MATHEMATICS

M. A. NAIMARK

ON THE DECOMPOSITION OF IRREDUCIBLE REPRESENTATIONS OF THE PRINCIPAL SERIES OF THE COMPLEX UNIMODULAR GROUP OF ORDER \(n\) INTO REPRESENTATIONS OF THE COMPLEX UNIMODULAR GROUP OF SECOND ORDER

(Presented by Academician S. L. Sobolev, 2 IV 1958)

Let \(g \to T_g\) be a continuous irreducible unitary representation of a topological group \(G\), and let \(H\) be a certain subgroup of the group \(G\). The restriction of the representation \(g \to T_g\) to the group \(H\) is a unitary representation \(h \to T_h\) of the group \(H\), generally speaking reducible. The problem arises of decomposing the representation \(h \to T_h\) into irreducible representations of the group \(H\).

The results of the author’s preceding paper \((^1)\) make it possible to solve this problem for the case when \(G\) is the complex unimodular group \(A_n\) of order \(n\), \(g \to T_g\) is a representation of the principal series of this group, and \(H\) is the group of all matrices \(g \in A_n\) such that \(g_{pq}=\delta_{pq}\) for \(p>2\) or \(q>2\) (\(\delta_{pq}\) is the Kronecker symbol), so that \(H\) is isomorphic to the group \(A_2\).

For simplicity of exposition we shall confine ourselves here to the case \(n=3\). In this case (see \((^2)\), item 2, § 5) the representations of the principal series are realized in the Hilbert space \(L^2(Z_3)\) of all measurable functions \(f(z)=f(z_{21},z_{31},z_{32})\) on the group \(Z_3\)—the group of all matrices

\[ z= \left\| \begin{array}{ccc} 1 & 0 & 0\\ z_{21} & 1 & 0\\ z_{31} & z_{32} & 1 \end{array} \right\|, \tag{1} \]

satisfying the condition*

\[ \|f\|^2=\int |f(z)|^2\,dz =\int |f(z_{21},z_{31},z_{32})|^2\,dz_{21}dz_{32}dz_{32}<\infty, \]

and the operators \(T_g\) of the representation are given by the formula

\[ T_g f(z)= |g_2^1|^{m_1+i\rho_1-2} (\overline{g_2^1})^{-m_1} |g_3^1|^{m_2-m_1+i(\rho_2-\rho_1)-2} (\overline{g_3^1})^{-m_2+m_1} f(z\overline{g}), \tag{2} \]

where \(g_p^1\) is the minor formed from the last \(p\) rows and columns of the matrix \(g^1=zg\); \(z\overline{g}\) is the matrix \(z^1\in Z_3\), determined by the relation \(zg=kz^1\); \(k_{pq}=0\) for \(p>q\); \(m_1,m_2\) are integers; \(\rho_1,\rho_2\) are real numbers determining the given representation \(g\to T_g\) (the parameters of the representation).

* As in the preceding paper \((^1)\), \(\int \varphi(z_1,z_2,\ldots,z_m)\,dz_1\cdots dz_m\), where \(z_1,\ldots,z_m\) are complex variables, denotes the \(2m\)-fold integral

\[ \int_{-\infty}^{\infty}\cdots\int_{-\infty}^{\infty} \varphi(x_1+iy_1,\ldots,x_m+iy_m)\,dx_1dy_1\cdots dx_mdy_m . \]

If, in particular, \(g\in H\), i.e.

\[ g=\left\|\begin{matrix} g_{11}&g_{12}&0\\ g_{21}&g_{22}&0\\ 0&0&1 \end{matrix}\right\|, \]

then, by virtue of (1),

\[ g^1=zg=\left\|\begin{matrix} g_{11}&g_{12}&0\\ g_{11}z_{21}+g_{21}&g_{12}z_{21}+g_{22}&0\\ g_{11}z_{31}+g_{21}z_{32}&g_{12}z_{31}+g_{22}z_{32}&1 \end{matrix}\right\|, \tag{3} \]

hence \(g_2^1=g_{12}z_{21}+g_{22}\), \(g_3^1=1\). Moreover, applying to (3) formula (5.22) §5 in \(({}^2)\), we obtain that

\[ z_{21}^1=\frac{g_{11}z_{21}+g_{21}}{g_{12}z_{21}+g_{22}},\qquad z_{31}^1=g_{11}z_{31}+g_{21}z_{32},\qquad z_{32}^1=g_{12}z_{31}+g_{22}z_{32}, \]

and substitution in (2) shows that the restriction of the given representation to the group \(H\) is given by the formula

\[ T_g f(z_{21},z_{31},z_{32})= \]

\[ =\left|g_{12}z_{21}+g_{22}\right|^{m_1+i\rho_1-2} \left(g_{12}z_{21}+g_{22}\right)^{-m_1}\times \]

\[ \times f\left( \frac{g_{11}z_{21}+g_{21}}{g_{12}z_{21}+g_{22}}, g_{11}z_{31}+g_{21}z_{32}, g_{12}z_{21}+g_{22}z_{32} \right). \tag{4} \]

Formula (4) means that the representation \(g\to T_g\) of the group \(H\) is the tensor product of the irreducible representation \(g\to T_g^{(1)}\) of the group \(A_2\), where

\[ T_g^{(1)}f_1(z)=\left|g_{12}z+g_{22}\right|^{m_1+i\rho_1-2} \left(g_{12}z+g_{22}\right)^{-m_1} f_1\left(\frac{g_{11}z+g_{21}}{g_{12}z+g_{22}}\right), \]

and the quasiregular representation (see \(({}^3)\), p. 421) \(g\to T_g^{(2)}\) of this group:

\[ T_g^{(2)}f_2(z_1,z_2)=f_2(g_{11}z_1+g_{21}z_2,\ g_{12}z_1+g_{22}z_2). \tag{5} \]

Therefore we shall obtain the decomposition of the representation (4) into irreducible representations if we first decompose into irreducible representations \(g\to T_g^\chi\) the quasiregular representation (5), and then decompose into irreducible representations the tensor products \(T_g^{(1)}\times T_g^\chi\).

To obtain the first decomposition, put

\[ \zeta=\frac{z_{31}}{z_{32}},\qquad f(z_{21},z_{31},z_{32})=f(z_{21},\zeta z_{32},z_{32}) =\varphi(z_{21},\zeta,z_{32})|z_{32}|^{-2}, \tag{6} \]

\[ \psi(z_{21},\zeta,\chi')=\int \varphi(z_{21},\zeta,z_{32})|z_{32}|^{-2}\chi'(z_{32})\,dz_{32}, \tag{7} \]

where \(\chi'(z)=|z|^{m'+i\rho'}z^{-m'}\). The correspondence \(f\to\psi\), established by formulas (6), (7), is an isometric mapping \(U\) of the space \(L^1(Z_3)\) onto the Hilbert space \(\mathfrak H_\psi\) of all measurable functions \(\psi(z_{21},\zeta,\chi')\) satisfying the condition

\[ \|\psi\|^2=\int |\psi(z_{21},\zeta,\chi')|^2\,dz_{21}\,d\zeta\,d\chi'<\infty; \]

here the integral with respect to \(d\chi'\) is taken over the whole group of characters \(X\) of the multiplicative group of complex numbers, and \(d\chi'\) denotes \(\dfrac{1}{(2\pi)^2}\,d\rho'\). This mapping \(U\) transforms the operators \(T_g\) into operators in the space \(\mathfrak H_\psi\), which we shall again denote by \(T_g\).

But, under the passage from \(f\) to \(T_g f\) according to formula (4), the function \(\varphi(z_{21},\zeta_1,z_{32})\) passes into

\[ |z_{32}|^2\left|g_{12}z_{21}+g_{22}\right|^{m_1+i\rho_1-2} (g_{12}z_{21}+g_{22})^{-m_1}+ \]

\[ +\varphi\left( \frac{g_{11}z_{21}+g_{21}}{g_{12}z_{21}+g_{22}}, \frac{g_{11}z_{31}+g_{21}z_{32}}{g_{12}z_{31}+g_{22}z_{32}}, g_{12}z_{31}+g_{22}z_{32} \right) |g_{12}z_{31}+g_{22}z_{32}|^{-2}= \]

\[ =\left|g_{12}z_{21}+g_{22}\right|^{m_1+i\rho_1-2} (g_{12}z_{21}+g_{22})^{-m_1} |g_{21}\zeta+g_{22}|^{-2}\times \]

\[ \times \varphi\left( \frac{g_{11}z_{21}+g_{21}}{g_{12}z_{21}+g_{22}}, \frac{g_{11}\zeta+g_{21}}{g_{12}\zeta+g_{22}}, (g_{12}\zeta+g_{22})z_{32} \right), \]

therefore, \(\psi(z_{21},\zeta,\chi')\) passes into

\[ \left|g_{12}z_{21}+g_{22}\right|^{m_1+i\rho_1-2} (g_{12}z_{21}+g_{22})^{-m_1}|g_{12}\zeta+g_{22}|^{-2}\times \]

\[ \times \int \varphi\left( \frac{g_{11}z_{21}+g_{21}}{g_{12}z_{21}+g_{22}}, \frac{g_{11}\zeta+g_{21}}{g_{12}\zeta+g_{22}}, (g_{12}\zeta+g_{22})z_{32} \right)|z_{32}|^{-2}\overline{\chi'(z_{32})}\,dz_{32}= \]

\[ =\left|g_{12}z_{21}+g_{22}\right|^{m_1+i\rho_1-2} (g_{12}z_{21}+g_{22})^{-m_1}|g_{12}\zeta+g_{22}|^{-2}\times \]

\[ \times \int \varphi\left( \frac{g_{11}z_{21}+g_{21}}{g_{12}z_{21}+g_{22}}, \frac{g_{11}\zeta+g_{21}}{g_{12}\zeta+g_{22}}, z_{32} \right)|z_{32}|^{-2} \overline{\chi'((g_{12}\zeta+g_{22})^{-1}z_{32})}\,dz_{32}= \]

\[ =\left|g_{12}z_{21}+g_{22}\right|^{m_1+i\rho_1-2} (g_{12}z_{21}+g_{22})^{-m_1} |g_{12}\zeta+g_{22}|^{m'+i\rho'-2}(g_{12}\zeta+g_{22})^{-m'}\times \]

\[ \times \psi\left( \frac{g_{11}z_{21}+g_{21}}{g_{12}z_{21}+g_{22}}, \frac{g_{11}\zeta+g_{21}}{g_{12}\zeta+g_{22}}, \chi' \right), \]

so that

\[ T_g\psi(z_{21},\zeta,\chi')= \left|g_{12}z_{21}+g_{12}\right|^{m_1+i\rho_1-2} (g_{12}z_{21}+g_{22})^{-m_1} |g_{12}\zeta+g_{22}|^{m'+i\rho'-2}\times \]

\[ \times (g_{12}\zeta+g_{22})^{-m'} \psi\left( \frac{g_{11}z_{21}+g_{21}}{g_{12}z_{21}+g_{22}}, \frac{g_{11}\zeta+g_{21}}{g_{12}\zeta+g_{22}}, \chi' \right). \tag{8} \]

Formula (8) means that the representation \(g\to T_g\) in \(\mathfrak{H}_{\psi}\) is the continuous sum over \(\chi'\) of tensor products \(T_g^{(1)}\times T_g^{\chi'}\), where

\[ T_g^{\chi'}f(\zeta)= |g_{12}\zeta+g_{22}|^{m'+i\rho'-2} (g_{12}\zeta+g_{22})^{-m'}f\left(\frac{g_{11}\zeta+g_{21}}{g_{12}\zeta+g_{22}}\right). \]

Therefore it remains to carry out the decomposition into irreducible representations of each of these tensor products according to the formulas of the article [1]. For this purpose denote by \(\mathfrak{F}\) the set of all pairs \((\chi,\chi')\in X\times X\) for which \(m_1+m'+m\) is an even integer, and put, for \((\chi,\chi')\in\mathfrak{F}\),

\[ F(z,\chi,\chi')=F(z,m,\rho,m'\rho')= \int \psi(z_{21},\zeta,\chi')|\zeta-z_{21}|^{-\frac{m+m_1+m'}{2}-i\frac{\rho+\rho_1+\rho'}{2}}\times \]

\[ \times(\zeta-z_{21})^{\frac{m+m_1+m'}{2}} |z-z_{21}|^{\frac{m-m_1+m'}{2}+i\frac{\rho-\rho_1+\rho'}{2}-1} (z-z_{21})^{-\frac{m-m_1+m'}{2}}\times \]

\[ \times|\zeta-z|^{\frac{m+m_1-m'}{2}+i\frac{\rho+\rho_1-\rho'}{2}-1} (\zeta-z)^{-\frac{m+m_1+m'}{2}}\,dz_{21}d\zeta; \tag{9} \]

the correspondence \(\psi\to F\), established by formula (9), is an isometric mapping of the space \(\mathfrak{H}_{\psi}\) onto the Hilbert space \(\mathfrak{H}_{F}\) of all measurable functions \(F(z,\chi,\chi')\), \((\chi,\chi')\in\mathfrak{F}\), satisfying the conditions:

1)

\[ \|F\|^2=\frac1{4\pi^2}\iint_{\mathfrak{F}}(m^2+\rho^2) \left[\int |F(z,\chi,\chi')|^2\,dz\right]d\chi\,d\chi'<\infty; \]

2) the Fourier transforms

\[ \widetilde F(w,\chi,\chi')=(2\pi)^{-2}\int F(z,\chi,\chi')\exp(-i\operatorname{Re}(z\overline w))\,dz \]

of the functions \(F(z,\chi,\chi')\) for almost all \(w,\chi,\chi'\), \((\chi,\chi')\in\mathfrak{F}\), satisfy

relation

\[ \widetilde F(w,\chi^{-1},\chi')=(-i)^{-m2-i\rho}\times \]

\[ \times \frac{ \Gamma\!\left(\frac{m+m_1-m'}{2}+\frac{1}{2}-i\frac{\sigma+\sigma_1-\sigma'}{2}\right) \Gamma\!\left(\frac{-m-m_1+m'}{2}+\frac{1}{2}-i\frac{\sigma-\sigma_1+\sigma'}{2}\right) }{ \Gamma\!\left(\frac{m+m_1+m'}{2}+\frac{1}{2}+i\frac{\sigma+\sigma_1-\sigma'}{2}\right) \Gamma\!\left(\frac{-m-m_1+m'}{2}+\frac{1}{2}+i\frac{\sigma-\sigma_1+\sigma'}{2}\right) } \times \]

\[ \times |w|^{-m+i\rho} w^m \widetilde F(w,\chi,\chi'). \]

Under this mapping the representation \(g\to T_g\) goes over into a representation in \(\mathfrak H_F\), which we shall again denote by \(g\to T_g\); from the preceding arguments and the results of paper \({}^{1}\) it follows that the representation \(g\to T_g\) in the space \(\mathfrak H_F\) is given by the formula

\[ T_gF(z,\chi,\chi')= |g_{12}z+g_{22}|^{m+i\rho-2}(g_{12}z+g_{22})^{-m} F\!\left(\frac{g_{11}z+g_{21}}{g_{12}z+g_{22}},\chi,\chi'\right), \]

i.e. this representation is a continuous sum of representations \(T_g^{(\chi)}\). Consequently, combining formulas (6), (7), and (9), we arrive at the following result:

Theorem. For every function \(f\in L^2(Z_3)\) the integral

\[ F(z,\chi,\chi')=\int f(z_{21},z_{31},z_{32}) |z_{31}-z_{32}z_{21}|^{-\frac{m+m_1+m'}{2}-i\frac{\rho+\rho_1+\rho'}{2}-1}\times \]

\[ \times (z_{31}-z_{32}z_{21})^{\frac{m+m_1+m'}{2}} |z-z_{21}|^{\frac{m-m_1+m'}{2}+i\frac{\rho-\rho_1+\rho'}{2}-1} (z-z_{21})^{-\frac{m-m_1+m'}{2}}\times \]

\[ \times |z_{21}-z_{32}z|^{\frac{m+m_1-m'}{2}+i\frac{\rho+\rho_1-\rho'}{2}-1} (z_{21}-z_{32}z)^{-\frac{m+m_1-m'}{2}} \,dz_{21}\,dz_{31}\,dz_{32} \tag{10} \]

converges in the sense of the norm in \(\mathfrak H_F\), and formula (10) realizes an isometric mapping of the space \(L^2(Z_3)\) onto the space \(\mathfrak H_F\). Under this mapping, the restriction to the group \(H\sim A_2\) of the representation \(g\to T_g\) of the principal series of the group \(A_3\), defined by formula (2), goes over into a continuous sum of irreducible representations of the principal series \(g\to T_g^{(\chi)}\) of the group \(A_2\), so that in passing from \(f\) to \(T_gf\) the function \(F(z,\chi,\chi')\) goes over into

\[ T_g^{(\chi)}F(z,\chi,\chi')= |g_{12}z+g_{22}|^{m+i\rho-2} (g_{12}z+g_{22})^{-m} F\!\left(\frac{g_{11}z+g_{12}}{g_{21}z+g_{22}},\chi,\chi'\right), \]

where \(\chi(\lambda)=|\lambda|^{m+i\rho}\lambda^{-m}\).

We note that the theorem proved is applicable to other subgroups of the group \(A_3\) that are isomorphic to the group \(A_2\). For example, in the case of the subgroup of matrices

\[ g= \left\| \begin{array}{ccc} 1&0&0\\ 0&g_{22}&g_{23}\\ 0&g_{32}&g_{33} \end{array} \right\| \]

the corresponding decomposition is obtained from the decomposition for the subgroup \(H\) and the relation \(sH's=H\), where

\[ s= \left\| \begin{array}{ccc} 0&0&1\\ 0&-1&0\\ 1&0&0 \end{array} \right\|. \]

Received
31 III 1958

CITED LITERATURE

\({}^{1}\) M. A. Naimark, DAN, 119, No. 5 (1958). \({}^{2}\) I. M. Gel'fand, M. A. Naimark, Tr. Matem. inst. im. V. A. Steklova, 36, 1 (1956). \({}^{3}\) I. M. Gel'fand, M. A. Naimark, Izv. AN SSSR, ser. matem., 11, 411 (1947).