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MATHEMATICS
B. D. ROMM
DECOMPOSITION INTO IRREDUCIBLE REPRESENTATIONS OF THE RESTRICTION OF REPRESENTATIONS OF THE PRINCIPAL SERIES OF THE PROPER LORENTZ GROUP TO THE REAL LORENTZ GROUP
(Presented by Academician L. S. Pontryagin on 14 III 1963)
I. The real Lorentz group \(G_0\) is a subgroup of the proper Lorentz group \(G_+\); \(G_0\) consists of all those transformations of 4-dimensional space (3 Cartesian coordinates and a time coordinate) that do not change one of the Cartesian coordinates. In the present work the question of decomposition is solved only for the principal series of irreducible representations of \(G_+\).
The group \(G_+\) is locally isomorphic to the group \(G\) of all complex matrices of the second order with determinant equal to one. \(G_+\) is a homomorphic image of \(G\) \(\left({}^{1}\right), \S 9\). Under this homomorphism, in \(G_0\) there passes the subgroup of all real matrices in \(G\). In representation theory, instead of \(G_+\) one considers \(G\), and \(G_0\) is replaced by the corresponding group of real matrices, which we shall also denote by \(G_0\).
The author uses the notation of the papers \(\left({}^{1,2}\right)\).
II. Let us explain the formulation of the problem. We consider the principal series of unitary representations of the group \(G\) \(\left({}^{1}\right), \S 10\) in the Hilbert space \(L_2(z)\) of functions \(f(z)\), defined on the complex plane \(z\), measurable, with summable square modulus with respect to the measure \(dz = dx\,dy\), where \(x = \operatorname{Re} z\), and \(y = \operatorname{Im} z\).
Let \(g=\begin{pmatrix}\alpha&\beta\\ \gamma&\delta\end{pmatrix}\in G\); \(T_g\) denotes the operator of the representation under consideration corresponding to \(g\). According to \(\left({}^{1}\right), \S 10\),
\[ T_g f(z)=(\beta z+\delta)^{-m}\,|\beta z+\delta|^{m+i\rho-2}\,f(\widetilde{zg}), \tag{1} \]
where \(f(z)\in L_2(z)\), and
\[ \widetilde{zg}=\frac{\alpha z+\gamma}{\beta z+\delta}, \tag{2} \]
\(m=0,\pm1,\pm2,\ldots\), and \(\rho\) is any real number. \(m\) and \(\rho\) define a representation from the principal series.
The totality of the operators \(T_g\) for \(g\in G_0\) defines a certain representation of the group \(G_0\), which is called the restriction of the initial representation to \(G_0\). We denote this representation of the group \(G_0\) by \(\mathfrak{S}_{m,\rho}\). The problem consists in decomposing \(\mathfrak{S}_{m,\rho}\) into irreducible representations.
III. The solution of this problem follows from the fact that the representation \(\mathfrak{S}_{m,\rho}\) in the case \(m\ne0\) is equivalent to a certain part of the regular representation of the group \(G_0\), i.e. \(L_2(z)\) can be mapped isometrically onto a certain subspace \(P\) of the space \(L_2(G_0)\) (\(P\) depends on \(m\)) in such a way that the representation \(\mathfrak{S}_{m,\rho}\) goes over into the representation by right translations on the group \(G_0\). The representation \(\mathfrak{S}_{0,\rho}\) is equivalent to the tensor product of two representations, each of which is a certain part of the regular representation.
It is clear that the given method of solution essentially uses the Plancherel formula for the group \(G_0\). We shall use the results of the paper \((^2)\), where this formula is actually obtained. In addition, we shall be interested in the irreducible representations of the principal series of the group \(G_0\), since the decomposition of the representation \(\mathfrak{S}_{m,\rho}\) is carried out with respect to them. Therefore, before formulating the final result, we shall describe these representations.
IV. In the paper \((^2)\) a realization of the group \(G_0\) is used in the form of the group \(G_1\) of complex matrices \(\begin{pmatrix}\alpha&\beta\\ \bar\beta&\bar\alpha\end{pmatrix}\), where \(\bar\alpha\) is the number complex-conjugate to \(\alpha\), \(\alpha\bar\alpha-\beta\bar\beta=1\). The connection between the groups \(G_0\) and \(G_1\) is established by the formula
\[ a_0=t^{-1}g_0t,\qquad \text{where } t=\begin{pmatrix}1&-i\\ -i&1\end{pmatrix},\qquad g_0\in G_0,\quad a_0\in G_1 . \tag{3} \]
The group \(G_1\) has two continuous and two discrete series of irreducible representations, which belong to the class of principal series. The continuous series are denoted respectively by \(C_q^0\) and \(C_q^{1/2}\); each representation of these series is uniquely determined by a number \(q\geq 1/4\).
The discrete series are denoted by \(D_k^+\) and \(D_k^-\); a representation from these series is determined by a number \(k=1,\ ^3/2,\ 2,\ldots\).
A representation of the series \(C_q^0\) and \(C_q^{1/2}\) is realized in the Hilbert space \(\mathfrak{H}\) of all functions \(f(\Phi)\), measurable on the interval \([-\pi,\pi]\) with square-integrable modulus with respect to the measure \(d\Phi/2\pi\). The scalar product of vectors \(g\) and \(f\in\mathfrak{H}\) has the form
\[ (g,f)=\frac1{2\pi}\int_{-\pi}^{\pi}\overline{g(\Phi)}\,f(\Phi)\,d\Phi . \tag{4} \]
The operator of the representation \(C_q^0\) \(\bigl(C_q^{1/2}\bigr)\) is denoted by \(T_{is}(a)\) \(\bigl(T'_{is}(a)\bigr)\), where \(s=\sqrt{q-1/4}\), and has the form
\[ T_{is}(a)f(\Phi)=|\bar\alpha-\beta e^{i\Phi}|^{-1-2is}f(a^{-1}\Phi), \]
\[ T'_{is}(a)f(\Phi)=(\bar\alpha-\beta e^{i\Phi})^{-1}|\bar\alpha-\beta e^{i\Phi}|^{-2is}f(a^{-1}\Phi), \tag{5} \]
where \(a=\begin{pmatrix}\alpha&\beta\\ \bar\beta&\bar\alpha\end{pmatrix}\), and the transformation \(\Phi\mapsto a\Phi\) on the interval \([-\pi,\pi]\) is determined from the condition
\[ e^{i(a\Phi)}=\frac{\bar\alpha e^{i\Phi}+\bar\beta}{\alpha+\beta e^{i\Phi}} \quad ((^2),\ \S 6,\ 7). \]
Representations of the series \(D_p^+\) and \(D_p^-\) are realized in the Hilbert space \(\mathfrak{H}_{2p}\) of functions \(f(w)\), analytic in the unit disk of the complex plane \((^2,\ \S 9)\). The scalar product of functions \(f\) and \(g\in\mathfrak{H}_{2p}\) is given by the formula
\[ (f,g)=\frac{2p-1}{\pi}\int_{|w|<1}(1-w\bar w)^{2p-2}\overline{f(w)}\,g(w)\,dw, \tag{6} \]
where \(dw=du\,dv\) for \(u=\operatorname{Im} w\) and \(v=\operatorname{Im} w\).
The operator \(T_p^+(a)\) of the representation \(D_p^+\) has the form
\[ T_p^+(a)f(w)=(-\beta w+\bar\alpha)^{-2p}f\left(\frac{\alpha w-\bar\beta}{-\beta w+\bar\alpha}\right). \tag{7} \]
The operator \(T_p^{-}(a)\) of the representation \(D_p^{-}\) is defined by the formula
\[ T_p^{-}(a)=T_p^{+}(\bar a) \tag{7a} \]
for \(\bar a=\begin{pmatrix}\bar\alpha&\bar\beta\\[2pt]\beta&\alpha\end{pmatrix}\), if \(a=\begin{pmatrix}\alpha&\beta\\[2pt]\bar\beta&\bar\alpha\end{pmatrix}\).
V. The representation \(\mathfrak S_{m,\rho}\) is uniquely determined by the number \(m\) and does not depend on \(\rho\), i.e. \(\mathfrak S_{m,\rho_1}\) and \(\mathfrak S_{m,\rho_2}\) are equivalent for \(\rho_1\ne\rho_2\). We indicate formulas that effect the decomposition of the representation \(\mathfrak S_{m,\rho}\) into irreducible representations. Put
\[ b=\frac{1}{2\sqrt{\operatorname{Im}z}} \begin{pmatrix} i\bar z+1&\bar z+i\\ z-i&-iz+1 \end{pmatrix}, \qquad \operatorname{Im}z>0. \tag{8} \]
It is easy to see that \(b\in G_1\). Denote
\[ K_m(z,\Phi,s)= \begin{cases} \dfrac{1}{2\sqrt{\pi}}\,|\operatorname{Im}z|^{\frac{i\rho}{2}-1}\,T_{is}(b^{-1})e^{i\frac{m}{2}\Phi}, & \text{for even }m,\\[8pt] \dfrac{1}{2\sqrt{\pi}}\,|\operatorname{Im}z|^{\frac{i\rho}{2}-1}\,T'_{is}(b^{-1})e^{i\frac{m-1}{2}\Phi}, & \text{for odd }m, \end{cases} \tag{9} \]
where \(m=0,\pm1,\ldots\), and \(T_{is}(a)\) and \(T'_{is}(a)\) are defined by (5);
\[ K_{mp}^{\pm}(z,w)=(-1)^{m-p} \sqrt{\frac{\pi(m+2p-1)!}{(2p-1)(m-p)!(p-1)!}}\, |\operatorname{Im}z|^{\frac{i\rho}{2}-1}T_p^{\pm}(b^{-1})w^{m-p}, \tag{10} \]
where \(m=p,p+1,\ldots;\quad p=1,\ ^3/_2,\ 2,\ldots;\)
\[ \Lambda= \begin{cases} +, & \text{if } m>0,\\ -, & \text{if } m<0, \end{cases} \qquad m=\pm2,\pm3,\ldots; \]
\[ p= \begin{cases} \left|\dfrac{m}{2}\right|,\ \left|\dfrac{m}{2}\right|-1,\ldots,1, & \text{if }m\text{ is even},\\[6pt] \left|\dfrac{m}{2}\right|,\ \left|\dfrac{m}{2}\right|-1,\ldots,\ ^3/_2, & \text{if }m\text{ is odd}. \end{cases} \tag{11} \]
Theorem. Let \(f(z)\in L_2(z)\). Then the integrals
\[ g_p(w)=\int_{\operatorname{Im}z>0}\overline{K_{\left|\frac{m}{2}\right|p}^{\Lambda}(z,w)}\,f(z)\,dz, \qquad \tilde g_p(w)=\int_{\operatorname{Im}z<0}\overline{K_{\left|\frac{m}{2}\right|p}^{\Lambda}(z,w)}\,f(z)\,dz \tag{12} \]
converge in \(\mathfrak H_{2p}\) for every \(p\) in (11). If \(\mathfrak H\) \((\mathfrak H')\) denotes the Hilbert space of all measurable functions \(f(\Phi;s)\) for \(\Phi\in[-\pi,\pi]\), \(s\in[0,+\infty)\) with summable square of the modulus with respect to the measure \(d\Phi\,ds\) with weight \(\dfrac{\operatorname{cth}\pi s}{s}\) \(\left(\dfrac{\operatorname{th}\pi s}{s}\right)\), then the integrals
\[ f(\Phi,s)=\frac{1}{2\sqrt{\pi}}\int_{\operatorname{Im}z>0}K_m(z,\Phi,s)f(z)\,dz, \tag{13} \]
\[ \bar f(\Phi,s)=\frac{1}{2\sqrt{\pi}}\int_{\operatorname{Im}z<0}K_m(\bar z,\Phi,s)f(z)\,dz \tag{14} \]
converge in \(\mathcal H'\) for odd \(m\) and converge in \(\mathcal H\) for even \(m\). Moreover, for almost all \(z\),
\[ f(z)= \begin{cases} \displaystyle \sum_p' \frac{2p-1}{\pi} \int_{|w|<1} K_{\left|\frac m2\right|p}^{\Lambda}(z,w) g_p(w)(1-\overline{w}w)^{2p-2}\,dw \\[6pt]\displaystyle\qquad +\,2\sqrt{\pi}\int_0^{+\infty}\int_{-\pi}^{\pi} K_m(z,\Phi,s) f(\Phi,s)\,d\Phi\,ds, & \text{for } \operatorname{Im} z>0; \\[12pt] \displaystyle \sum_p' \frac{2p-1}{\pi} \int_{|w|<1} K_{\left|\frac m2\right|p}^{\Lambda}(\overline z,w)\widetilde g_p(w)(1-\overline{w}w)^{2p-2}\,dw \\[6pt]\displaystyle\qquad +\,2\sqrt{\pi}\int_0^{+\infty}\int_{-\pi}^{\pi} K_m(\overline z,\Phi,s)\widetilde f(\Phi,s)\,d\Phi\,ds, & \text{for } \operatorname{Im} z<0, \end{cases} \tag{15} \]
where \(\sum_p'\) denotes summation over all \(p\) in (11):
\[ \int |f(z)|^2\,dz = \sum_p' \left[ (g_p(w),g_p(w))+ (\widetilde g_p(w),\widetilde g_p(w)) \right] + (f(\Phi,s),f(\Phi,s)) + (\widetilde f(\Phi,s),\widetilde f(\Phi,s)). \tag{16} \]
When \(f(z)\) passes into \(T_{g_0}f(z)\), \(g_p(w)\) and \(\widetilde g_p(w)\) are transformed respectively into \(T_p^\Lambda(a_0)g\) and \(T_p^{-\Lambda}(a_0)\widetilde g\), while \(f(\Phi,s)\) and \(\widetilde f(\Phi,s)\)—respectively into \(T_{is}(a_0)f\), \(T_{is}(a_0)\widetilde f\) or \(T'_{is}(a_0)f\), \(T'_{is}(a_0)\widetilde f\).
Here \(f\) and \(\widetilde f\in\mathcal H\) \((\mathcal H')\) for almost every fixed \(s\in[0,+\infty)\), if the function \(f\) \((\widetilde f)\) is regarded as a function of \(\Phi\). Thus, in the space \(\mathcal H\) \((\mathcal H')\), a direct continuous sum of the representations \(C_q^0\) \((C_q^{1/2})\) is realized. Formula (16) is the Plancherel formula for the representation \(\mathfrak S_{m,\rho}\). This formula may be rewritten in the form
\[ L_2(z) = \mathfrak h_{\left|m\right|} \oplus \mathfrak h_{\left|m\right|-2} \oplus\cdots\oplus \mathfrak h_2 \oplus \mathcal H \oplus \mathfrak h_{\left|m\right|} \oplus \mathfrak h_{\left|m\right|-2} \oplus\cdots\oplus \mathfrak h_2 \oplus \mathcal H \quad \text{for even }m; \tag{17} \]
\[ L_2(z) = \mathfrak h_{\left|m\right|} \oplus \mathfrak h_{\left|m\right|-2} \oplus\cdots\oplus \mathfrak h_3 \oplus \mathcal H' \oplus \mathfrak h_{\left|m\right|} \oplus \mathfrak h_{\left|m\right|-2} \oplus\cdots\oplus \mathfrak h_3 \oplus \mathcal H' \quad \text{for odd }m. \]
Received
26 II 1963
REFERENCES CITED
- M. A. Naimark, Linear Representations of the Lorentz Group, Moscow, 1958.
- V. Bargmann, Ann. Math., 48, 568 (1947).