MATHEMATICS

B. D. ROMM

AN ANALOGUE OF THE PLANCHEREL FORMULA FOR THE REAL UNIMODULAR GROUP OF THE THIRD ORDER

(Presented by Academician I. M. Vinogradov on 7 IX 1964)

For the first time, the problem of decomposing the regular representation into irreducible representations for the real unimodular group was solved in papers (¹, ²) for the case of the second order. In paper (²) the Plancherel formula was obtained. In the present note, for simplicity and brevity of exposition, the Plancherel formula is given for the group of the third order \(G_3\). However, the method used by the author extends to the real unimodular group of any order.

The irreducible representations of the principal series were constructed in paper (³) for the group of any order. We briefly describe these representations in the case of the group \(G_3\).

Let \(K\) denote the subgroup in the group \(G_3\) consisting of all matrices \(k=\|a_{ij}\|\), \(i,j=1,2,3\), where \(a_{31}=a_{32}=0\), and let \(X\) be the subgroup for whose matrices \(x=\|\xi_{ij}\|\) one has \(\xi_{11}=\xi_{22}=\xi_{33}=1\), \(\xi_{21}=0\), \(\xi_{ij}=0\) for \(i<j\). For every \(g\in G_3\) and almost all \(x\in X\) there is a decomposition \(xg=k_1x_1\) with

\[ k_1\in K,\quad x_1\in X. \]

Let \(k_1=\|a'_{ij}\|\), \(z_1=\dfrac{a'_{11}z+a'_{21}}{a'_{12}z+a'_{22}}\). Denote

\[ \dot z= \begin{pmatrix} 1&0&0\\ z&1&0\\ 0&0&1 \end{pmatrix}, \qquad \operatorname{Im} z\ne 0. \]

\(\dot z_1\) denotes the same as \(\dot z\), except that \(z\) is replaced by \(z_1\); \(\Lambda=a'_{11}a'_{22}-a'_{12}a'_{21}\); \(\beta(K)\) is the ratio of the left-invariant measure on the group \(K\) to the right-invariant one, normalized so that \(\beta(e)=1\), where \(e\) is the identity in the group \(G_3\).

Consider the totality of all functions \(f(\dot z x)\) on the manifold of products \(\dot z x\), analytic in \(z\) for \(\operatorname{Im} z\ne 0\) and measurable with respect to the parameters \(\xi_{31}\) and \(\xi_{32}\) of the matrix \(x\), for which the integral

\[ \int |f(\dot z x)|^2 |\operatorname{Im} z|^{\,n-2}\,d\xi_{31}d\xi_{32}dz,\qquad dz=dx\,dy,\quad z=x+iy,\quad n=2,3,4,\ldots, \]

converges for some fixed \(n\). Let \(\rho\in(-\infty,+\infty)\), \(\varepsilon=0,1\); then the operator

\[ T_g f(\dot z x)=(a'_{12}z+a'_{22})^{-n}|\Lambda|^{i\rho+n/2}(\operatorname{sgn}\Lambda)^\varepsilon\beta^{-1/2}(K_1)f(\dot z_1x_1) \tag{1} \]

defines a unitary representation, irreducible in the Hilbert space \(\mathscr H_n'\) of functions \(f(\dot z x)\) of the indicated type, where the scalar product is determined by the formula

\[ (f,f_1)=\int f(\dot z x)\overline{f_1(\dot z x)}|\operatorname{Im} z|^{\,n-2}d\xi_{31}d\xi_{32}. \]

In paper (³) \(\varepsilon=0\). This series of irreducible representations is denoted there by \(d_1\).

Let us construct one more series of irreducible unitary representations of the group \(G_3\). Let \(K_0\) denote the subgroup of all matrices \(k_0=\|v_{ij}\|\) in \(G_3\), where \(v_{ij}=0\) if \(i>j\); \(X_0\), the subgroup of all matrices \(x=\|\theta_{ij}\|\), where \(\theta_{ij}=0\) for \(i<j\), \(\theta_{11}=\theta_{22}=\theta_{33}=1\). By analogy with the preceding case, for almost all \(x_0\in X_0\) and for each fixed \(g\in G_3\) we have \(x_0g=k_0'x_0'\), with \(k_0'\in K_0\), \(x_0'\in X_0\); \(\beta_0(K_0)\) is the ratio of the left-invariant measure on the group \(K_0\) to the right-invariant one, with \(\beta_0(e)=1\); \(\varepsilon_1,\varepsilon_2=0,1\); \(\rho_1,\rho_2\in(\infty,+\infty)\), \(k_0'=\|v_{ij}'\|\). The operator

\[ T_g f(x_0)=|v_{11}'|^{i\rho_1}(\operatorname{sgn} v_{11}')^{\varepsilon_1}|v_{22}'|^{i\rho_2}(\operatorname{sgn} v_{22}')^{\varepsilon_2}\beta^{-1/2}(k_0')f(x_0') \tag{2} \]

in the Hilbert space \(L_2(X_0)\) of functions \(f(x_0)\) measurable on the group \(X_0\) and with summable square of the modulus with respect to the invariant measure on \(X_0\), defines an irreducible unitary representation of the group \(G_3\). The totality of all these representations in (3) is denoted by \(d_0\).

The series \(d_0\) and \(d_1\) exhaust all representations that enter into the decomposition of the regular representation into irreducibles.

Let us indicate the normalization of the invariant measure in the Plancherel formula. If \(d\) denotes the sign of differentiation, then the exterior product of the differential forms \(g_{ij}\), where \((i,j)\ne(3,3)\), \(dg\cdot g^{-1}=\|g_{ij}\|\), \(g\in G_3\), forms an invariant measure \(d\mu(g)\) on \(G_3\).

As is known, for a finite infinitely differentiable function \(x(g)\) on the group \(G_3\) and an irreducible unitary representation \(g\mapsto T_g\), the operator

\[ T_x=\int x(g)T_g\,d\mu(g) \]

has a trace in the corresponding representation space. When \(T_g\) is defined by formula (1), we shall denote this trace by \(\operatorname{Sp}(T_{x,n,\rho,\varepsilon})\), and in the case (2), by \(\operatorname{Sp}(T_{x,\rho_1,\rho_2,\varepsilon_1,\varepsilon_2})\).

The Plancherel formula has the form

\[ \begin{aligned} x(e)=\frac{1}{2^8\pi^5}\Bigg\{& \sum_{k=2}^{\infty}(k-1)\int\big[(k-1)^2+4\rho^2\big] \big[\operatorname{Sp}(T_{x,k,\rho,0})+\operatorname{Sp}(T_{x,k,\rho,1})\big]\,d\rho+ \\ &+2\iint_{\rho_1>\rho_2>0}\rho_1\rho_2(\rho_1-\rho_2) \left[\operatorname{th}\frac{\pi(\rho_1-\rho_2)}{2}+\operatorname{th}\frac{\pi\rho_2}{2}-\operatorname{th}\frac{\pi\rho_1}{2}\right]\times \\ &\qquad\times \operatorname{Sp}(T_{x,\rho_1,\rho_2,0,0})\,d\rho_1d\rho_2 +2\iint_{\rho_1>\rho_2>0}\rho_1\rho_2(\rho_1-\rho_2) \left[\operatorname{cth}\frac{\pi(\rho_1-\rho_2)}{2}+\right. \\ &\qquad\left.+\operatorname{th}\frac{\pi\rho_2}{2}-\operatorname{cth}\frac{\pi\rho_1}{2}\right] \operatorname{Sp}(T_{x,\rho_1,\rho_2,1,0})\,d\rho_1d\rho_2 +2\iint_{\rho_1>\rho_2>0}\rho_1\rho_2(\rho_1-\rho_2)\times \\ &\qquad\times\left[\operatorname{cth}\frac{\pi(\rho_1-\rho_2)}{2}+\operatorname{cth}\frac{\pi\rho_2}{2}-\operatorname{th}\frac{\pi\rho_1}{2}\right] \operatorname{Sp}(T_{x,\rho_1,\rho_2,0,1})\,d\rho_1d\rho_2+ \\ &+2\iint_{\rho_1>\rho_2>0}\rho_1\rho_2(\rho_1-\rho_2) \left[\operatorname{th}\frac{\pi(\rho_1-\rho_2)}{2}+\operatorname{cth}\frac{\pi\rho_2}{2}-\operatorname{cth}\frac{\pi\rho_1}{2}\right]\times \\ &\qquad\times \operatorname{Sp}(T_{x,\rho_1,\rho_2,1,1})\,d\rho_1d\rho_2 \Bigg\}. \end{aligned} \]

Moscow Institute of Physics and Technology

Received
27 VIII 1964

REFERENCES

1. V. Bargmann, Ann. Math., 48, 568 (1947).
2. Harish-Chandra, Proc. Nat. Acad. Sci., 38, No. 4 (1952).
3. I. M. Gelfand, M. I. Graev, Izv. AN SSSR, ser. matem., 17, 189 (1953).