MATHEMATICS

M. A. NAIMARK

ON THE DECOMPOSITION INTO IRREDUCIBLE REPRESENTATIONS OF THE TENSOR PRODUCT OF TWO REPRESENTATIONS OF THE COMPLEMENTARY SERIES OF THE PROPER LORENTZ GROUP

(Presented by Academician A. N. Kolmogorov, 17 IX 1959)

In previous papers \((^{4-6})\) the author solved the problem of decomposing into irreducible representations the tensor product of two irreducible unitary representations of the proper Lorentz group in the two cases when both factors belong to the principal series, or when one of these representations belongs to the principal series and the other to the complementary series.

The present paper is devoted to the remaining case, when both factors belong to the complementary series. Namely, we determine here into which irreducible representations the tensor product \(\mathfrak D_{\nu_1}\times \mathfrak D_{\nu_2}\) of two representations \(\mathfrak D_{\nu_1}, \mathfrak D_{\nu_2}\) of the complementary series decomposes. Throughout this paper the notation and results of papers \((^4,^5)\) are used.

The representation \(\mathfrak D_{\nu_1}\times \mathfrak D_{\nu_2}\) is realized in the space \(\mathfrak H_{\nu_1}\times \mathfrak H_{\nu_2}\)—the completion with respect to the scalar product

\[ (f_1,f_2)=\int |z_1-z'_1|^{-2+\nu_1}|z_2-z'_2|^{-2+\nu_2} f_1(z_1,z_2)\overline{f_2(z'_1,z'_2)}\,dz_1\,dz_2\,dz'_1\,dz'_2 \]

of the space \(\mathfrak H'_{\nu_1}\times \mathfrak H'_{\nu_2}\) of all measurable functions \(f(z_1,z_2)\) for which

\[ \int |z_1-z'_1|^{-2+\nu_1}|z_2-z'_2|^{-2+\nu_2} |f(z_1,z_2)|\,|f(z'_1,z'_2)|\,dz_1\,dz'_1\,dz_2\,dz'_2<\infty; \]

the operators \(T_g\) of the representation \(\mathfrak D_{\nu_1}\times \mathfrak D_{\nu_2}\) on \(f\in \mathfrak H'_{\nu_1}\times \mathfrak H'_{\nu_2}\) are given by the formula

\[ T_g f(z_1,z_2)=|\beta z_1+\delta|^{-\nu_1-2} |\beta z_2+\delta|^{-\nu_2-2} f\left(\frac{\alpha z_1+\gamma}{\beta z_1+\delta}, \frac{\alpha z_2+\gamma}{\beta z_2+\delta}\right) \tag{1} \]

\[ \left(\text{for } g=\begin{pmatrix}\alpha&\beta\\ \gamma&\delta\end{pmatrix}\right) \]

and are then extended by continuity to all of \(\mathfrak H_{\nu_1}\times \mathfrak H_{\nu_2}\). The spaces \(\mathfrak H_\nu,\ \mathfrak H_{\nu_1}\times \mathfrak H_{\nu_2}\) can also be described as follows.

Let \(K(Z)\) be the set of all infinitely differentiable finite functions of \(z\) (i.e., of \(x=\operatorname{Re}z,\ y=\operatorname{Im}z\)). Clearly, \(K(Z)\) is dense in \(\mathfrak H'_\nu\), and hence also in \(\mathfrak H_\nu\). The Fourier transform

\[ \widetilde\varphi(w)=\frac{1}{2\pi}\int \varphi(z)e^{i\operatorname{Re}(z\overline w)}\,dz \]

is an isometric mapping \(F:\varphi\to \widetilde\varphi\) of the set \(K(Z)\subset \mathfrak H_\nu\) onto a dense subset of the Hilbert space \(\widetilde{\mathfrak H}_\nu\) of all measurable func-

functions \(\widetilde\varphi(w)\) for which \(\int |w|^{-\nu}|\widetilde\varphi(w)|^2\,dw<\infty\), with scalar product

\[ (\widetilde\varphi_1,\widetilde\varphi_2) = a(\nu)\int |w|^{-\nu}\widetilde\varphi_1(w)\overline{\widetilde\varphi_2(w)}\,dw, \]

where
\[ a(\nu)=2^\nu\pi\Gamma\left(\frac{\nu}{2}\right) \left[\Gamma\left(1-\frac{\nu}{2}\right)\right]^{-1} \]
(see (3), pp. 2 and 3, § 12). Consequently, \(F\) extends uniquely to an isometric mapping, which we again denote by \(F\), of the whole space \(\mathfrak H_\nu\) onto \(\widetilde{\mathfrak H}_\nu\). Therefore every element \(f\in\mathfrak H_\nu\) may be realized as a generalized function on \(K(Z)\), putting
\[ (f;\varphi)=\int \widetilde f(w)\widetilde\varphi(w)\,dw \]
for \(\varphi\in K(Z)\), \(\widetilde f=Ff,\ \widetilde\varphi=F\varphi\). It follows from this that \(\mathfrak H_\nu\) consists of precisely those generalized functions on \(K(Z)\) whose Fourier transforms are ordinary functions belonging to \(\widetilde{\mathfrak H}_\nu\).

Similarly one constructs an isometric mapping \(F\) of the space \(\mathfrak H_{\nu_1}\times\mathfrak H_{\nu_2}\) onto the space \(\widetilde{\mathfrak H}_{\nu_1}\times\widetilde{\mathfrak H}_{\nu_2}\) of all measurable functions \(\widetilde\varphi(w_1,w_2)\) for which
\[ \int |w_1|^{-\nu_1}|w_2|^{-\nu_2} |\widetilde\varphi(w_1,w_2)|^2\,dw_1\,dw_2<\infty \]
with scalar product

\[ (\widetilde\varphi_1,\widetilde\varphi_2) = a(\nu_1)a(\nu_2)\int |w_1|^{-\nu_1}|w_2|^{-\nu_2} \widetilde\varphi_1(w_1,w_2)\overline{\widetilde\varphi_2(w_1,w_2)}\,dw_1\,dw_2 . \]

The mapping \(F\) is the extension by continuity of the Fourier transform of functions \(\varphi(z_1,z_2)\in K(Z\times Z)\), where \(K(Z\times Z)\) is the totality of all finite infinitely differentiable functions of \(z_1,z_2\).

The space \(\mathfrak H_{\nu_1}\times\mathfrak H_{\nu_2}\) consists of precisely those generalized functions on \(K(Z\times Z)\) whose Fourier transforms are ordinary functions belonging to \(\widetilde{\mathfrak H}_{\nu_1}\times\widetilde{\mathfrak H}_{\nu_2}\).

Theorem. If \(\nu_1+\nu_2\leqslant 2\), then \(\mathfrak D_{\nu_1}\times\mathfrak D_{\nu_2}\) is unitarily equivalent to \(\mathfrak S_{m_1\sigma_1}\times\mathfrak S_{m_2\sigma_2}\) for even \(m_1+m_2\) (in particular, it is unitarily equivalent to \(\mathfrak S_{00}\times\mathfrak S_{00}\)) and therefore there is a continuous sum of representations \(\mathfrak S_{m\sigma}\) with even \(m\); if, however, \(\nu_1+\nu_2>2\), then \(\mathfrak D_{\nu_1}\times\mathfrak D_{\nu_2}\) is unitarily equivalent to the direct sum of the representation \(\mathfrak D_{\nu_1+\nu_2-2}\) and \(\mathfrak S_{m_1\sigma_1}\times\mathfrak S_{m_2\sigma_2}\) for even \(m_1+m_2\), and therefore there is a direct sum of the representation \(\mathfrak D_{\nu_1+\nu_2-2}\) and a continuous sum of representations \(\mathfrak S_{m\sigma}\) with even \(m\).

We outline the proof of this theorem. First consider the case \(\nu_1=\nu_2\). Put \(\nu=\nu_1=\nu_2\) and define an operator \(A\) from \(\mathfrak H_\nu\times\mathfrak H_\nu\) into \(L^2(Z\times Z)\) with domain
\[ D=\{f:f\in\mathfrak H_\nu'\times\mathfrak H_\nu',\ |z_1-z_2|^\nu f\in L^2(Z\times Z)\}, \]
by setting, for \(f\in D\), \(Af=f'\), where
\[ f'(z_1,z_2)=|z_1-z_2|^\nu f(z_1,z_2); \]
\(D\) is dense in \(\mathfrak H_\nu\times\mathfrak H_\nu\), since \(D\supset K(Z\times Z)\). It is easy to verify that \(D\) is invariant with respect to the operators \(T_g\) of the representation \(\mathfrak D_\nu\times\mathfrak D_\nu\), and that for \(f\in D\)

\[ AT_g f=T'_g Af, \tag{2} \]

where \(g\mapsto T'_g\) is the representation \(\mathfrak S_{00}\times\mathfrak S_{00}\). It can be shown that:

1) \(A^*\) is defined on a set \(D^*\) dense in \(L^2(Z)\) (namely, \(D^*\) contains the set \(K_0\) of all functions \(\varphi\in K(Z\times Z)\) whose support does not contain the diagonal \(z_1=z_2\)), and therefore \(A\) admits a closure.

2) The orthogonal complement \(R^{*\perp}\) of the range \(R^*\) of the operator \(A^*\) consists of precisely those generalized functions \(f\in\mathfrak H_\nu\times\mathfrak H_\nu\) which vanish on \(K_0\), hence which are concentrated on the diagonal \(z_1=z_2\) and therefore have the form
\[ f(z_1,z_2)=\sum_{kj}\varphi_{kj}(z_2)D_j^{(k)}\delta(z_1-z_2), \]
where \(D_j^{(k)}\) denote derivatives of \(k\)-th order with respect to \(x=\operatorname{Re}z,\ y=\operatorname{Im}z\) (see (1), Chapter II, §§ 4 and 5). Hence, from the condition \(f\in\mathfrak H_\nu\times\mathfrak H_\nu\) (therefore,

\(\tilde f \in \tilde{\mathfrak H}_\nu \times \tilde{\mathfrak H}_\nu\) it follows that, for \(\nu \leq 1\), \(R^{*\perp}=(0)\), and therefore \(R^*\) is dense in \(\mathfrak H_\nu \times \mathfrak H_\nu\), while for \(\nu>1\), \(R^{*\perp}\) consists of*

\[ f(z_1,z_2)=\delta(z_1-z_2)\psi(z_2),\qquad \psi(z_2)\in \mathfrak H_{2\nu-2}. \tag{3} \]

It follows from (3) that, for \(\nu>1\): a) \(R^{*\perp}\) is invariant with respect to the operators of the representation \(\mathfrak D_\nu\times \mathfrak D_\nu\); b) the correspondence \(f\to\psi\), established by formula (3), defines an isometric mapping of \(R^{*\perp}\) onto \(\mathfrak H_{2\nu-2}\), under which the restriction of \(\mathfrak D_\nu\times \mathfrak D_\nu\) to \(R^{*\perp}\) passes into \(\mathfrak D_{2\nu-2}\). From these listed properties of the operator \(A\) and the generalized Schur lemma (see \((^2)\), theorem 1, item 2, § 21) we conclude that the assertion of the theorem is valid for \(\nu_1=\nu_2\).

Let now \(\nu_1\ne\nu_2\), and, for definiteness, let \(\nu_1<\nu_2\); put \(\nu=\nu_2-\nu_1\). Define an operator \(A\) from \(\mathfrak H_{\nu_1}\times \mathfrak H_{\nu_2}\) into \(L^2(Z)\times \mathfrak H_\nu\) with domain of definition
\[ D=\{f;\ f\in \mathfrak H'_{\nu_1}\times \mathfrak H'_{\nu_2};\ |z_1-z_2|^{\nu_1}f\in L^2(Z)\times \mathfrak H'_\nu\}, \]
putting, for \(f\in D\), \(Af=f'\), where \(f'(z_1,z_2)=|z_1-z_2|^{\nu_1}f(z_1,z_2)\). Again \(D\) is invariant with respect to the operators \(T_g\), and again (2) is satisfied, where now \(g\to T_g\) is the representation \(\mathfrak S_{00}\times \mathfrak D_\nu\). The operator \(A^*\) is defined on a dense set \(D^*\), and for the orthogonal complement \(R^{*\perp}\) of the range \(R^*\) of the operator \(A\) we have:

\(1')\) \(R^{*\perp}=(0)\), and therefore \(R^*\) is dense in \(\mathfrak H_{\nu_1}\times \mathfrak H_{\nu_2}\) for \(\nu_1+\nu_2\leq 2\);

\(2')\) \(R^{*\perp}\) consists of all functions

\[ f(z_1,z_2)=\delta(z_1-z_2)\psi(z_2),\qquad \psi(z_2)\in \mathfrak H_{\nu_1+\nu_2-2} \tag{4} \]

for \(\nu_1+\nu_2>2\).

In case \(2')\): \(a')\) \(R^{*\perp}\) is invariant with respect to the operators of the representation \(\mathfrak D_{\nu_1}\times \mathfrak D_{\nu_2}\); \(b')\) the correspondence \(f\to\psi\), established by formula (4), is an isometric mapping of \(R^{*\perp}\) onto \(\mathfrak H_{\nu_1+\nu_2-2}\), under which the restriction of \(\mathfrak D_{\nu_1}\times \mathfrak D_{\nu_2}\) to \(R^{*\perp}\) passes into \(\mathfrak D_{\nu_1+\nu_2-2}\). Therefore, applying the generalized Schur lemma, we conclude that, for \(\nu_1+\nu_2\leq 2\), the representation \(\mathfrak D_{\nu_1}\times \mathfrak D_{\nu_2}\) is unitarily equivalent to the representation \(\mathfrak S_{00}\times \mathfrak D_\nu\), while for \(\nu_1+\nu_2>2\) it is equivalent to the direct sum of the representations \(\mathfrak S_{00}\times \mathfrak D_\nu\) and \(\mathfrak D_{\nu_1+\nu_2-2}\).

But, on the other hand, by virtue of the principal result in \((^5)\), the representation \(\mathfrak S_{00}\times \mathfrak D_\nu\) is unitarily equivalent to the representation \(\mathfrak S_{00}\times \mathfrak S_{00}\). Hence the assertion of the theorem follows for \(\nu_1\ne\nu_2\).

Remark 1. Let \(\nu_1=\nu_2=\nu\).

Denote by \(\mathcal H_\nu\) the Hilbert space of all measurable functions \(f(z_1,z_2)\) for which
\[ \int |z_1-z_2|^{2\nu}|f(z_1,z_2)|^2\,dz_1\,dz_2<\infty \]
with scalar product
\[ (f_1,f_2)_1=\int |z_1-z_2|^{2\nu} f_1(z_1,z_2)\overline{f_2(z_1,z_2)}\,dz_1\,dz_2. \]
Define in \(\mathcal H_\nu\) the representation \(\mathfrak E_\nu\) by the same formula (1); it is easy to verify that \(\mathfrak E_\nu\) is a unitary representation of the group \(\mathfrak G\).

The correspondence \(f(z_1,z_2)\to |z_1-z_2|^\nu f(z_1,z_2)\) is an isometric mapping of \(\mathcal H_\nu\) onto \(L^2(Z)\), carrying \(\mathfrak E_\nu\) into \(\mathfrak S_{00}\times \mathfrak S_{00}\); therefore, from the theorem proved, we conclude that, for \(\nu\leq 1\), the representation \(\mathfrak D_\nu\times \mathfrak D_\nu\) is unitarily equivalent to \(\mathfrak E_\nu\), while for \(\nu>1\) the representation \(\mathfrak D_\nu\times \mathfrak D_\nu\) is unitarily equivalent to the direct sum of the representations \(\mathfrak E_\nu\) and \(\mathfrak D_{2\nu-2}\).

Let now \(\nu_1\ne\nu_2\), and, for definiteness, let \(\nu_1<\nu_2\). Denote by \(\mathcal H'_{\nu_1\nu_2}\) the totality of all measurable functions \(f(z_1,z_2)\) for which

\[ \int |z_1-z'_1|^{-2+\nu_2-\nu_1}|z_1-z_2|^{\nu_1}|z'_1-z'_2|^{\nu_1}|f(z_1,z_2)|\,|f(z'_1,z'_2)|\,dz_1\,dz_2\,dz'_2<\infty \]

* (3) means that, for \(\varphi\in K(Z\times Z)\),
\[ (f;\varphi(z_1,z_2))=(\psi;\varphi(z,z)). \]

with the scalar product

\[ (f_1,f_2)_2=\int |z_1-z'_1|^{-2+\nu_2-\nu_1}|z'_1-z_2|^{\nu_1}|z_1-z'_2|^{\nu_1} f_1(z_1,z_2)f_2(z_1,z'_2)\,dz_1\,dz_2\,dz'_2, \]

and by \(\mathscr H'_{\nu_1\nu_2}\) the completion of \(\mathscr H_{\nu_1\nu_2}\) with respect to the scalar product \((f_1,f_2)_2\). Define in \(\mathscr H_{\nu_1\nu_2}\) the representation \(\mathfrak S_{\nu_1\nu_2}\) by the same formula (1); it is easy to verify that \(\mathfrak S_{\nu_1\nu_2}\) is a unitary representation of the group \(\mathfrak G\). The correspondence \(f(z_1,z_2)\to |z_1-z_2|^{\nu_1} f(z_1,z_2)\) is an isometric mapping of \(\mathscr H_{\nu_1\nu_2}\) onto \(L^2(Z)\times \mathfrak H_{\nu_2-\nu_1}\), carrying \(\mathfrak S_{\nu_1\nu_2}\) into \(\mathfrak S_{00}\times \mathfrak D_{\nu_2-\nu_1}\); therefore, from the theorem proved above we conclude that for \(\nu_1+\nu_2\leq 2\) the representation \(\mathfrak D_{\nu_1}\times \mathfrak D_{\nu_2}\) is unitarily equivalent to \(\mathfrak S_{\nu_1\nu_2}\), while for \(\nu_1+\nu_2>2\) the representation \(\mathfrak D_{\nu_1}\times \mathfrak D_{\nu_2}\) is unitarily equivalent to the direct sum of the representations \(\mathfrak S_{\nu_1\nu_2}\) and \(\mathfrak D_{\nu_1+\nu_2-2}\).

Remark 2. The fact that for \(\nu_1+\nu_2>2\) the representation \(\mathfrak D_{\nu_1}\times \mathfrak D_{\nu_2}\) must contain \(\mathfrak D_{\nu_1+\nu_2-2}\) was noted by I. M. Gel'fand, on the basis of the following consideration.

Let the representation \(\mathfrak D_{\nu_1}\times \mathfrak D_{\nu_2}\) decompose into a direct sum of representations \(\mathfrak S_{m\sigma}\), and let \(f_1\in\mathfrak H_{\nu_1}\), \(f_2\in\mathfrak H_{\nu_2}\) be normalized elements invariant with respect to the restrictions of \(\mathfrak D_{\nu_1}\), \(\mathfrak D_{\nu_2}\) to the unitary subgroup. Then \((T_\varepsilon(f_1\times f_2),f_1\times f_2)\) is equal to the product of the spherical functions of the representations \(\mathfrak D_{\nu_1}\), \(\mathfrak D_{\nu_2}\), i.e.

\[ (T_\varepsilon(f_1\times f_2),f_1\times f_2) =\frac{4}{\nu_1\nu_2}\frac{\operatorname{sh}\nu_1t\cdot\operatorname{sh}\nu_2t}{\operatorname{sh}^2 2t} \quad\text{for }\varepsilon= \begin{pmatrix} e^{-t}&0\\ 0&e^t \end{pmatrix}. \]

On the other hand, applying to \(\mathfrak D_{\nu_1}\times\mathfrak D_{\nu_2}\) the decomposition into the representations \(\mathfrak S_{m\sigma}\), we obtain that

\[ \frac{4\operatorname{sh}\nu_1t\,\operatorname{sh}\nu_2t}{\nu_1\nu_2\operatorname{sh}^2 2t} =(T_\varepsilon(f_1\times f_2),f_1\times f_2) =\int_0^\infty \frac{2}{\rho}\frac{\sin\rho t}{\operatorname{sh}2t}\,d\mu(\rho), \]

where \(\mu\) is some nondecreasing function on \([0,\infty)\) and

\[ \int_0^\infty d\mu(\rho)=1, \]

while \(\dfrac{2}{\rho}\dfrac{\sin\rho t}{\operatorname{sh}2t}\) is the spherical function of the representation \(\mathfrak S_{0\rho}\). Hence

\[ \frac{4\operatorname{sh}\nu_1t\,\operatorname{sh}\nu_2t}{\nu_1\nu_2\operatorname{sh}2t} =\int_0^\infty \frac{2}{\rho}\sin\rho t\,d\mu(\rho), \tag{5} \]

but this is impossible for \(\nu_1+\nu_2>2\), since then the left-hand side is unbounded as \(t\to\infty\), whereas the right-hand side is \(\leq 2/\rho\). If, however, from

\[ \frac{4\operatorname{sh}\nu_1t\,\operatorname{sh}\nu_2t}{\nu_1\nu_2\operatorname{sh}^2 2t} \]

one subtracts the product by

\[ 2(\nu_1+\nu_2-2)\nu_1^{-1}\nu_2^{-1} \]

of the spherical function

\[ \frac{2}{\nu_1+\nu_2-2}\, \frac{\operatorname{sh}(\nu_1+\nu_2-2)t}{\operatorname{sh}2t} \]

of the representation \(\mathfrak D_{\nu_1+\nu_2-2}\), then the product by \(\operatorname{sh}2t\) of the resulting difference will already be bounded, and for it a representation in the form of the integral on the right-hand side of (5) is already possible.

Moscow Institute of Physics and Technology

Received
16 IX 1959

CITED LITERATURE

¹ I. M. Gel'fand, G. E. Shilov, Generalized Functions, vol. 2, Moscow, 1958.
² M. A. Naimark, Normed Rings, Moscow, 1956.
³ M. A. Naimark, Linear Representations of the Lorentz Group, Moscow, 1958.
⁴ M. A. Naimark, DAN, 119, No. 5, 782 (1958).
⁵ M. A. Naimark, DAN, 125, No. 6 (1959).
⁶ M. A. Naimark, Transactions of the Moscow Mathematical Society, 8, 121 (1959).