hilbert–Schmidt operator.

5) If t ∈ \sqrt{−1}ℝ, then π_{+}(e^{tZ}) is a unitary operator.

Remark 3.6.2. We define a subset Γ_{+} := \{tZ : t > 0\} in \sqrt{−1}𝔤. Proposition 3.6.1 indicates how the unitary representation π_{+} of G extends to a holomorphic semigroup on the complex domain G · exp Γ_{+} · G of G_{ℂ}. Our results may be regarded as a part of the Gelfand–Gindikin program, which tries to understand unitary representations of a real semisimple Lie group by means of holomorphic objects on an open subset of G_{ℂ} \ G, where G_{ℂ} is a complexification of G (see [10, 27, 29]).

Remark 3.6.3 (Weil representation). In the case of the Weil representation (ϖ, L^{2}(ℝ^{m})) of G′ = Sp(m, ℝ), the functions f′_{a,l} (see Remark 3.2.4) play the same role as f_{a,l} because of the following facts:

1) {f′_{a,l} : 0 ≤ l ≤ a} spans a complete orthogonal basis of L^{2}(ℝ_{+}, r^{m−1}dr) (cf. Remark 3.2.2).

2) For any ϕ ∈ ℋ^{l}(ℝ^{m}), f′_{a,l}(r)ϕ(ω) (0 ≤ l ≤ a) are eigenfunctions of dϖ(Z′) with negative eigenvalues −(\frac{m}{4} + \frac{l}{2} + \frac{a}{2}).

Owing to these facts, we obtain a holomorphic semigroup of contraction operators ϖ(e^{tZ′}) := exp t(dϖ(Z′)) (Re t > 0) on L^{2}(ℝ^{m}). Since dϖ(Z′) coincides with the Hermite operator 𝒟 (see Remark 3.4.5), this holomorphic semigroup is nothing but the Hermite semigroup (see [16, §5]).

4 Radial part of the semigroup

This section gives an explicit integral formula for the ‘radial part’ of the holomorphic semigroup π_{+}(e^{tZ}) in the ‘Schrödinger model’ L^{2}(ℝ^{m}, |x|^{−1}dx). The main result of Section 4 is Theorem 4.1.1. As its applications, we see that the semigroup law π_{+}(e^{(t_{1}+t_{2})Z}) = π_{+}(e^{t_{1}Z})∘π(e^{t_{2}Z}) gives a simple and representation theoretic proof of the classical Weber’s second exponential integral formula on Bessel functions (Corollary 4.5.1), and that taking the boundary value lim_{s↓0} π_{+}(e^{sZ}) = id provides an example of a Dirac sequence

(Corollary 4.6.1).

Theorem 4.1.1 will play a key role in Section 5, where we complete the proof of the main theorem of this article, namely, Theorem 5.1.1 that gives an integral formula of the holomorphic semigroup π_{+}(e^{tZ}) on L^{2}(ℝ^{m}, |x|^{−1}dx).

4.1 Result of the section

For a complex parameter t ∈ ℂ with Re t > 0, we have defined a contraction operator π_{+}(e^{tZ}) : L^{2}(ℝ^{m}, \frac{dx}{|x|}) → L^{2}(ℝ^{m}, \frac{dx}{|x|}) in Proposition 3.6.1.

We recall that Z is a central element in 𝔨_{ℂ} (see Subsection 3.4) and that R is a subgroup of K. Therefore, π_{+}(e^{tZ}) intertwines with the R-action. On the other hand, the natural action of R ≃ SO(m) gives a direct sum decomposition of the Hilbert space:

L^{2}(ℝ^{m}, \frac{dx}{|x|}) ≃ \begin{matrix}∞ \\ ∑ & ⊕ \\ l=0 \end{matrix} L^{2}(ℝ_{+}, r^{m−2}dr) ⊗ ℋ^{l}(ℝ^{m}).  (4.1.1)

Hence, by Schur’s lemma, there exists a family of continuous operators parametrized by l ∈ ℕ:

π_{+,l}(e^{tZ}) : L^{2}(ℝ_{+}, r^{m−2}dr) → L^{2}(ℝ_{+}, r^{m−2}dr)  (4.1.2)

such that π_{+}(e^{tZ}) is diagonalized according to the direct sum decomposition (4.1.1) as follows:

π_{+}(e^{tZ}) = \begin{matrix}∞ \\ ∑ & ⊕ \\ l=0 \end{matrix} π_{+,l}(e^{tZ}) ⊗ id .  (4.1.3)

The goal of this section is to give an explicit integral formula of π_{+,l}(e^{tZ}) on L^{2}(ℝ_{+}, r^{m−2}dr) for l ∈ ℕ. We note that π_{+,l}(e^{tZ}) is a unitary operator if Re t = 0 because so is π_{+}(e^{tZ}). Likewise, π_{+,l}(e^{tZ}) is a Hilbert–Schmidt operator if Re t > 0 because so is π_{+}(e^{tZ}).

We now introduce the following subset of ℂ:

Ω := \{t ∈ ℂ : Re t ≥ 0\} \ 2π\sqrt{−1}ℤ,  (4.1.4)

and define a family of analytic functions K_{l}^{+}(r, r′; t) on ℝ_{+} × ℝ_{+} × Ω by the formula: for l = 0, 1, 2, …,

K_{l}^{+}(r, r′; t) := \frac{2e^{−2(r+r′) coth \frac{t}{2}}}{sinh \frac{t}{2}} (rr′)^{− \frac{m−2}{2}} I_{m−2+2l}(\frac{4\sqrt{rr′}}{sinh \frac{t}{2}} )

= \frac{2^{m−1+2l}e^{−2(r+r′) coth \frac{t}{2}} (rr′)^{l}}{(sinh \frac{t}{2})^{m−1+2l}} Ĩ_{m−2+2l}(\frac{4\sqrt{rr′}}{sinh \frac{t}{2}} ).  (4.1.5)

Here, Ĩ_{ν}(z) := (\frac{z}{2})^{−ν}I_{ν}(z), and I_{ν}(z) denotes the I-Bessel function (see Subsection 8.5). We note that the denominator sinh \frac{t}{2} is nonzero for t ∈ Ω.

We are ready to state the integral formula of π_{+,l}(e^{tZ}) for t ∈ Ω:

Theorem 4.1.1 (Radial part of the semigroup). 1) For Re t > 0, the Hilbert–Schmidt operator π_{+,l}(e^{tZ}) on L^{2}(ℝ_{+}, r^{m−2}dr) is given by the following integral transform:

(π_{+,l}(e^{tZ})f)(r) = ∫_{0}^{∞} K_{l}^{+}(r, r′; t)f(r′)r′^{m−2}dr′.  (4.1.6)

The right-hand side converges absolutely for f ∈ L^{2}(ℝ_{+}, r^{m−2}dr).

2) If t ∈ \sqrt{−1}ℝ but t ∉ 2π\sqrt{−1}ℤ, then the integral formula (4.1.6) for the unitary operator π_{+,l}(e^{tZ}) holds in the sense of L^{2}-convergence. Furthermore, the right-hand side converges absolutely if f is a finite linear combination of f_{a,l} (a = l, l + 1, ⋯).

Remark 4.1.2. Let us compare Theorem 4.1.1 with the corresponding result for the Weil representation ϖ of G′ = Sp(m, ℝ) realized as the Schrödinger model L^{2}(ℝ^{m}). According to the direct sum decomposition of the Hilbert space:

L^{2}(ℝ^{m}) ≃ \begin{matrix}∞ \\ ∑ & ⊕ \\ l=0 \end{matrix} L^{2}(ℝ_{+}, r^{m−1}dr) ⊗ ℋ^{l}(ℝ^{m}),

there exists a family of continuous operators ϖ_{l}(e^{tZ′}) (Re t > 0) such that the holomorphic semigroup ϖ(e^{tZ′}) has the following decomposition:

ϖ(e^{tZ′}) = \begin{matrix}∞ \\ ∑ & ⊕ \\ l=0 \end{matrix} ϖ_{l}(e^{tZ′}) ⊗ id .

Then, by an analogous computation to Theorem 4.1.1, we find the kernel function of the semigroup ϖ_{l}(e^{tZ′}) is given by

𝒦_{l}(r, r′; t) := \frac{e^{− \frac{1}{2} (r^{2}+r′^{2}) coth \frac{t}{2}}}{sinh \frac{t}{2}} (rr′)^{− \frac{m−2}{2}} I_{\frac{m−2+2l}{2}} (\frac{rr′}{sinh \frac{t}{2}} ).

The relation between the kernel function 𝒦 (Mehler kernel) of ϖ(e^{tZ′}) and 𝒦_{l}(r, r′; t) will be discussed in Remark 5.7.2.

This section is organized as follows. In Subsection 4.3, we give a proof of Theorem 4.1.1 for the case Re t > 0, which is based on a computation of the kernel function by means of the infinite sum of the eigenfunctions. In Subsection 4.4, by taking the analytic continuation, the case Re t = 0 is proved. Applications of Theorem 4.1.1 to special function theory are discussed in Subsections 4.5 and 4.6.

4.2 Upper estimate of the kernel function

In this subsection, we shall give an upper estimate of the kernel function K_{l}^{+}(r, r′; t).

For t = x + \sqrt{−1}y, we set

α(t) := \frac{sinh x}{cosh x − cos y} ,  (4.2.1)
β(t) := \begin{matrix}cos & \frac{y}{2} \\ cosh & \frac{x}{2} \end{matrix} .  (4.2.2)

Then, an elementary computation shows

Re coth \frac{t}{2} = α(t),  (4.2.3)
Re \frac{1}{sinh \frac{t}{2}} = α(t)β(t).  (4.2.4)

For t ∈ Ω (see (4.1.4) for definition), we have cosh x − cos y > 0, and then,

α(t) ≥ 0 and |β(t)| < 1.  (4.2.5)

If Re t > 0, then

α(t) > 0.  (4.2.6)

For later purposes, we prepare:

Lemma 4.2.1. If y ∈ ℝ satisfies |y| ≤ 4\sqrt{rr′}, then, for t ∈ Ω, we have the following estimate for some constant C:

|e^{−2(r+r′) coth \frac{t}{2}} Ĩ_{ν}(\frac{y}{sinh \frac{t}{2}} )| ≤ Ce^{−2α(t)(1−|β(t)|)(r+r′)}.  (4.2.7)

Proof. Using the upper estimate of the I-Bessel function (see Lemma 8.5.1),

|Ĩ_{ν}(\frac{y}{sinh \frac{t}{2}} )| ≤ Ce^{|y|| Re _{\frac{1}{sinh \frac{t}{2}}} |},

we have

|e^{−2(r+r′) coth \frac{t}{2}} Ĩ_{ν}(\frac{y}{sinh \frac{t}{2}} )| ≤Ce^{−2(r+r′) Re coth \frac{t}{2} +|y| Re|_{\frac{1}{sinh \frac{t}{2}}} |}
≤Ce^{−2(r+r′)α(t)+4\sqrt{rr′}α(t)|β(t)|}
≤Ce^{−2α(t)(1−|β(t)|)(r+r′)}.

Here, the last inequality follows from

r + r′ − 2|β(t)|\sqrt{rr′} ≥ (1 − |β(t)|)(r + r′)  (4.2.8)

for t ∈ Ω. Thus Lemma is proved.

Now we state a main result of this subsection:

Lemma 4.2.2. Let l ∈ ℕ and m ≥ 2.

1) There exists a constant C > 0 such that

|K_{l}^{+}(r, r′; t)| ≤ \frac{C(rr′)^{l}e^{−2α(t)(1−|β(t)|)(r+r′)}}{| sinh \frac{t}{2}|^{m−1+2l}} .  (4.2.9)

for any r, r′ ∈ ℝ_{+} and t ∈ Ω.

2) If Re t > 0, then K_{l}^{+}(·, ·; t) ∈ L^{2}((ℝ_{+})^{2}, (rr′)^{m−2}drdr′).

3) If Re t > 0, then for a fixed r > 0, we have K_{l}^{+}(r, ·; t) ∈ L^{2}(ℝ_{+}, r^{′m−2}dr′).

Proof. 1) By the definition of K_{l}^{+} (see (4.1.5)), we have

|K_{l}^{+}(r, r′; t)| =\frac{2^{m−1+2l}(rr′)^{l}}{| sinh \frac{t}{2}|^{m−1+2l}} |e^{−2(r+r′) coth \frac{t}{2}} Ĩ_{m−2+2l}(\frac{4\sqrt{rr′}}{sinh \frac{t}{2}} )|.

Now (4.2.9) follows from Lemma 4.2.1 by substituting ν = m − 2 + 2l and y = 4\sqrt{rr′}.

Since α(t) > 0 for Re t > 0, the statements 2) and 3) hold by 1).

4.3 Proof of Theorem 4.1.1 (Case Re t > 0)

We recall from Remark 3.2.2 that

f_{a,l}(r) = L_{a−l}^{m−2+2l}(4r)r^{l}e^{−2r} (a = l, l + 1, …)

forms a complete orthogonal basis of L^{2}(ℝ_{+}, r^{m−2}dr). Further, by the orthogonal relation of the Laguerre polynomials L_{m}^{α}(x) (see (8.1.3)), we have the normalization of {f_{a,l}} as follows:

Lemma 4.3.1. For integers a, b ≥ l, we have

∫_{0}^{∞} f_{a,l}(r)f_{b,l}(r)r^{m−2}dr = \begin{cases}0 & if a ≠ b, \\ \frac{Γ(m−1+a+l)}{4^{m−1+2l}Γ(a−l+1)} & if a = b. \end{cases}  (4.3.1)

We rewrite (3.2.1) by using Proposition 3.2.1 as follows:

L^{2}(ℝ^{m}, \frac{dx}{|x|})_{K} = ⨁_{a=0}^{∞}(⨁_{l=0}^{a} W_{a,l})

= ⨁_{l=0}^{∞}(⨁_{a=l}^{∞} W_{a,l})

= ⨁_{l=0}^{∞}((⨁_{a=l}^{∞} ℂf_{a,l}) ⊗ ℋ^{l}(ℝ^{m})).  (4.3.2)

It follows from Proposition 3.6.1 (2) and the definition (4.1.3) of π_{+,l}(e^{tZ}) that

π_{+,l}(e^{tZ})f_{a,l} = e^{−(a+ \frac{m−1}{2} )t}f_{a,l} (a = l, l + 1, ⋯).

Therefore, the kernel function K_{l}^{+}(r, r′; t) of π_{+,l}(e^{tZ}) can be written as the infinite sum:

K_{l}^{+}(r, r′; t) = ∑_{a=l}^{∞} \frac{e^{−(a+ \frac{m−1}{2} )t}f_{a,l}(r)\overline{f_{a,l}(r′)}}{||f_{a,l}||_{L^{2}(ℝ_{+},r^{m−2}dr)}^{2}}.  (4.3.3)

Since π_{+,l}(e^{tZ}) is a Hilbert–Schmidt operator if Re t > 0, the right-hand side converges in L^{2}((ℝ_{+})^{2}, (rr′)^{m−2}drdr′), and therefore converges for almost all (r, r′) ∈ (ℝ_{+})^{2}.

Let us compute the infinite sum (4.3.3). For this, we set

κ(r, r′; t) := ∑_{a=l}^{∞} \frac{Γ(a − l + 1)}{Γ(m − 1 + a + l)}L_{a−l}^{m−2+2l}(4r)L_{a−l}^{m−2+2l}(4r′)e^{−(a−l)t}.

Then, it follows from (4.3.1) that we have

K_{l}^{+}(r, r′; t) = 4^{m−1+2l}(rr′)^{l}e^{−2(r+r′)}e^{−(l+ \frac{m−1}{2} )t}κ(r, r′; t).  (4.3.4)

Now, we apply the Hille–Hardy formula (see (8.1.4)) with α = m−2+2l, n = a − l, x = 4r, y = 4r′, and w = e^{−t}. We note that |w| = e^{− Re t} < 1 by the assumption Re t > 0. Then we have

κ(r, r′; t) = \frac{e^{\frac{(4r+4r′)e^{−t}}{1−e^{−t}}} (−16rr′e^{−t})^{− \frac{m−2}{2} +l}}{1 − e^{−t}} J_{m−2+2l}(\frac{2\sqrt{−16rr′e^{−t}}}{1 − e^{−t}} )

= \frac{(rr′)^{− \frac{m−1}{2} +l}e^{−2(r+r′) \frac{2e^{−t}}{1−e^{−t}}} e^{( \frac{m−2}{2} +l)t}}{4^{m−2+2l}(1 − e^{−t})} I_{m−2+2l}(\frac{4\sqrt{rr′}}{sinh \frac{t}{2}} ).

Hence, the formula (4.1.5) is proved.

Therefore, the right-hand side of (4.1.6) converges absolutely by the Cauchy–Schwarz inequality because K_{l}^{+}(r, r′; t) ∈ L^{2}(ℝ_{+}, r′^{m−2}dr′) for any r > 0 and Re t > 0 (see Lemma 4.2.2 (3)).

Remark 4.3.2. The special functions and related formulas that arise in the analysis of the radial part have a scheme of generalization from SL(2, ℝ) to G = SO_{0}(m + 1, 2) and G′ = Sp(m, ℝ). This scheme is illustrated as follows:

SL(2, ℝ)  ⇒  G or G′
Segal–Shale–Weil representation ⇒ minimal representation
Hermite polynomials  ⇒ Laguerre polynomials
Mehler’s formula  ⇒ Hille–Hardy formula

See [17, p 116, Exercise 5 (d)] for the SL(2, ℝ) case. Owing to the reduction formula of Laguerre polynomials to Hermite polynomials (see (8.2.1) and (8.2.2)), the radial part f′_{a,l}(r) (see Remark 3.2.4) for the Weil representation

of G′ = Sp(m, ℝ) collapses to a constant multiple of H_{2a−l}(r)e^{− \frac{r^{2}}{2}} (l = 0, 1) if m = 1.

Remark 4.3.3. In [24, Chapter 2], W. Myller-Lebedeff proved the following integral formula:

∫_{0}^{∞} K_{l}(r, r′; t)f_{a,l}(r′)r^{′m−2}dr′ = e^{−(a+ \frac{m−1}{2} )t}f_{a,l}(r) for a ≥ l ≥ 0. (4.3.5)

In view of (4.1.3) and Proposition 3.6.1 (2), the formula (4.1.6) in Theorem 4.1.1 implies (4.3.5) and vice versa. The proof of [24] is completely different from ours. Here is a brief sketch: For the partial differential operator

L := \frac{∂^{2}}{∂x^{2}} + \frac{α + 1}{x} \frac{∂}{∂x} − \frac{1}{x} \frac{∂}{∂t} α > 0,

one has the following identity using Green’s formula,

∫∫_{D}(vLu − uL^{∗}v)dtdx = ∫_{∂D}(v \frac{∂u}{∂x} − u\frac{∂v}{∂x} + \frac{α + 1}{x}uv)dt + \frac{1}{x}uvdx, (4.3.6)

for a domain D ⊂ ℝ^{2}. Here, L^{∗} denotes the (formal) adjoint of L. Now, we take u(x, t) := t^{n}L_{n}^{α}(\frac{x}{t}), v(x, t) := (\frac{x}{ξ})^{\frac{α}{2}} \frac{x}{τ−t}e^{− \frac{x+ξ}{τ−t}} I_{α}(\frac{2\sqrt{xξ}}{τ−t}), n ∈ ℕ, τ, ξ ∈ ℝ as solutions to Lu = 0, L^{∗}v = 0 respectively, and the domain D as a rectangular domain D := {(x, t) ∈ ℝ^{2} : 0 < x < ∞, t_{1} < t < t_{2}} for some t_{1}, t_{2} ∈ ℝ. Then by the decay properties of u and v, the integrands in the right-hand side of (4.3.6) vanish on x = 0 and x = ∞. Since the integral of the left-hand side of (4.3.6) vanishes, the integral ∫_{0}^{∞} \frac{1}{x}u(x, t)v(x, t)dx becomes constant with respect to t. By taking the limit t → τ, we have lim_{t→τ} ∫_{0}^{∞} \frac{1}{x}u(x, t)v(x, t)dx = u(ξ, τ) since v(x, τ) is proved to be a Dirac delta function. Hence we obtain

∫_{0}^{∞} \frac{1}{x}u(x, t)v(x, t)dx = u(ξ, τ),

which coincides with (4.3.5) by a suitable change of variables.

4.4 Proof of Theorem 4.1.1 (Case Re t = 0)

Suppose t ∈ \sqrt{−1}ℝ. Then, π_{+,l}(e^{tZ}) is a unitary operator on L^{2}(ℝ_{+}, r^{m−2}dr). Suppose furthermore t ∉ 2π\sqrt{−1}ℤ. For ε > 0 and f ∈ L^{2}(ℝ_{+}, r^{m−2}dr), we have from Theorem 4.1.1 (1)

π_{+,l}(e^{(ε+t)Z})f = ∫_{0}^{∞} K_{l}^{+}(r, r′; ε + t)f(r′)r′^{m−2}dr′.

By Proposition 3.6.1 (1), the left-hand side converges to π_{+,l}(e^{tZ})f in L^{2}(ℝ_{+}, r^{m−2}dr) as ε tends to 0. For the right-hand side, we have:

Claim 4.4.1. For t ∈ \sqrt{−1}ℝ \ 2π\sqrt{−1}ℤ,

lim_{ε↓0} ∫_{0}^{∞} K_{l}^{+}(r, r′; ε + t)f_{a,l}(r′)r′^{m−2}dr′ = ∫_{0}^{∞} K_{l}^{+}(r, r′; t)f_{a,l}(r′)r′^{m−2}dr′

and the right-hand side converges absolutely.

Proof. If ε ∈ ℝ and t ∈ \sqrt{−1}ℝ then

|sinh \frac{ε + t}{2}|^{2} = |sinh \frac{ε}{2}|^{2} + |sinh \frac{t}{2}|^{2} ≥ |sinh \frac{t}{2}|^{2}.

Therefore, it follows from (4.2.9) that

|K_{l}^{+}(r, r′; ε + t)| ≤ \frac{C(rr′)^{l}}{| sinh \frac{t}{2}|^{m−1+2l}}

if ε > 0 and t ∈ \sqrt{−1}ℝ \ 2π\sqrt{−1}ℤ, because ε + t ∈ Ω implies α(ε + t) ≥ 0 and |β(ε + t)| < 1. Therefore, we have

|K_{l}^{+}(r, r′; ε + t)f_{a,l}(r′)r′^{m−2}| ≤ \frac{Cr^{l}r′^{l+m−2}e^{−2r′}|L_{a−l}^{m−2+4l}(4r′)|}{| sinh \frac{t}{2}|^{m−1+2l}}.

By the Lebesgue convergence theorem, we have proved Claim.

Since linear combinations of f_{a,l} span a dense subspace of L^{2}(ℝ_{+}, r^{m−2}dr),

2) is proved.

4.5 Weber’s second exponential integral formula

From the semigroup law:

π_{+}(e^{(t_{1}+t_{2})Z}) = π_{+}(e^{t_{1}Z}) ∘ π_{+}(e^{t_{2}Z}) (Re t_{1}, Re t_{2} > 0),  (4.5.1)

we get a representation theoretic proof of classical Weber’s second exponential integral for Bessel functions (see [33, §13.31 (1)]):

Corollary 4.5.1. (Weber’s second exponential integral) Let ν be a positive integer, and ρ, α, β > 0. We have the following integral formula

∫_{0}^{∞} e^{−ρx^{2}}J_{ν}(αx)J_{ν}(βx)xdx = \frac{1}{2ρ} exp(−\frac{α^{2} + β^{2}}{4ρ} )I_{ν}(\frac{αβ}{2ρ} ).  (4.5.2)

Proof. It follows from the semigroup law (4.5.1) that the integral kernels for π_{+}(e^{(t_{1}+t_{2})Z}) and π_{+}(e^{t_{1}Z}) ∘ π_{+}(e^{t_{2}Z}) must coincide. Then, from Theorem 4.1.1, we have

∫_{0}^{∞} K_{l}^{+}(r, s; t_{1})K_{l}^{+}(s, r′; t_{2})s^{m−2}ds = K_{l}^{+}(r, r′; t_{1} + t_{2}).  (4.5.3)

In view of (4.1.5), the formula (4.5.2) is obtained by (4.5.3) by the change of variables,

x = \sqrt{s}, α = \frac{4e^{\frac{π\sqrt{−1}}{2}} \sqrt{r}}{sinh \frac{t_{1}}{2}} , β = \frac{4e^{\frac{π\sqrt{−1}}{2}} \sqrt{r′}}{sinh \frac{t_{2}}{2}} , ν = m − 2 + 2l,

ρ = 2(coth \frac{t_{1}}{2} + coth \frac{t_{2}}{2} ).

4.6 Dirac sequence operators

We shall state another corollary to Theorem 4.1.1. Let ν be a positive integer, x, y ∈ ℝ, s ∈ ℂ such that Re s > 0. For a function f on ℝ, let 𝒯_{s} be an operator defined by

𝒯_{s} : f(x) ↦ ∫_{0}^{∞} A(x, y; s)f(y)dy,

with the kernel function

A(x, y; s) := (xy)^{\frac{1}{2}} \frac{e^{− \frac{1}{2} (x^{2}+y^{2}) coth s}}{sinh s} I_{ν}(\frac{xy}{sinh s}).  (4.6.1)

Then we have the following corollary.

Corollary 4.6.1. 1) The operators {𝒯_{s} : Re s > 0} form a semigroup of contraction operators on L^{2}(ℝ_{+}, dx).

2) (Dirac sequence) lim_{s→0} ||𝒯_{s}h − h||_{L^{2}(ℝ_{+},dx)} = 0 holds for all h ∈ L^{2}(ℝ_{+}, dx).

Remark 4.6.2. For sufficiently small s, the semigroup {𝒯_{s} : Re s > 0} behaves like the Hermite semigroup (see [16]) whose kernel is given by the following Gaussian (cf. (1.2.2)):

κ(x, y; s) = \frac{1}{\sqrt{2π sinh s}}e^{− \frac{x^{2}}{2} coth s+ \frac{xy}{sinh s} − \frac{y^{2}}{2} coth s}

because I_{ν}(z) ∼ \frac{1}{\sqrt{2πz}} e^{z} for sufficiently large z (see [33, §7.23] for the asymptotic behavior of I_{ν}(z)). Note that it is stated in [16, §5.5] that the Hermite semigroup forms a ‘Dirac sequence’.

Proof. Since we assume ν is a positive integer, ν = m − 2 + 2l for some m > 3 and l ∈ ℤ. We change the variables r = \frac{x^{2}}{4} , r′ = \frac{y^{2}}{4} , t = 2s and define a unitary map

Φ : L^{2}(ℝ_{+}, r^{m−2}dr) → L^{2}(ℝ_{+}, dx), (Φf)(x) := (\frac{x}{2})^{\frac{2m−3}{2}} f(\frac{x^{2}}{4} ). (4.6.2)

Comparing (4.6.1) with (4.1.5), we have

Φ^{−1} ∘ 𝒯_{s} ∘ Φf = π_{+,l}(e^{2sZ})f  (4.6.3)

Thus by Theorem 4.1.1 (1), 1) is proved.

2) We take a limit s ↓ 0 of (4.6.3). By Proposition 3.6.1 (1), the righthand side equals f. Hence by putting h := Φf, we have lim_{s↓0} 𝒯_{s}h = h.

5 Integral formula for the semigroup

In this section, we shall give an explicit integral formula for the holomorphic semigroup exp(tdπ_{+}(Z)) = π_{+}(e^{tZ}) on L^{2}(ℝ^{m}, \frac{dx}{|x|}) for Re t > 0, or more precisely, for t ∈ Ω = \{t ∈ ℂ : Re t ≥ 0\} \ 2π\sqrt{−1}ℤ (see (4.1.4)). The main result of this section is Theorem 5.1.1. In particular, we give a proof of Theorem A in Introduction.

5.1 Result of the section

Let ⟨x, x′⟩ be the standard inner product of ℝ^{m}, |x| := \sqrt{⟨x, x⟩} be the norm. We recall the notation from Subsection 1.1:

ψ(x, x′) := 2\sqrt{2(|x||x′| + ⟨x, x′⟩)} = 4|x|^{\frac{1}{2}} |x′|^{\frac{1}{2}} cos \frac{θ}{2},  (5.1.1)

where θ ≡ θ(x, x′) is the angle between x and x′ in ℝ^{m}. Let us define a kernel function K^{+}(x, x′; t) on ℝ^{m} × ℝ^{m} × Ω by the following formula as in Introduction:

K^{+}(x, x′; t) := \frac{2^{\frac{m−1}{2}} e^{−2(|x|+|x′|) coth \frac{t}{2}}}{π^{\frac{m−1}{2}} sinh^{\frac{m+1}{2}} \frac{t}{2}} ψ(x, x′)^{− \frac{m−3}{2}} I_{\frac{m−3}{2}} (\frac{ψ(x, x′)}{sinh \frac{t}{2}})

= \frac{2e^{−2(|x|+|x′|) coth \frac{t}{2}}}{π^{\frac{m−1}{2}} sinh^{m−1} \frac{t}{2}} Ĩ_{\frac{m−3}{2}} (\frac{ψ(x, x′)}{sinh \frac{t}{2}}),  (5.1.2)

where I_{ν}(z) is the modified Bessel function of the first kind and Ĩ_{ν}(z) := (\frac{z}{2})^{−ν}I_{ν}(z) is an entire function (see Subsection 8.5). We note that sinh \frac{t}{2} in the denominator is non-zero because t ∉ 2π\sqrt{−1}ℤ. Therefore, K^{+}(x, x′; t) is a continuous function on ℝ^{m} × ℝ^{m} × Ω.

We recall from Proposition 3.6.1 (3) that π_{+}(e^{tZ}) is a contraction operator with operator norm ∥π_{+}(e^{tZ})∥ = e^{− \frac{m−1}{2} Re t}. Here is an integral formula of the holomorphic semigroup π_{+}(e^{tZ}):

Theorem 5.1.1 (Integral formula for the semigroup). 1) For Re t > 0, π_{+}(e^{tZ}) is a Hilbert–Schmidt operator on L^{2}(ℝ^{m}, \frac{dx}{|x|}), and is given by the following integral transform:

(π_{+}(e^{tZ})u)(x) = ∫_{ℝ^{m}} K^{+}(x, x′; t)u(x′)\frac{dx}{|x|} for u ∈ L^{2}(ℝ^{m}, \frac{dx}{|x|}). (5.1.3)

Here, the right-hand side converges absolutely.

2) For t ∈ \sqrt{−1}ℝ, π_{+}(e^{tZ}) is a unitary operator on L^{2}(ℝ^{m}, \frac{dx}{|x|}). If t ∈ \sqrt{−1}ℝ but t ∉ 2π\sqrt{−1}ℤ, then the right-hand side of (5.1.3) converges absolutely for any u ∈ L^{1}(ℝ^{m}, \frac{dx}{|x|}) ∩ L^{2}(ℝ^{m}, \frac{dx}{|x|}), in particular, for any K-finite vectors in L^{2}(ℝ^{m}, \frac{dx}{|x|}). Since K-finite vectors span a dense subspace of L^{2}(ℝ^{m}, \frac{dx}{|x|}), the integral formula (5.1.3) holds in the sense of L^{2}-convergence.

Remark 5.1.2 (Realization on the cone C_{+}). Via the isomorphism L^{2}(C_{+}) ≃ L^{2}(ℝ^{m}, \frac{dx}{|x|}) (see Subsection 3.1), the above formula for L^{2}(ℝ^{m}, \frac{dx}{|x|}) can be readily transferred to the formula of the holomorphic extension π_{+}(e^{tZ}) on L^{2}(C_{+}). For this, we define a continuous function K̃^{+}(ζ, ζ′; t) on C_{+} × C_{+} × Ω by the following formula:

K̃^{+}(ζ, ζ′; t) := \frac{2e^{−\sqrt{2}(|ζ|+|ζ′|) coth \frac{t}{2}}}{π^{\frac{m−1}{2}} sinh^{m−1} \frac{t}{2}} Ĩ_{\frac{m−3}{2}} (\frac{2\sqrt{2⟨ζ, ζ′⟩}}{sinh \frac{t}{2}}),  (5.1.4)

where |ζ| := \sqrt{⟨ζ, ζ⟩} = (ζ_{1}^{2} + ⋯ + ζ_{m+1}^{2})^{\frac{1}{2}} . Then,

(π_{+}(e^{tZ})u)(ζ) = ∫_{C_{+}} K̃^{+}(ζ, ζ′; t)u(ζ′)dμ(ζ′) for u ∈ L^{2}(C_{+}). (5.1.5)

Remark 5.1.3 (Weil representation). For the Weil representation ϖ of G′, the corresponding semigroup of contraction operators is the Hermite semigroup {ϖ(e^{tZ′}) : Re t > 0} (see Remark 3.6.3), whose kernel function is given by the Mehler kernel 𝒦(x, x′; t) (see Fact C in Subsection 1.2; see also [16, §5]).

The rest of this section is devoted to the proof of Theorem 5.1.1. Let us mention briefly a naive idea of the proof. We observe that the action of K(= SO(m + 1) × SO(2) ) on L^{2}(ℝ^{m}, \frac{dx}{|x|}) is hard to describe because K does not act on ℝ^{m}. However, the action of its subgroup R = SO(m) has a simple feature, that is, we have the following direct sum decomposition:

L^{2}(ℝ^{m}, \frac{dx}{|x|}) ≃ \begin{matrix}∞ \\ ∑ & ⊕ \\ l=0 \end{matrix} L^{2}(ℝ_{+}, r^{m−2}dr) ⊗ ℋ^{l}(ℝ^{m}).  (5.1.6)

We have already proved in Theorem 4.1.1 that K_{l}^{+}(r, r′; t) is the kernel of π_{+,l}(e^{tZ}) which is the restriction of π_{+}(e^{tZ}) in each l-component of the righthand side of (5.1.6). Theorem 5.1.1 will be proved if we decompose K^{+} into K_{l}^{+}. This will be carried out in Lemma 5.6.1. An expansion formula of K^{+} by K_{l}^{+} is not used in the proof of Theorem 5.1.1, but might be of interest of its own. We shall give it in Subsection 5.7.

5.2 Upper estimates of the kernel function

In this subsection, we give an upper estimate of the kernel function K^{+}(x, x′; t). That parallels Lemma 4.2.2.

Lemma 5.2.1. Let m ≥ 2. 1) There exists a constant C > 0 such that

|K^{+}(rω, r′ω′; t)| ≤ \frac{C}{| sinh \frac{t}{2}|^{m−1}} e^{−2α(t)(1−|β(t)|)(r+r′)},  (5.2.1)

for any r, r′ ∈ ℝ_{+}, ω, ω′ ∈ S^{m−1}, and t ∈ Ω. Here, α(t), β(t) are defined in (4.2.1) and (4.2.2), respectively.

2) If Re t > 0, then

∫_{ℝ^{m}} ∫_{ℝ^{m}} |K^{+}(x, x′; t)|^{2} \frac{dx}{|x|} \frac{dx′}{|x′|} < ∞.

3) If Re t > 0, then for a fixed x ∈ ℝ^{m}, we have K^{+}(x, ·; t) ∈ L^{2}(ℝ^{m}, \frac{dx′}{|x′|}).

Proof. By the definition (5.1.2) of K^{+}(x, x′; t), we have

|K^{+}(rω, r′ω′; t)| = \frac{2e^{−2(r+r′) coth \frac{t}{2}}}{π^{\frac{m−1}{2}} | sinh \frac{t}{2}|^{m−1}} |Ĩ_{\frac{m−3}{2}} (\frac{ψ(rω, r′ω′)}{sinh \frac{t}{2}})|.

By (5.1.1), we have |ψ(rω, r′ω′)| ≤ 4\sqrt{rr′}. Applying Lemma 4.2.1 with ν = \frac{m−3}{2}, we have

|K^{+}(rω, r′ω′; t)| ≤ \frac{2Ce^{−α(t)(1−|β(t)|)(r+r′)}}{π^{\frac{m−1}{2}} | sinh \frac{t}{2}|^{m−1}} .

Replacing C with a new constant, we get (5.2.1).

The second and third statements follow from (5.2.1) because α(t) > 0 and |β(t)| < 1 if Re t > 0.

5.3 Proof of Theorem 5.1.1 (Case Re t > 0)

Suppose Re t > 0. We set

(S_{t}u)(x) := ∫_{ℝ^{m}} K^{+}(x, x′; t)u(x′)\frac{dx′}{|x′|}.  (5.3.1)

By Lemma 5.2.1 (2), we observe that S_{t} is a Hilbert–Schmidt operator on L^{2}(ℝ^{m}, \frac{dx}{|x|}), and the right-hand side of (5.3.1) converges absolutely for u ∈ L^{2}(ℝ^{m}, \frac{dx}{|x|}) by the Cauchy–Schwarz inequality and by Lemma 5.2.1 (3).

The remaining assertion of Theorem 5.1.1 (1) is the equality π_{+}(e^{tZ}) = S_{t}. To see this, we observe from the definition (5.1.2) of K^{+}(x, x′; t) that

K^{+}(kx, kx′; t) = K^{+}(x, x′; t) for all k ∈ R.

Therefore, the operator S_{t} intertwines the R-action, and preserves each summand of (4.1.1). In light of the decomposition (4.1.3) of the operator π_{+}(e^{tZ}), the equality π_{+}(e^{tZ}) = S_{t} will follow from:

Lemma 5.3.1. Let Re t > 0. For every l ∈ ℕ, we have

π_{+,l}(e^{tZ}) ⊗ id = S_{t}|_{L^{2}(ℝ_{+},r^{m−2}dr)⊗ℋ^{l}(ℝ^{m})} .  (5.3.2)

We postpone the proof of Lemma 5.3.1 until Subsection 5.6.

Thus, the proof of Theorem 5.1.1 (1) is completed by admitting Lemma 5.3.1.

5.4 Proof of Theorem 5.1.1 (Case Re t = 0)

Suppose Re t = 0. Then, by Lemma 5.2.1 (1), we have

|K^{+}(x, x′; t)| ≤ \frac{C}{| sinh \frac{t}{2}|^{m−1}}

because α(t) = 0. Therefore, the right-hand side of (5.1.3) converges absolutely for any u ∈ L^{1}(ℝ^{m}, \frac{dx}{|x|}) ∩ L^{2}(ℝ^{m}, \frac{dx}{|x|}), as is seen by

∫_{ℝ^{m}} |K^{+}(x, x′; t)u(x′)|\frac{dx′}{|x′|} ≤ \frac{C}{| sinh \frac{t}{2}|^{m−1}} ∫_{ℝ^{m}} |u(x′)|\frac{dx′}{|x′|} < ∞.

By Proposition 3.2.1 (1), we have L^{2}(ℝ^{m}, \frac{dx}{|x|})_{K} ⊂ L^{1}(ℝ^{m}, \frac{dx}{|x|}). Hence, the right-hand side of (5.1.3) converges absolutely, in particular, for K-finite functions.

Finally, let us show the last statement of (2). Since W_{a,l}(0 ≤ l ≤ a) spans L^{2}(ℝ^{m}, \frac{dx}{|x|})_{K} (see (3.2.1)), it is sufficient to prove

π_{+}(e^{tZ}) = S_{t} on W_{a,l} (0 ≤ l ≤ a).  (5.4.1)

We recall from Proposition 3.2.1 that every vector u ∈ W_{a,l} is of the form

u(rω) = f_{a,l}(r)ϕ(ω)  (5.4.2)

for some ϕ ∈ ℋ^{l}(ℝ^{m})(see (3.2.4) for the definition). Suppose ε > 0 and t ∈ \sqrt{−1}ℝ \ 2π\sqrt{−1}ℤ. As in the proof of Claim 4.4.1, we have

|K^{+}(x, x′; ε + t)| ≤ \frac{C′}{| sinh \frac{t}{2}|^{m−1}}

and therefore

|K^{+}(x, x′; ε + t)f_{a,l}(r′)ϕ(ω)r′^{m−2}| ≤ \frac{Cr′^{m−2+l}e^{−2r′}|L_{a−l}^{m−2+2l}(4r′)|}{| sinh \frac{t}{2}|^{m−1}}.

Here, C := C′ max_{ω∈S^{m−1}} |ϕ(ω)|. Hence, by the dominated convergence theorem, we have

lim_{ε↓0} S_{ε+t}u = S_{t}u

for any u ∈ W_{a,l}.

On the other hand, by Proposition 3.6.1 (1), we have

lim_{ε↓0} π_{+}(e^{(ε+t)Z})u = π_{+}(e^{tZ})u.

Since π_{+}(e^{(ε+t)Z}) = S_{ε+t} for ε > 0, we have now proved (5.4.1).

5.5 Spectra of an O(m)-invariant operator

The rest of this section is devoted to the proof of Lemma 5.3.1.

The orthogonal group O(m) acts on L^{2}(S^{m−1}) as a unitary representation, and decomposes it into irreducible representations as follows:

L^{2}(S^{m−1}) ≃ \begin{matrix}∞ \\ ∑ & ⊕ \\ l=0 \end{matrix} ℋ^{l}(ℝ^{m}).

Since this is a multiplicity-free decomposition, any O(m)-invariant operator S on L^{2}(S^{m−1}) acts on each irreducible component as a scalar multiplication.

The next lemma gives an explicit formula of the spectrum for an O(m)- invariant integral operator on C(S^{m−1}) in the general setting. This should be known to experts, but for the convenience of the readers, we present it in the following form:

Lemma 5.5.1. For a continuous function h on the closed interval [−1, 1], we consider the following integral transform:

S_{h} : L^{2}(S^{m−1}) → L^{2}(S^{m−1}), ϕ(ω) ↦ ∫_{S^{m−1}} h(⟨ω, ω′⟩)ϕ(ω′)dω′. (5.5.1)

Then, S_{h} acts on ℋ^{l}(ℝ^{m}) by a scalar multiplication of c_{l,m}(h) ∈ ℂ. The constant c_{l,m}(h) is given by

c_{l,m}(h) = \frac{2^{m−2}π^{\frac{m−2}{2}} l!}{Γ(m − 2 + l)} ∫_{0}^{π} h(cos θ)C̃l^{\frac{m−2}{2}} (cos θ) sin^{m−2} θdθ,  (5.5.2)

where C̃l^{\frac{m−2}{2}} (x) denotes the normalized Gegenbauer polynomial (see (8.3.2)).

Example 5.5.2 (see [15, Introduction, Lemma 3.6]). For h(x) := e^{\sqrt{−1}λx}, c_{l,m}(h) amounts to

c_{l,m}(h) = (2π)^{\frac{m}{2}} e^{\frac{\sqrt{−1}}{2} πl}λ^{− \frac{m−2}{2}} J_{\frac{m−2}{2} +l}(λ).

Example 5.5.3. We set Ĩ_{ν}(z) = (\frac{z}{2})^{−ν} I_{ν}(z) (see (8.5.6)). For h(s) := Ĩ_{\frac{m−3}{2}} (α\sqrt{1 + s}), we have

c_{l,m}(h) = 2^{\frac{3m−4}{2}} π^{\frac{m−1}{2}} α^{−m+2}I_{m−2+2l}(\sqrt{2}α)  (5.5.3)

Proof of Example 5.5.3. We apply Lemma 8.5.2 with ν = \frac{m−3}{2}. Then, we have

∫_{0}^{π} Ĩ_{\frac{m−3}{2}} (α\sqrt{1 + cos θ})C̃l^{\frac{m−2}{2}} (cos θ) sin^{m−2} θ dθ

= \frac{2^{\frac{m}{2}} \sqrt{π} Γ(m − 2 + l)}{α^{m−2}l!} I_{m−2+2l}(\sqrt{2}α).

Hence, (5.5.3) follows from Lemma 5.5.1.

Proof of Lemma 5.5.1. 1) The operator S_{h} intertwines the O(m)-action because h(⟨kω, kω′⟩) = h(⟨ω, ω′⟩) for k ∈ O(m). Hence it follows from Schur’s lemma that S_{h} acts on each irreducible O(m)-subspace ℋ^{l}(ℝ^{m}) by the multiplication of a constant, which we shall denote by c_{l,m}(h) for l = 0, 1, 2, . . .. Thus, we have

(S_{h}ϕ)(ω) = c_{l,m}(h)ϕ(ω) for ϕ ∈ ℋ^{l}(ℝ^{m}).  (5.5.4)

To compute the constant c_{l,m}(h), we use the following coordinate:

[0, π) × S^{m−2} → S^{m−1}, (θ, η) ↦ ω = (cos θ, sin θ · η).

With this coordinate, we have dω = sin^{m−2} θdθdη.

We set ω_{0} = (1, 0, ⋯, 0). Now, we take ϕ(ω) := C̃l^{\frac{m−2}{2}} (⟨ω, ω_{0}⟩) ∈ ℋ^{l}(ℝ^{m}), which is an O(m − 1)-invariant spherical harmonics. Then the equation (5.5.4) for ω = ω_{0} amounts to

\frac{2π^{\frac{m−1}{2}}}{Γ(\frac{m−1}{2})} ∫_{0}^{π} h(cos θ)C̃l^{\frac{m−2}{2}} (cos θ) sin^{m−2} θdθ = \frac{\sqrt{π} Γ(m − 2 + l)}{2^{m−3}l! Γ(\frac{m−1}{2})}c_{l,m}(h),

because vol(S^{m−2}) = \frac{2π^{\frac{m−1}{2}}}{Γ( \frac{m−1}{2} )} and ϕ(ω_{0}) = C̃l^{\frac{m−2}{2}} (1) = \frac{\sqrt{π} Γ(m−2+l)}{2^{m−3}l! Γ( \frac{m−1}{2} )} (see

(8.3.3)). Hence, (5.5.2) is proved.

5.6 Proof of Lemma 5.3.1

This subsection gives a proof of Lemma 5.3.1.

We recall from Theorem 4.1.1 (1) that the kernel function of π_{+,l}(e^{tZ}) is given by K_{l}^{+}(r, r′; t) (see (4.1.5) for definition). Therefore, the equation (5.3.2) is equivalent to the following equation between kernel functions:

Lemma 5.6.1. For ϕ ∈ ℋ^{l}(ℝ^{m}), we have

\frac{1}{2} ∫_{S^{m−1}} K^{+}(rω, r′ω′; t)ϕ(ω′)dω′ = K_{l}^{+}(r, r′; t)ϕ(ω).  (5.6.1)

Proof of Lemma 5.6.1. We set

h(r, r′, t, s) := \frac{2e^{−2(r+r′) coth \frac{t}{2}}}{π^{\frac{m−1}{2}} sinh^{m−1} \frac{t}{2}} Ĩ_{\frac{m−3}{2}} (\frac{2\sqrt{2rr′(1 + s)}}{sinh \frac{t}{2}}) .  (5.6.2)

By the definition (5.1.2) of K^{+}(x, x′; t), we have

K^{+}(rω, r′ω′; t) = h(r, r′, t, ⟨ω, ω′⟩).

Consider now the integral transform S_{h(r,r′,t,·)} on L^{2}(S^{n−1}) with kernel h(r, r′, t, ⟨ω, ω′⟩). Then it follows from Example 5.5.3 with α = \frac{2\sqrt{2rr′}}{sinh \frac{t}{2}} that

c_{l,m}(h(r, r′, t, ·)) = 2K_{l}^{+}(r, r′; t).  (5.6.3)

Then, (5.6.1) is a direct consequence of Lemma 5.5.1.

5.7 Expansion formulas

We recall that the kernel functions for the semigroups π_{+}(e^{tZ}) and π_{+,l}(e^{tZ}) are given by K^{+}(rω, r′ω′; t) and K_{l}^{+}(r, r′; t). In this subsection, we shall give expansion formulas for K^{+}(rω, r′ω′; t) arising from the decomposition (see (4.1.3))

π_{+}(e^{tZ}) = \begin{matrix}∞ \\ ∑ & ⊕ \\ l=0 \end{matrix} π_{+,l}(e^{tZ}) ⊗ id .

Proposition 5.7.1 (Expansion formulas). Let m > 1.

1) The kernel function K^{+}(x, x′; t) (see (5.1.2)) has the following expansion:

K^{+}(rω, r′ω′; t) = \frac{1}{π^{\frac{m}{2}}} ∑_{l=0}^{∞}(\frac{m − 2}{2} + l)K_{l}^{+}(r, r′; t)C̃l^{\frac{m−2}{2}} (⟨ω, ω′⟩). (5.7.1)

2) The special value t = π\sqrt{−1} for (5.7.1) yields the expansion formula for the Bessel function:

J̃_{ν− \frac{1}{2}} (\sqrt{z} cos \frac{θ}{2}) = \frac{2^{4ν}}{\sqrt{π}} ∑_{l=0}^{∞}(ν + l)(−1)^{l} \frac{J_{2ν+2l}(\sqrt{z})}{z^{ν}} C̃_{l}^{ν}(cos θ)  (5.7.2)

for z ∈ ℝ_{+}, ν ∈ \frac{1}{2}ℤ.

Remark 5.7.2 (Weil representation, Gegenbauer’s expansion). Let us compare the above result with the case of the Weil representation of G′. Then, by a similar argument to the proof of Proposition 5.7.1, we can show that the Mehler kernel 𝒦 (see Subsection 1.2) has the following decomposition:

𝒦(rω, r′ω′; t) = \frac{1}{2π^{\frac{m}{2}}} ∑_{l=0}^{∞}(\frac{m − 2}{2} + l)𝒦_{l}(r, r′; t)C̃l^{\frac{m−2}{2}} (⟨ω, ω′⟩) (5.7.3)

if m > 1.

In light of the formula (see Subsection 1.2)

𝒦(rω, r′ω′; π\sqrt{−1}) = \frac{1}{(2π\sqrt{−1})^{\frac{m}{2}}} e^{−\sqrt{−1}rr′⟨ω,ω′⟩},

the special value at t = π\sqrt{−1} for (5.7.3) yields the following expansion formula for the exponential function known as Gegenbauer’s expansion ([9], see also [33, Chapter XI, §11.5]):

e^{\sqrt{−1}z cos ϕ} = 2^{ν} ∑_{m=0}^{∞}(ν + m)\sqrt{−1}^{m} \frac{J_{ν+m}(z)}{z^{ν}} C̃_{m}^{ν}(cos ϕ)  (5.7.4)

for z ∈ ℝ, ν ∈ \frac{1}{2}ℤ. This formula corresponds to (5.7.2).

If m = 1, we have

𝒦(x, x′; t) = \frac{1}{2}(𝒦_{0}(|x|, |x′|; t) id(x) + 𝒦_{1}(|x|, |x′|; t) sgn(x)),  (5.7.5)

which is immediately verified by the formulas for 𝒦_{0}, 𝒦_{1} and 𝒦 (see Remark 4.1.2).

Proof of Proposition 5.7.1. 1) First we prepare a lemma.

Lemma 5.7.3. Let f ∈ C[−1, 1]. Then f is expanded into the infinite series:

f(x) = \frac{1}{2π^{\frac{m}{2}}} ∑_{l=0}^{∞}(\frac{m − 2}{2} + l)c_{l,m}(f)C̃l^{\frac{m−2}{2}} (x),  (5.7.6)

where c_{l,m}(f) is the constant defined in (5.5.2).

Proof of Lemma 5.7.3. Applying the expansion formula f(x) = ∑_{l=0}^{∞} α_{l}^{ν}(f)C̃_{l}^{ν}(x) (see (8.3.5)) with ν = \frac{m−2}{2}, we have

αl^{\frac{m−2}{2}} (f) = \frac{(\frac{m−2}{2} + l)}{2π^{\frac{m}{2}}}c_{l,m}(f),  (5.7.7)

from (5.5.2) and (8.3.6). Hence, we get the lemma.

Let h(r, r′, t, s) be as in (5.6.2). We recall from (5.6.3)

c_{l,m}(h(r, r′, t, ·)) = 2K_{l}^{+}(r, r′; t).

Therefore, by Lemma 5.7.3, we get (5.7.1).

2) Substitute t = π\sqrt{−1} and put

z := 16rr′, cos θ := ⟨ω, ω′⟩, ν = \frac{m − 2}{2},

we get (5.7.2).

6 The unitary inversion operator

6.1 Result of the section

We define the ‘inversion’ element w_{0} ∈ G by

w_{0} := e^{π\sqrt{−1}Z}.

Then, clearly, w_{0} has the following properties:

1) w_{0} is of order four.

2) w_{0} normalizes M^{max}A and Ad(w_{0})𝔫^{max} = \overline{𝔫^{max}}.

3) The group G is generated by \overline{P^{max}} and w_{0}.

We note that if m is odd then e^{π\sqrt{−1}Z} is equal to (\begin{matrix}I_{m+1} & 0 \\ 0 & −I_{2} \end{matrix}) in

SO_{0}(m + 1, 2) = G/{1, η}.

This section gives an explicit integral formula of the unitary operator π_{+}(w_{0}) on L^{2}(ℝ^{m}, \frac{dx}{|x|}). In light of w_{0} = e^{π\sqrt{−1}Z} and π\sqrt{−1} ∈ Ω, we define the following kernel functions by substituting t = π\sqrt{−1} into (5.1.2) and (4.1.5), respectively:

K^{+}(x, x′) := K^{+}(x, x′; π\sqrt{−1})
= \frac{2}{e^{\frac{m−1}{2} π\sqrt{−1}}π^{\frac{m−1}{2}}} \sqrt{2⟨ζ, ζ′⟩}^{− \frac{m−3}{2}} J_{\frac{m−3}{2}} (2\sqrt{2⟨ζ, ζ′⟩}), (6.1.1)
K_{l}^{+}(r, r′) := K_{l}^{+}(r, r′; π\sqrt{−1})
= 2(−1)^{l}e^{− \frac{m−1}{2} π\sqrt{−1}}(rr′)^{− \frac{m−2}{2}} J_{m−2+2l}(4\sqrt{rr′}).  (6.1.2)

Then, the following result is a direct consequence of Theorems 5.1.1 and 4.1.1.

Theorem 6.1.1 (Integral formula for the unitary inversion operator). The unitary operator π_{+}(w_{0}) : L^{2}(ℝ^{m}, \frac{dx}{|x|}) → L^{2}(ℝ^{m}, \frac{dx}{|x|}) is given by the following integral transform:

(T u)(x) := ∫_{ℝ^{m}} K^{+}(x, x′)u(x′)\frac{dx}{|x|}, u ∈ L^{2}(ℝ^{m}, \frac{dx}{|x|}).  (6.1.3)

Here, the integral (6.1.3) converges absolutely for any u ∈ L^{1}(ℝ^{m}, \frac{dx}{|x|}) ∩ L^{2}(ℝ^{m}, \frac{dx}{|x|}), in particular, for any K-finite vector, hence the equation (6.1.3) holds in the sense of L^{2}-convergence.

The substitution of t = π\sqrt{−1} into (4.1.3) gives the decomposition

π_{+}(w_{0}) = \begin{matrix}∞ \\ ∑ & ⊕ \\ l=0 \end{matrix} π_{+,l}(w_{0}) ⊗ id .

As π_{+}(w_{0}) is given by the kernel K^{+}(x, x′), so is π_{+,l}(w_{0}) by K_{l}^{+}(r, r′). Thus, we have:

Theorem 6.1.2 (Radial part of the unitary inversion operator). The unitary operator π_{+,l}(w_{0}) : L^{2}(ℝ_{+}, r^{m−2}dr) → L^{2}(ℝ_{+}, r^{m−2}dr) is given by the following integral transform:

(T_{l}f)(r) := ∫_{0}^{∞} K_{l}^{+}(r, r′)f(r′)r′^{m−2}dr′.  (6.1.4)

Here, the integral (6.1.4) converges absolutely for any f ∈ L^{1}(ℝ_{+}, r^{m−2}dr)∩ L^{2}(ℝ_{+}, r^{m−2}dr). In particular, the equation (6.1.4) holds in the sense of L^{2}-convergence.

Remark 6.1.3 (Weil representation). In the case of the Weil representation ϖ of G′, the counterparts of Theorem 6.1.1 and Theorem 6.1.2 can be stated as follows: let ω_{0} := e^{π\sqrt{−1}Z′}.

We define kernel functions 𝒦 and 𝒦_{l} by the formulas

𝒦(x, x′) :=𝒦(x, x′; π\sqrt{−1}) = \frac{1}{(2π\sqrt{−1})^{\frac{m}{2}}} e^{−\sqrt{−1}⟨x,x′⟩},
𝒦_{l}(r, r′) :=𝒦_{l}(r, r′; π\sqrt{−1}) = \sqrt{−1}^{− \frac{m−1+2l}{2}} (rr′)^{− \frac{m−2}{2}} J_{\frac{m−2+2l}{2}} (rr′).

Then, 1) The unitary operator ϖ(w_{0}) : L^{2}(ℝ^{m}) → L^{2}(ℝ^{m}) is given by

(ϖ(w_{0})u)(x) = ∫_{ℝ^{m}} 𝒦(x, x′)u(x′)dx′.

Hence we see that ϖ(w_{0}) is nothing but the Fourier transform.

2) The ‘radial’ part of ϖ(w_{0}), namely, the unitary operator ϖ_{l}(w_{0}) : L^{2}(ℝ_{+}, r^{m−1}dr) → L^{2}(ℝ_{+}, r^{m−1}dr) (see Remark 4.1.2) is given by

(ϖ_{l}(w_{0})f)(r) = ∫_{0}^{∞} 𝒦_{l}(r, r′)f(r′)r′^{m−1}dr′.

6.2 Inversion and Plancherel formula

It follows from (3.6.3) that π(w_{0})^{2} = π_{+}(e^{2π\sqrt{−1}Z}) = (−1)^{m+1} id. Thus, as an immediate consequence of Theorem 6.1.1, we have:

Corollary 6.2.1. The integral transform

T : u(x) ↦ ∫_{ℝ^{m}} K^{+}(x, x′)\frac{u(x′)}{|x′|}dx′,
K^{+}(x, x′) := \frac{2^{\frac{m−1}{2}} ψ(x, x′)^{− \frac{m−3}{2}}}{e^{\frac{m−1}{2} π\sqrt{−1}}π^{\frac{m−1}{2}}} J_{\frac{m−3}{2}} (ψ(x, x′))

is a unitary operator on L^{2}(ℝ^{m}, \frac{dx}{|x|}) of order two (m: odd) and of order four (m: even), that is, we have:

(Inversion formula) T^{−1} = (−1)^{m+1}T,

(Plancherel formula) ∥T u∥_{L^{2}(ℝ^{m}, \frac{dx}{|x|})} = ∥u∥_{L^{2}(ℝ^{m}, \frac{dx}{|x|})} for all u ∈ L^{2}(ℝ^{m}, \frac{dx}{|x|}).

Remark 6.2.2 (Weil representation, see [16, Corollaries 5.7.3 and 5.7.4])). Let us compare Corollary 6.2.1 with the corresponding result for the Schrödinger model L^{2}(ℝ^{m}) of the Weil representation ϖ. The unitary operator ϖ(w_{0}) corresponding to the “inversion element” w_{0} is given by the (ordinary) Fourier transform ℱ (see Fact D in Subsection 1.2). As is wellknown, ℱ is a unitary operator of order four. This reflects the fact that w_{0}^{4} = e in Sp(m, ℝ).

6.3 The Hankel transform

Similar to Corollary 6.2.1, from the equality π_{+}(w_{0})^{2} = (−1)^{m+1}id and Theorem 6.1.2, we have:

Corollary 6.3.1. Let ν be a positive integer, and x, y ∈ ℝ. Then the integral transform

𝒯_{ν} : f(x) ↦ ∫_{0}^{∞} J_{ν}(xy)f(y)\sqrt{xy}dy

is a unitary operator on L^{2}((0, ∞), dx) of order two. Hence we have:

(Inversion formula) 𝒯_{ν}^{−1} = 𝒯_{ν},
(Plancherel formula) ∥𝒯_{ν}f∥_{L^{2}(ℝ_{+},dx)} = ∥f∥_{L^{2}(ℝ_{+},dx)}.

Remark 6.3.2. The unitary operator 𝒯_{ν} coincides with the Hankel transform (see [7, Chapter VIII]) and the property 𝒯_{ν}^{2} = id in the corollary corresponds to its classically known reciprocal formula due to Hankel [13] (see also [7, §8.1 (1)], [33, §14.3 (3)]). The Parseval-Plancherel formula for the Hankel transform goes back to Macaulay-Owen [25].

Proof of Corollary 6.3.1. Since we assume ν is a positive integer, ν = m − 2 + 2l for some m > 3 and l ∈ ℤ. We change the variables r = \frac{x^{2}}{4} , r′ = \frac{y^{2}}{4} and define a unitary map Φ : L^{2}(ℝ_{+}, r^{m−2}dr) → L^{2}(ℝ), (Φf)(x) :=

(\frac{x}{2})^{\frac{2m−3}{2}} f(\frac{x^{2}}{4} ). Then, we have 𝒯_{ν} = e^{−(l+ \frac{m−1}{2} )π\sqrt{−1}}Φ ∘ T_{l} ∘ Φ^{−1}. Since T_{l} = π_{+,l}(w_{0}) by Theorem 6.1.2, T_{l} has its inverse as T_{l}^{−1} = (−1)^{m+1}T_{l}, 𝒯_{ν} has its inverse given by

𝒯_{ν}^{−1} = e^{(l+ \frac{m−1}{2} )π\sqrt{−1}}Φ ∘ (−1)^{m+1}T_{l} ∘ Φ^{−1} = 𝒯_{ν}.

Hence, the corollary follows.

6.4 Forward and backward light cones

So far, we have discussed only the irreducible unitary representation π_{+} realized on L^{2}(C_{+}) for the forward light cone. In this subsection, let us briefly comment on (π_{−}, L^{2}(C_{−})) for the backward light cone and (π, L^{2}(C)) for C = C_{+} ∪ C_{−}.

Since dπ_{−}(Z) = −dπ_{+}(Z) in the polar coordinate representation, we can define a semigroup of contraction operators {π_{−}(e^{tZ}) : Re t < 0} similarly on L^{2}(C_{−}). All the statements about the semigroup π_{+}(e^{tZ}) also hold by changing the signature t → −t and replacing C_{+} by C_{−}.

We define a function K(ζ, ζ′) on C × C by

K(ζ, ζ′) := \begin{cases}K^{+}(ζ, ζ′) & if ⟨ζ, ζ′⟩ ≥ 0 \\ 0 & if ⟨ζ, ζ′⟩ < 0. \end{cases}  (6.4.1)

Here we note:

{(ζ, ζ′) ∈ C × C : ⟨ζ, ζ′⟩ ≥ 0} = (C_{+} × C_{+}) ∪ (C_{−} × C_{−}).

Originally, K^{+}(ζ, ζ′) was defined on C_{+} × C_{+} (see (6.1.1)). Since K^{+}(ζ, ζ′) depends only on the inner product ⟨ζ, ζ′⟩, we can define K^{+}(ζ, ζ′) also on C_{−} × C_{−}.

Corollary 6.4.1. The unitary operator π(w_{0}) : L^{2}(C) → L^{2}(C) coincides with the integral transform defined by

T : L^{2}(C) → L^{2}(C), u ↦ ∫_{C} K(ζ, ζ′)u(ζ′)dμ(ζ′).  (6.4.2)

Remark 6.4.2. We note that the kernel function K(ζ, ζ′) is supported on the proper subset {(ζ, ζ′) ∈ C × C : ⟨ζ, ζ′⟩ ≥ 0} of C ×C. More generally, in the case of the minimal representation of O(p, q) with p + q : even, ≥ 8, we shall see in [20] that the integral kernel K_{p,q}(ζ, ζ′) representing the inversion element is also supported on the proper subset {(ζ, ζ′) ∈ C × C : ⟨ζ, ζ′⟩ ≥ 0} if both integers p, q are even. This feature fails if both p, q are odd. In fact, the support of K_{p,q} is the whole space C × C then.

7 Explicit actions of the whole group on L^{2}(C)

Building on the explicit formula of π(w_{0}) = π(e^{π\sqrt{−1}Z}) on L^{2}(C) (see Corollary 6.4.1) and that of π(g) (g ∈ \overline{P^{max}}) (see Subsection 2.2), we can find an explicit formula of the minimal representation for the whole group G.

For simplicity, this section treats the case where m is odd. The main result of this section is Theorem 7.2.1 for the action of O(m + 1, 2).

7.1 Bruhat decomposition of O(m + 1, 2)

We recall the notation in Subsection 2.1. In particular, \overline{P^{max}} = M^{max}A\overline{N^{max}} is a maximal parabolic subgroup of O(m + 1, 2). Hence, O(m + 1, 2) is expressed as the disjoint union:

O(m + 1, 2) = \overline{P^{max}} ⨿ \overline{P^{max}}w_{0}\overline{P^{max}}
= \overline{P^{max}} ⨿ \overline{N^{max}}AM^{max}w_{0}\overline{N^{max}}.

We begin by finding a, b ∈ ℝ^{m+1}, t ∈ ℝ and m ∈ M^{max} such that

g = \overline{n}_{b}e^{tE}mw_{0}\overline{n}_{a} ∈ \overline{N^{max}}AM^{max}w_{0}\overline{N^{max}}  (7.1.1)

holds for g ∉ \overline{P^{max}}.

Suppose g is of the form (7.1.1), and we write m = δm_{+} (δ = ±1, m_{+} ∈ M_{+}^{max}). Then, in light of the formulas (2.1.3)–(2.1.5), we have

g(e_{0} − e_{m+2}) = \overline{n}_{b}e^{tE}(δm_{+})w_{0}\overline{n}_{a}(e_{0} − e_{m+2})
= \overline{n}_{b}e^{tE}(δm_{+})w_{0}(e_{0} − e_{m+2})
= \overline{n}_{b}e^{tE}(δm_{+})w_{0}(e_{0} − e_{m+2})
= δe^{t}\overline{n}_{b}(e_{0} + e_{m+2})
= δe^{t} ^{t}(1 − Q(b), 2b_{1}, …, 2b_{m+1}, 1 + Q(b)).  (7.1.2)

Now, we set

x ≡ ^{t}(x_{0}, …, x_{m+2}) := g(e_{0} − e_{m+2}).  (7.1.3)

Then, solving (7.1.2), we have

x_{0} + x_{m+2} = 2δe^{t},

\frac{x_{j}}{x_{0} + x_{m+2}} = b_{j} (1 ≤ j ≤ m + 1).

Likewise, if we set

y ≡ ^{t}(y_{0}, …, y_{m+2}) := g^{−1}(e_{0} − e_{m+2}),  (7.1.4)

the following relation:

\frac{y_{j}}{y_{0} + y_{m+2}} = −a_{j} (1 ≤ j ≤ m + 1)

must hold, because g^{−1} = \overline{n}_{−a}e^{tE}(w_{0}m^{−1}w_{0}^{−1})w_{0}\overline{n}_{−b}.

Let ⟨ , ⟩ denote the standard positive definite inner product on ℝ^{m+3}. It follows from g ∈ O(m + 1, 2) that

x_{0} + x_{m+2} = ⟨e_{0} + e_{m+2}, g(e_{0} − e_{m+2})⟩
= ⟨^{t}g(e_{0} + e_{m+2}), e_{0} − e_{m+2}⟩
= ⟨I_{m+1,2}g^{−1}I_{m+1,2}(e_{0} + e_{m+2}), e_{0} − e_{m+2}⟩
= ⟨y, e_{0} + e_{m+2}⟩
= y_{0} + y_{m+2}.

If x_{0} + x_{m+2} ≠ 0, we set

a := (a_{1}, …, a_{m+1}), a_{j} := \frac{−y_{j}}{y_{0} + y_{m+2}} (1 ≤ j ≤ m + 1), (7.1.5) (a)
b := (b_{1}, …, b_{m+1}), b_{j} := \frac{x_{j}}{x_{0} + x_{m+2}} (1 ≤ j ≤ m + 1), (7.1.5) (b)
δ := sgn(x_{0} + x_{m+2}),  (7.1.5) (c)
t := log |\frac{x_{0} + x_{m+2}}{2} | ,  (7.1.5) (d)
m_{+} := δe^{−tE}\overline{n}_{−b}g\overline{n}_{−a}w_{0}^{−1}.  (7.1.5) (e)

Lemma 7.1.1. Retain the above notation.

1) For g ∈ O(m + 1, 2), the following three conditions are equivalent:

i) g ∉ \overline{P^{max}}, ii) x_{0} + x_{m+2} ≠ 0, iii) y_{0} + y_{m+2} ≠ 0.

2) Suppose one of (therefore, all of) the above conditions holds. Then, the element m_{+} defined by (7.1.5) (e) belongs to M_{+}^{max} ≃ O(m, 1).

7.2 Explicit action of the whole group

For g ∈ O(m + 1, 2), we set

x ≡ ^{t}(x_{0}, …, x_{m+2}) := g(e_{0} − e_{m+2}) (see (7.1.3)),
y ≡ ^{t}(y_{0}, …, y_{m+2}) := g^{−1}(e_{0} − e_{m+2}) (see (7.1.4)).

For g such that x_{0} + x_{m+2} ≠ 0, we also set a = (a_{1}, …, a_{m+1}), b = (b_{1}, …, b_{m+1}), δ ∈ {±1}, and m_{+} as in (7.1.5).

If x_{0}+x_{m+2} = 0, then g ∈ \overline{P^{max}} and (π(g)ψ)(ζ) is obtained readily by the formulas (2.2.2)–(2.2.5). For generic g such that x_{0} + x_{m+2} ≠ 0, the unitary operator π(g) can be written by means of the kernel function K(ζ, ζ′) (see (6.4.1) for the definition), the above x, y ∈ ℝ^{m+3} and m_{+} ∈ M_{+}^{max} ≃ O(m, 1) as follows:

Theorem 7.2.1. For g ∈ O(m + 1, 2) such that x_{0} + x_{m+2} ≠ 0, the unitary operator π(g) is given by the following integral formula: for ψ ∈ L^{2}(C),

(π(g)ψ)(ζ) = (\frac{2}{x_{0} + x_{m+2}} )^{\frac{m−1}{2}} e^{\frac{2\sqrt{−1}(x_{1}ζ_{1}+···+x_{m+1}ζ_{m+1})}{x_{0}+x_{m+2}}}

∫_{C} K (\frac{2ζ}{|x_{0} + x_{m+2}|}, m_{+}ζ′) e^{\frac{−2\sqrt{−1}(y_{1}ζ′_{1}+···+y_{m+1}ζ{′ \atop m+1})}{y_{0}+y_{m+2}}} ψ(ζ′)du(ζ′).

Proof. We write g = \overline{n}_{b}e^{t}(δm_{+})w_{0}\overline{n}_{a} as in the form (7.1.1). Then, we have

(π(g)ψ)(ζ)
= π(\overline{n}_{b})π(e^{tE})π(δm_{+})(π(w_{0}\overline{n}_{a})ψ)(ζ)
= e^{2\sqrt{−1}⟨b,ζ⟩}e^{− \frac{m−1}{2} t}δ^{\frac{m−1}{2}} (π(w_{0}\overline{n}_{a})ψ)(e^{−t} ^{t}m_{+}ζ) by (2.2.2)–(2.2.5).

Now, by using Corollary 6.4.1,

= e^{2\sqrt{−1}⟨b,ζ⟩}e^{− \frac{m−1}{2} t}δ^{\frac{m−1}{2}} ∫_{C} K(e^{−t} ^{t}m_{+}ζ, ζ′)(π(\overline{n}_{a})ψ)(ζ′)du(ζ′)
= e^{2\sqrt{−1}⟨b,ζ⟩}e^{− \frac{m−1}{2} t}δ^{\frac{m−1}{2}} ∫_{C} K(e^{−t}ζ, m_{+}ζ′)e^{2\sqrt{−1}⟨a,ζ′⟩}ψ(ζ′)du(ζ′).

8 Appendix: special functions

For the convenience of the reader, we collect here basic formulas of special functions in a way that we use in this article.

8.1 Laguerre polynomials

For α ∈ ℂ, n ∈ ℕ, the Laguerre polynomials L_{n}^{α}(x) are defined by the formula (see [1, §6.2], for example):

L_{n}^{α}(x) := \frac{x^{−α}e^{x}}{n!} \frac{d^{n}}{dx^{n}} (e^{−x}x^{n+α})  (8.1.1)
= \frac{(α + 1)_{n}}{n!} ∑_{k=0}^{n} \frac{(−n)_{k}x^{k}}{(α + 1)_{k}k!}

= \frac{(−1)^{n}}{n!} x^{n} + ⋯ + \frac{(α + 1)(α + 2) ⋯ (α + n)}{n!}.

Here, we write β_{n} for β(β + 1) ⋯ (β + n − 1). The Laguerre polynomial solves the linear ordinary differential equation of second order:

xu′′ + (α + 1 − x)u′ + nu = 0.  (8.1.2)

Suppose α ∈ ℝ and α > −1. Then the Laguerre polynomials {L_{n}^{α}(x) : n = 0, 1, ⋯} are complete in L^{2}((0, ∞), x^{α}e^{−x}dx), and satisfy the orthogonality relation (see [1, §6.5]):

∫_{0}^{∞} L_{m}^{α}(x)L_{n}^{α}(x)x^{α}e^{−x}dx = \frac{Γ(α + n + 1)}{n!}δ_{mn} (α > −1).  (8.1.3)

The Hille–Hardy formula gives the bilinear generating function of Laguerre polynomials (see [6, §10.12 (20)], [14, §1 (3)]):

∑_{n=0}^{∞} \frac{Γ(n + 1)}{Γ(n + α + 1)}L_{n}^{α}(x)L_{n}^{α}(y)w^{n}

= \frac{1}{1 − w} exp(−\frac{(x + y)w}{1 − w} )(−xyw)^{− \frac{α}{2}} J_{α}(\frac{2\sqrt{−xyw}}{1 − w} ), x > 0, y > 0, |w| < 1. (8.1.4)

Here the left-hand side converges absolutely.

8.2 Hermite polynomials

Hermite polynomials H_{n}(x) are given as special values of Laguerre polynomials. We recall from [6, II, §10.13] that

H_{2m}(x) = (−1)^{m}2^{2m}m! L_{m}^{− \frac{1}{2}} (x^{2}),  (8.2.1)
H_{2m+1}(x) = (−1)^{m}2^{2m+1}m! xLm^{\frac{1}{2}}(x^{2}),  (8.2.2)

This reduction formula is reflected by the fact that Hermite polynomials appear in the analysis of the Weil representation of SL(2, ℝ), while Laguerre polynomials appear in the analysis of the minimal representation of SO_{0}(m+ 1, 2) and Sp(m, ℝ) (see Remark 4.3.2).

8.3 Gegenbauer polynomials

For ν ∈ ℂ and l ∈ ℕ, the Gegenbauer polynomials C_{l}^{ν}(x) are the polynomials of x of degree l given by the Rodrigues formula (see [1, §6.4]):

C_{l}^{ν}(x) : = (−\frac{1}{2})^{l} \frac{(2ν)_{l}}{l! (ν + \frac{1}{2})_{l}} (1 − x^{2})^{−ν+ \frac{1}{2}} \frac{d^{l}}{dx^{l}} ((1 − x^{2})^{l+ν− \frac{1}{2}} ). (8.3.1)

It then follows from (8.3.1) that

C_{l}^{ν}(1) = \frac{Γ(2ν + l)}{l! Γ(2ν)}.

We renormalize the Gegenbauer polynomial by

C̃_{l}^{ν}(x) := Γ(ν)C_{l}^{ν}(x).  (8.3.2)

Then by the duplication formula of the Gamma function:

Γ(2ν) = \frac{2^{2ν−1}}{\sqrt{π}} Γ(ν)Γ(n + \frac{1}{2}),

we have

C̃_{l}^{ν}(1) = \frac{\sqrt{π} Γ(2ν + l)}{2^{2ν−1}l! Γ(ν + \frac{1}{2})}.  (8.3.3)

The special value at ν = 0 is given by the limit formula (see [6, I, §3.15.1 (14)])

C̃_{l}^{0}(cos θ) = lim_{ν→0} Γ(ν)C_{l}^{ν}(cos θ) = \frac{2 cos(lθ)}{l}.

Suppose Re ν > −\frac{1}{2}. Then, \{C̃_{l}^{ν}(x) : l = 0, 1, 2, …\} forms a complete orthogonal basis of L^{2}((−1, 1), (1 − x^{2})^{ν− \frac{1}{2}} dx), and the norm of C̃_{l}^{ν}(x) is given by

∫_{−1}^{1} C̃_{l}^{ν}(x)^{2}(1 − x^{2})^{ν− \frac{1}{2}} dx = \frac{2^{1−2ν}πΓ(2ν + l)}{l! (l + ν)},  (8.3.4)

(see [6, I, §3.15.1 (17)]). Therefore, f ∈ L^{2}((−1, 1), (1 − x^{2})^{ν− \frac{1}{2}} dx) has the following expansion:

f(x) = ∑_{l=0}^{∞} α_{l}^{ν}(f)C̃_{l}^{ν}(x),  (8.3.5)

where we set

α_{l}^{ν}(f) := \frac{l! (l + ν)}{2^{1−2ν}πΓ(2ν + l)} ∫_{−1}^{1} f(x)C̃_{l}^{ν}(x)(1 − x^{2})^{ν− \frac{1}{2}} dx.  (8.3.6)

The following integral formula is used in the proof of Lemma 8.5.2 (see [7, §16.3 (3)]): Suppose Re β > −1 and Re ν > −\frac{1}{2}.

∫_{−1}^{1}(1 − x)^{ν− \frac{1}{2}} (1 + x)^{β}C̃_{n}^{ν}(x)dx =

\frac{2^{β−ν+ \frac{3}{2}} \sqrt{π} Γ(β + 1)Γ(2ν + n)Γ(β − ν + \frac{3}{2})}{n! Γ(β − ν − n + \frac{3}{2})Γ(β + ν + n + \frac{3}{2})}. (8.3.7)

8.4 Spherical harmonics and Gegenbauer polynomials

Let Δ_{S^{n−1}} be the Laplace–Beltrami operator on the (n−1)-dimensional unit sphere S^{n−1}. Then the spherical harmonics on S^{n−1} are defined as

ℋ^{l}(ℝ^{n}) := {f ∈ C^{∞}(S^{n−1}) : Δ_{S^{n−1}}f = −l(l + n − 2)f}, l = 0, 1, 2, ….

The following facts are well-known:

(1) ℋ^{l}(ℝ^{n}) is an irreducible representation space of O(n).

(2) It is still irreducible as an SO(n)-module if n ≥ 3.

(3) ℋ^{l}(ℝ^{2}) = ℂe^{\sqrt{−1}lθ} ⊕ ℂe^{−\sqrt{−1}lθ}, l ≥ 1 as SO(2)-modules, where θ = tan^{−1} \frac{x_{2}}{x_{1}} , (x_{1}, x_{2}) ∈ ℝ^{2}.

(4) ℋ^{l}(ℝ^{n})|_{O(n−1)} ≃ ⨁_{k=0}^{l} ℋ^{k}(ℝ^{n−1}) gives an irreducible decomposition of O(n−1)-modules. This is also an irreducible decomposition as SO(n−1)- modules (n ≥ 4).

We regard O(n − 1) as the isotropy subgroup of O(n) at (1, 0, ⋯, 0) ∈ ℝ^{n}. We write (x_{1}, ⋯, x_{n}) for the standard coordinate of ℝ^{n}. Then, O(n − 1)-invariant spherical harmonics are unique up to scalar, and we have:

ℋ^{l}(ℝ^{n})^{O(n−1)} ≃ ℂC̃l^{\frac{n−2}{2}} (x_{1}).  (8.4.1)

8.5 Bessel functions

For ν ∈ ℂ, z ∈ ℂ \ ℝ_{≤0}, Bessel functions J_{ν}(z) are defined by

J_{ν}(z) := (\frac{z}{2})^{ν} ∑_{m=0}^{∞} \frac{(−1)^{m}(\frac{z}{2})^{2m}}{m! Γ(ν + m + 1)}.  (8.5.1)

It solves the Bessel’s differential equation of second order:

u′′ + \frac{1}{z}u′ + (1 − \frac{ν^{2}}{z^{2}} )u = 0.

The modified Bessel functions I_{ν}(z) are defined by

I_{ν}(z) := \begin{cases}e^{− \frac{πν\sqrt{−1}}{2}} J_{ν}(\sqrt{−1}z) & (−π < arg z ≤ \frac{π}{2} ), \\ e^{\frac{3πν\sqrt{−1}}{2}} J_{ν}(\sqrt{−1}z) & (\frac{π}{2} < arg z < π). \end{cases}  (8.5.2)

=(\frac{z}{2})^{ν} ∑_{n=0}^{∞} \frac{(\frac{z}{2})2m}{m! Γ(ν + m + 1)} (z ∈ ℂ \ ℝ_{≤0}).  (8.5.3)

For a special value ν = ±\frac{1}{2}, I_{ν}(z) reduces to

I_{\frac{1}{2}} (z) = \sqrt{\frac{2}{πz}} sinh z, I_{− \frac{1}{2}} (z) = \sqrt{\frac{2}{πz}} cosh z.  (8.5.4)

We set

J̃_{ν}(z) := (\frac{z}{2})^{−ν} J_{ν}(z) = ∑_{m=0}^{∞} \frac{(−1)^{m} (\frac{z}{2})^{2m}}{m! Γ(ν + m + 1)},  (8.5.5)

Ĩ_{ν}(z) := (\frac{z}{2})^{−ν} I_{ν}(z) = ∑_{m=0}^{∞} \frac{(\frac{z}{2})2m}{m! Γ(ν + m + 1)}.  (8.5.6)

We note that J̃_{ν}(z) and Ĩ_{ν}(z) are entire functions of z, and

J̃_{ν}(\sqrt{−1}z) = Ĩ_{ν}(z).

The following lemma on the estimate of I-Bessel functions are used in Subsections 4.4 and 5.2. We need an estimate of Ĩ_{ν}(z) for ν ≥ −\frac{1}{2}:

Lemma 8.5.1. There exists a constant C > 0 such that the following estimate holds for all z ∈ ℂ:

|Ĩ_{ν}(z)| ≤ Ce^{| Re z|}.  (8.5.7)

Proof. First, suppose ν = −\frac{1}{2}. Then |Ĩ_{ν}(z)| = \frac{1}{\sqrt{π}} | cosh z| ≤ \frac{1}{\sqrt{π}} e^{| Re z|}.

Next, suppose ν > −\frac{1}{2}. By an integral representation of the Bessel function [33, §6.15 (2)]:

Ĩ_{ν}(z) = \frac{1}{Γ(ν + \frac{1}{2})Γ(\frac{1}{2})} ∫_{−1}^{1} e^{−zt}(1 − t^{2})^{ν− \frac{1}{2}} dt,

we have

|Ĩ_{ν}(z)| ≤\frac{1}{Γ(ν + \frac{1}{2})Γ(\frac{1}{2})} ∫_{−1}^{1} e^{−t Re z}(1 − t^{2})^{Re ν− \frac{1}{2}} dt
≤\frac{e^{| Re z|}}{Γ(ν + \frac{1}{2})Γ(\frac{1}{2})} ∫_{−1}^{1}(1 − t^{2})^{Re ν− \frac{1}{2}} dt ≤ Ce^{| Re z|}

for some constant C independent of z.

The following lemma is used in Subsection 5.5, where we set ν = \frac{m−3}{2}, α = \frac{2\sqrt{2rr′}}{sinh \frac{t}{2}} .

Lemma 8.5.2. Assume α ∈ ℂ, ν ≥ −\frac{1}{2}, and l ∈ ℕ. Then we have:

∫_{0}^{π} I_{ν}(α\sqrt{1 + cos θ})C̃_{l}^{ν+ \frac{1}{2}} (cos θ)(1 + cos θ)^{− \frac{ν}{2}} sin^{2ν+1} θdθ

= \frac{2^{\frac{3}{2}} \sqrt{π} Γ(2ν + l + 1)}{α^{ν+1}l!}I_{2ν+2l+1}(\sqrt{2}α). (8.5.8)

We could not find this formula in the literature, and so we give its proof here.

Proof. First we note that the integral (8.5.8) converges since I_{ν}(α(\sqrt{1 + cos θ}))· (1+cos θ)^{− \frac{ν}{2}} is continuous on the closed interval [0, π]. Furthermore, we have a uniformly convergent expansion

I_{ν}(α(\sqrt{1 + cos θ})) · (1 + cos θ)^{− \frac{ν}{2}} = ∑_{j=0}^{∞} \frac{(\frac{α}{2})^{ν+2j}(1 + cos θ)^{j}}{j! Γ(j + ν + 1)} .

Now the left-hand side of (8.5.8) equals

(\frac{α}{2})^{ν} ∑_{j=0}^{∞} \frac{(\frac{α}{2})2j}{j! Γ(j + ν + 1)} ∫_{0}^{π}(1 + cos θ)^{j}C̃_{l}^{ν+ \frac{1}{2}} (cos θ) sin^{2ν+1} θdθ

= (\frac{α}{2})^{ν} ∑_{j=0}^{∞} \frac{(\frac{α}{2})2j}{j! Γ(j + ν + 1)} \frac{2^{j+1}\sqrt{π} Γ(j + ν + 1)Γ(2ν + l + 1)Γ(j + 1)}{l! Γ(j − l + 1)Γ(2ν + j + l + 2)}

= (\frac{α}{2})^{ν} \frac{2\sqrt{π} Γ(2ν + l + 1)}{l!} ∑_{j=0}^{∞} \frac{(\frac{α}{\sqrt{2}})2j}{Γ(j − l + 1)Γ(2ν + j + l + 2)}

= (\frac{α}{2})^{ν} \frac{2\sqrt{π} Γ(2ν + l + 1)}{l!}(\frac{α}{\sqrt{2}})^{2l} ∑_{n=0}^{∞} \frac{(\frac{α}{\sqrt{2}})n}{Γ(n + 1)Γ(2ν + 2l + n + 2)}

= the right-hand side of (8.5.8).

Here the first equality follows from the formula (8.3.7). In fact, the substitution of x = cos θ into (8.3.7) yields

∫_{0}^{π}(1 + cos θ)^{j}C̃_{l}^{ν+ \frac{1}{2}} (cos θ) sin^{2ν+1} θdθ

= ∫_{−1}^{1}(1 − x)^{ν}(1 + x)^{j+ν}C̃_{l}^{ν+ \frac{1}{2}} (x)dx

= \frac{2^{j+1}\sqrt{π} Γ(j + ν + 1)Γ(2ν + l + 1)Γ(j + 1)}{l! Γ(j − l + 1)Γ(2ν + j + l + 2)}.

Thus, the lemma has been proved.

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