B. D. ROMM

DECOMPOSITION INTO IRREDUCIBLE REPRESENTATIONS OF THE TENSOR PRODUCT OF TWO IRREDUCIBLE REPRESENTATIONS OF THE REAL UNIMODULAR GROUP OF SECOND ORDER (THE CASE OF TWO DISCRETE SERIES)

(Presented by Academician P. S. Novikov, 12 VI 1963)

The real unimodular group of second order \(G\) is realized, as in \((^{1})\), in the form of the group \(G_0\) of all unimodular complex matrices of second order leaving invariant the form \(z_1\bar z_1 - z_2\bar z_2\). The groups \(G\) and \(G_0\) are isomorphic. The isomorphism is established by the formula

\[ g = tat^{-1}, \tag{1} \]

where \(a \in G_0,\ g \in G\),

\[ t = \begin{pmatrix} 1 & -i\\ -i & 1 \end{pmatrix}. \tag{2} \]

The irreducible representations of the discrete series are constructed in spaces of analytic functions on the unit disk of the complex plane. Let \(p = \tfrac12, 1, \tfrac32,\ldots\); let \(\mathfrak{H}\) denote the set of all polynomials \(f(z)=a_0+a_1z+\cdots+a_nz^n\) of arbitrary degrees. On the linear set \(\mathfrak{H}\) we define a system of scalar products \((f,g)_p\) for \(f\) and \(g \in \mathfrak{H}\) by

\[ (f,g)_p = \lim_{k\to p+0} \frac{2k-1}{\pi} \int_{|z|<1} \overline{f(z)}\,g(z)\,(1-z\bar z)^{2k-2}\,dz; \tag{3} \]

\[ dz = dx\,dy,\quad \text{for } x=\operatorname{Re} z,\ y=\operatorname{Im} z,\quad p=\tfrac12,1,\tfrac32,\ldots . \]

The closure of the set \(\mathfrak{H}\) in the scalar product (3) forms a Hilbert space, which we denote by \(\mathfrak{H}_l,\ l=2p\). In the space \(\mathfrak{H}_l\) we define a unitary representation \(D_p^+\) of the group \(G_0\) by the formula \((^{1})\)

\[ T^+(a)f(z) = (-\beta z+\bar\alpha)^{-2p} f\!\left(\frac{\alpha z-\bar\beta}{-\beta z+\bar\alpha}\right), \quad a= \begin{pmatrix} \alpha & \beta\\ \bar\beta & \bar\alpha \end{pmatrix} \in G_0. \tag{4} \]

If \(a\in G_0\), then denote by \(\bar a\) the matrix
\[ \begin{pmatrix} \bar\alpha & \bar\beta\\ \beta & \alpha \end{pmatrix}. \]
Put

\[ T^-(a)=T^+(\bar a). \tag{5} \]

The operators \(T^-(a)\) in the space \(\mathfrak{H}_l\) define a certain unitary representation of the group \(G_0\), which we denote by \(D_p^-\).

In the paper we consider tensor products of two representations of the type \(D_p^+\) and \(D_{p_1}^+\), as well as \(D_p^-\) and \(D_{p_1}^-\). The first of these we denote by \(D_{pp_1}^+\), and the second by \(D_{pp_1}^-\). The representations \(D_{pp_1}^+\) and \(D_{pp_1}^-\) are realized in the tensor product \(\mathfrak{H}_{ll_1}\) of the spaces \(\mathfrak{H}_l\) and \(\mathfrak{H}_{l_1}\), \(l_1=2p_1\), which is obtained as a com-

the closure of the set \(\mathfrak H''\) of all polynomials in \(z\) and \(z_1\) in the scalar product

\[ (f,g)_{pp_1}=\lim_{\substack{k\to p+0\\ k_1\to p_1+0}} \frac{(2k-1)(2k_1-1)}{\pi^2} \iint_{\substack{|z|<1\\ |z_1|<1}} \overline{f(z_1,z_2)}\,g(z_1,z_2)(1-z_1\bar z_1)^{2k-2}\times \]

\[ {}\times(1-z_2\bar z_2)^{2k_1-2}\,dz_1dz_2. \tag{6} \]

The series of operators

\[ K_m(z,z_1,z_2)=\frac{1}{m!} \left\{(l+l_1+2m-1)!\left[(l-1)!(l_1-1)!\times\right.\right. \]

\[ \left.\left.{}\times\sum_{p=0}^{m} \frac{1}{(l-1+p)!(l_1-1+m+p)!\,p!\,(m-p)!} \right]^{-1}\right\}^{1/2} \exp\left\{z\left(\frac{\partial}{\partial z_1}+\frac{\partial}{\partial z_2}\right)\right\} \tag{7} \]

converges strongly in the space \(\mathfrak H_{l_1}\) on the set \(\mathfrak H''\) for any \(m=0,1,2,\ldots\) for each fixed \(z\).

For any fixed \(z_1\) and \(z_2\), the series of operators

\[ K'_m(z,z_1,z_2)= \frac{(l+l_1+2m-1)!}{m!} \left\{(l+l_1+2m-1)!\left[(l-1)!(l_1-1)!\times\right.\right. \]

\[ \left.\left.{}\times\sum_{k=0}^{m} \frac{1}{(l-1+k)!(l_1-1+m+k)!\,k!\,(m-k)!} \right]^{-1}\right\}^{1/2}\times \]

\[ {}\times\sum_{n=0}^{\infty} \frac{(z_1^2\partial/\partial z_1+z_2^2\partial/\partial z_2+l z_1+l_1 z_2)^n\,(z_1-z_2)^m} {(l+l_1+2m+n-1)!\,n!}\, \frac{\partial^n}{\partial z^n} \tag{8} \]

converges strongly in the space \(\mathfrak H_p\) on \(\mathfrak H\), \(p=\tfrac12,1,\tfrac32,\ldots\).

Theorem. Let \(f(z_1,z_2)\in\mathfrak H''\). Then all functions \(\varphi_m(z)\), where \(m=0,1,2,\ldots\), of the form

\[ \varphi_m(z)=\frac{(l-1)(l_1-1)}{\pi^2} \iint_{\substack{|z_1|<1\\ |z_2|<1}} \overline{(z_1-z_2)}^{\,m} (1-z_1\bar z_1)^{l-2}(1-z_2\bar z_2)^{l_1-2}\times \]

\[ {}\times K_m(z,z_1,z_2)f(z_1,z_2)\,dz_1dz_2 \tag{9} \]

belong to \(\mathfrak H\). Conversely, let a finite number of the polynomials \(\varphi_m(z)\), \(m=0,1,2,\ldots\), be different from zero. Then the formula

\[ f(z_1,z_2)=\sum_{m=0}^{\infty} \frac{l+l_1+2m-1}{\pi} \int_{|z|<1} (1-z\bar z)^{l+l_1+2m-2} K'_m(z,z_1,z_2)\varphi_m(z)\,dz \tag{10} \]

is the inverse of the mapping (9). This mapping admits a closure, where \(f(z_1,z_2)\in\mathfrak H_{ll_1}\), \(\varphi_m(z)\in\mathfrak H_{2(p+p_1+m)}\), and

\[ (f,f)_{pp_1}=\sum_{m=0}^{\infty}(\varphi_m,\varphi_m)_{p+p_1+m}. \tag{11} \]

When \(f(z_1,z_2)\) is transformed according to the representation \(D^+_{pp_1}\) \((D^-_{pp_1})\), \(\varphi_m(z)\) is transformed according to \(D^+_{p+p_1+m}\) \((D^-_{p+p_1+m})\).

Formula (11) is the Plancherel formula.

The author expresses deep gratitude to D. P. Zhelobenko and M. A. Naimark for valuable comments.

Received
30 V 1963

REFERENCES

1. V. Bargmann, Ann. Math., 48, 568 (1947).