UDC 519.46

MATHEMATICS

V. F. MOLCHANOV

HARMONIC ANALYSIS ON THE ONE-SHEETED HYPERBOLOID

(Presented by Academician I. M. Vinogradov on 6 III 1966)

1. Let a bilinear form be given in the (n)-dimensional real space (R^n):
   ([x,y] = -x_1y_1 + x_2y_2 + \cdots + x_ny_n); let (G) be the connected group of linear transformations with determinant 1 that preserve this form. (G) acts transitively on the one-sheeted hyperboloid (X: [x,x] = 1). The shifts (f(x) \to f(xg)) form a unitary representation of (G) in (L^2(X)) with respect to the invariant measure (dx = |x_1|^{-1} dx_2 \cdots dx_n). (X) is an example of a homogeneous space with a noncompact stationary subgroup. Harmonic analysis on such spaces has been studied only in special cases. An analogue of the Plancherel formula on the one-sheeted hyperboloid for (n=4) ((G) is the Lorentz group) was obtained by the horisphere method in ((^1)) (see also ((^2))).

In the present work (L^2(X)) is decomposed into irreducibles with the aid of spherical functions.

Let (W) be the stationary subgroup of the point (x^0=(0,1,0,\ldots,0)); let (T_g) be an irreducible unitary representation of (G) in a Hilbert space (H), and let (\theta) be a linear functional defined on an everywhere dense set (D) and invariant with respect to (W). Then the expression ((T_g\theta,\theta)), understood in the regularized sense (see § 5), defines a function on (X):
(\Phi(x)=(T_g\theta,\theta)), (x=x^0g), which is called spherical. The main problem consists in expanding the (\delta)-function on (X) in spherical functions.

1. Representations of (G) associated with the cone. Representations of the group (G) were studied in ((^{3-5})) (see also ((^6)), Ch. X, § 2).

Let (\tau) be a complex number; (D_\tau) the space of infinitely differentiable functions (\varphi(\xi)) on the upper part of the cone (X_0^+: [x,x]=0), (x_1>0), homogeneous of degree (\sigma=(2-n+\tau)/2): (\varphi(t\xi)=t^\sigma\varphi(\xi)), (t>0). In (D_\tau) there acts a representation (T^\tau) of the group (G): (\varphi(\xi)\to\varphi(\xi g)). A function (\varphi\in D_\tau) is determined by its values on the set (Y:\xi_1-\xi_2=1); a point (\eta\in Y) has the form
[
\eta=(\eta_1,\eta_2,\eta)=\left(\frac{1}{2}(|\eta|^2+1),\frac{1}{2}(|\eta|^2-1),\eta_3,\ldots,\eta_n\right),
]
where (|\eta|^2=\eta_3^2+\cdots+\eta_n^2). (D_\tau) is realized as a certain aggregate of functions from (C^\infty(Y)).

The operator
[
A_\tau:\varphi(\eta)\to \frac{1}{\Gamma(-\tau/2)}\int |\eta-\eta'|^{\,2-n-\tau}\varphi(\eta')\,d\eta'
\qquad (d\eta=d\eta_3\cdots d\eta_n)
]
maps (D_\tau) into (D_{-\tau}) and commutes with the action of (G).

Continuous series. (\tau=i\rho), (-\infty<\rho<\infty). (T^\tau) is irreducible and preserves the scalar product
[
(\varphi,\varphi)=\int |\varphi(\eta)|^2\,d\eta,
]
(T^\tau) and (T^{-\tau}) are equivalent.

Discrete series. For (\tau=2-n-2l), (l=0,1,2,\ldots), the operator (A_\tau) has in (D_\tau) its own null subspace (F_\tau), which is irreducible with respect to the restriction of (T^\tau) (except for (n=3)) and in which there exists an invariant scalar product
[
B(\varphi,\varphi)=c_{nl}\int |\eta-\eta'|^{2l}\nu_n|\eta-\eta'|\varphi(\eta)\overline{\varphi(\eta')}\,d\eta\,d\eta'.
]

The constant (c_{nl}) here is chosen equal to
((-1)^{l+1}2\pi^{2-n}\Gamma(l+n-2)/\Gamma(l+1)\Gamma(-\tau/2)). The representations in (F_\tau') form a discrete series.

(D_{ip}) and (F_{2-n-2l}) are completed to Hilbert spaces (H_p) and (H_l) with respect to the scalar products already present.

3. Functions invariant with respect to the stationary subgroup. The representation (T^\tau) is extended from (D_\tau) to the space (D_\tau') of generalized functions (\theta(\eta)) on (X_0^+), homogeneous of degree (\sigma), (\theta:\varphi(\eta)\to\int \theta(\eta)A_\tau\varphi(\eta)\,d\eta). For (\tau=2-n-2l) denote by (F_\tau') the null subspace of the operator (A_\tau). In (D_\tau') there exist two linearly independent functions invariant with respect to (W), namely ((\xi_2)\pm^\sigma), if (\sigma\ne -m), (m) natural, and (\xi_2^{-m}), (\delta^{(m-1)}(\xi_2)), if (\sigma=-m). Only one of the functions (\xi_2^{2-n-l}), (\delta^{(n-3+l)}(\xi_2)) belongs to (F\tau'): the first for odd (n), the second for even (n). Put (\theta_l(\eta)=\beta_l(\eta_2)), where (\beta(t)=(-1)^{(n-3)/2}t^{2-n-l}) for odd (n), (\beta(t)=(-1)^t l\Gamma^{-1}(l+n-2)\delta^{(n-3+l)}(t)) for even (n).

4. Integral representation of spherical functions. Let
[
V={v\in R^n:[v,v]=0,\ v_2=1},\qquad V^\pm=V\cap{v_1\gtrless0},\qquad \tilde\eta=\eta_2v.
]

Continuous series. To the generalized functions (\theta_p^\pm(\eta)=(\eta_2)\pm^\sigma) there correspond (see §1) the spherical functions
[
\Phi\rho^\pm(x)=\int_Y [x^0,\tilde\eta]\pm^\sigma [x,\tilde\eta]\pm^{\bar\sigma}\,d\eta
=\int_{V^\pm}[x,v]_+^{\bar\sigma}\,dv
\qquad
\left(dv=\frac{dv_3\ldots dv_n}{|v_1|}\right).
\tag{1}
]

Discrete series. From (\theta_l(\eta)) we obtain (see §1) the spherical functions
[
\Psi_l(x)=p_{nl}\int_Y [x_0,\tilde\eta]+^l\,\beta_l([x,\tilde\eta])\,d\eta
=p\beta_l([x,v])\,dv,}\int_{V^+
\tag{2}
]
where (p_{nl}=2^{-\tau}\pi^{(4-n)/2}/\Gamma(l+1)). (For (n=3), (\Psi_l) is the sum of two spherical functions.) The integrals (1), (2) are understood in the sense of the regularized value, see §5.

5. Regularization of integrals. Consider the integral
[
R_\lambda(x)=\int_{V^+}[x,v]+^\lambda\,dv,
]
where (\lambda) is complex, (x\in X). If (|x_2|>1), then it converges absolutely for values of (\lambda) lying in a certain domain, and is analytically continued to the points (\sigma) of interest to us from (1). If (|x_2|<1), then (R\lambda(x)) diverges for all (\lambda). However, then the integral
[
R_{\lambda,\mu}(x)=\int_{V^+}[x,v]_+^\lambda |v_1|^\mu\,dv
]
converges absolutely for (\operatorname{Re}(\lambda+\mu)<3-n), (\operatorname{Re}\lambda>-1), and is analytically continued to the point (\lambda=\sigma,\ \mu=0). We proceed analogously with the integrals (2) and (3).

6. Expression of the spherical functions in terms of Legendre functions.

Continuous series. Let (\nu=(n-3)/2), (E(\rho)=\dfrac{2}{\pi}(2\pi)^\nu|\Gamma(\sigma+1)|^2). Then
[
\frac{\Phi_\rho^+(x)}{E(\rho)}
=
\begin{cases}
-\,e^{-i\nu\pi}\sin\sigma\pi\,(x_2^2-1)^{-\nu/2}Q_{\nu+\sigma}^{\nu}(x_2),
& \text{for } x_2>1,\ x_1>0,\[2mm]
-\,e^{-i\pi\nu}\sin\bar\sigma\pi\,(x_2^2-1)^{-\nu/2}Q_{-\nu-\bar\sigma-1}^{\nu}(x_2),
& \text{for } x_2>1,\ x_1<0,\[2mm]
\dfrac{\pi}{2}(1-x_2^2)^{-\nu/2}P_{\nu+\sigma}^{\nu}(-x_2),
& \text{for } |x_2|<1,\[2mm]
0,
& \text{for } x_2<-1,\ x_1>0,\[2mm]
\pi\cos\pi\nu\,(x_2^2-1)^{-\nu/2}P_{\nu+\sigma}^{\nu}(-x_2),
& \text{for } x_2<-1,\ x_1<0.
\end{cases}
]

The expressions for (\Phi_\rho^{-}(x)) are obtained by replacing (x_1) by (-x_1).

Discrete series. If we denote
[
f(z)=(z^2-1)^{-\nu/2}Q_{l+\nu}^{\nu}(z),
]
then
[
\Psi_l(x)=2^{\nu-l}\overline{\gamma_l}\Gamma^{-1}(l+n-2)\,[f(x_2+i0)-(-1)^n f(x_2-i0)].
]
All spherical functions are locally integrable.

7. Expansion of the (\delta)-function in spherical functions. Let (\delta(x)) denote the (\delta)-function on (X), concentrated at (x^0). For the expansion of (\delta(x)) in spherical functions we use the method of M. Riesz, as in ((^1,^2)). Take the generalized function of one variable (a)
[
s(a,\lambda)=\frac{1}{2^{2\lambda}\Gamma(\lambda+\tfrac12)}\,|a-1|^{2\lambda}a_{+}^{-\lambda-\nu},\qquad \nu=(n-3)/2,
]
and put
[
S(x,\lambda)=\int_Y s([x,v],\lambda)\,dv,\qquad x\in X. \tag{3}
]
The integral is understood in the sense of the regularization of Sect. 5. It is computed explicitly:
[
S(x,\lambda)={\cos\pi\lambda\cdot(1-x_2){-}^{\lambda}+(1-x_2)}^{\lambda}}S_1(x)S_2(\lambda), \tag{4
]
where (S_1(x)=(x_2+1)^{-\nu}) for odd (n); (S_1(x)=(x_2+1){+}^{-\nu}) for even (n);
[
S_2(\lambda)=\pi^{-1/2-\nu}2^{-\lambda+\nu+1}\Gamma(-\nu-\lambda).
]
Let (K(\lambda,\rho)) be such that
[
s(a,\lambda)=\intK(\lambda,\rho)\,d\rho.}^{\infty} a_{+}^{(2-n+i\rho)/2
]
Then, denoting (\Phi_\rho=\Phi_\rho^{+}+\Phi_\rho^{-}), we obtain from (1), (2)
[
S(x,\lambda)=\int_{-\infty}^{\infty}K(\lambda,\rho)\overline{\Phi_\rho(x)}\,d\rho. \tag{5}
]
Put (\lambda=-\nu-1). Using the results of (7), Ch. III, § 2, item 2, and also the relation (\Phi_\rho=\Phi_{-\rho}), we obtain from (4) and (5)
[
\varepsilon_n\delta(x)=\tfrac12\pi^{-\nu-2}\Gamma(\nu+1)L(x)+\int_{-\infty}^{\infty}\omega(\rho)\overline{\Phi_\rho(x)}\,d\rho,
]
where
[
L(x)=(x_2-1)^{-1}(1-x_2^2)^{-\nu}\quad\text{for odd }n;\qquad
L(x)=(x_2-1)^{-1}(1-x_2^2){+}^{-\nu}\quad\text{for even }n;
]
(\varepsilon_n=\tfrac12) or (1), respectively, and
[
2^{2n-3}\pi^{\,n-1}\omega(\rho)=
\begin{cases}
\rho\tanh\dfrac{\pi\rho}{2}\cdot\displaystyle\prod,\[1.2em]}^{\nu}\bigl[(2k-1)^2+\rho^2\bigr], & n\ \text{odd
\displaystyle\prod_{k=0}^{[\nu]}(4k^2+\rho^2), & n\ \text{even}.
\end{cases}
]
We expand (L(x)) in (\Psi_l(x)), for which purpose we use the expressions of (\Psi_l) in terms of Legendre functions. Finally:
[
\delta(x)=\sum_{l=0}^{\infty}a_l\overline{\Psi_l(x)}+\int_{-\infty}^{\infty}\omega(\rho)\overline{\Phi_\rho(x)}\,d\rho,
]
where
[
a_l=\tfrac12\pi^{-n/2-1}(2l+n-2)\Gamma(l+n-2)\,4^{\,2-n-l}.
]

8. An analogue of the Plancherel formula. Let (C_c^\infty(X)) be the space of compactly supported infinitely differentiable functions on (X). We shall call the following functions on (Y) the Fourier components of a function (f\in C_c^\infty(X)):
[
F_\rho^{\pm}(\eta)=\int f(x)[x,\eta]_{\pm}^{\sigma}\,dx,\qquad
\sigma=\frac{2-n+i\rho}{2},
]
[
F_l(\eta)=\int f(x)\beta_l([x,\eta])\,dx.
]

For each (\rho) or (l), these formulas define a mapping of (C_c^\infty(X)) into (D_\tau), (\tau=i\rho) or (\tau=2-n-2l), respectively. Under this, the shifts by (g) pass into (T_g^\tau).

From the expansion of (\delta(x)) and the integral representations (1), (2), it follows that (f\in C_c^\infty(X)) is reconstructed from its Fourier components as follows:

[
f(x)=\sum_{l=0}^{\infty} a_l B\bigl(F_l,\beta_l,([x,\widetilde{\eta}])\bigr)
+\int_{-\infty}^{\infty}\omega(\rho){(F_\rho^+,[x,\widetilde{\eta}]+^\sigma)+(F\rho^-[x,\widetilde{\eta}]_-^\sigma)}\,d\rho.
]

The correspondence
[
f\to {F_\rho^+,\,F_\rho^-,\,F_l}
]
can be extended to
[
L^2(X)\to {H_\rho,\,H_\rho,\,H_l}.
]
Thus, the representation in (L^2(X)) of the group (G) decomposes into representations of the continuous series with multiplicity 2 and representations of the discrete series with multiplicity 1. There is an analogue of the Plancherel formula

[
\int |f(x)|^2\,dx
=
\sum_{l=0}^{\infty} a_l B(F_l,F_l)
+
\int_{-\infty}^{\infty}\omega(\rho){(F_\rho^+,F_\rho^+)+(F_\rho^-,F_\rho^-)}\,d\rho,
]

where (a_l,\omega_\rho) are determined by the formulas preceding (7).

The author expresses gratitude to his adviser F. A. Berezin for useful discussions of the work.

Moscow State University
named after M. V. Lomonosov

Received
3 III 1966

REFERENCES

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