UDC 513.882

MATHEMATICS

I. Ya. VAKHUTINSKII

UNITARY REPRESENTATIONS OF THE GROUP \(GL(3,R)\) OF REAL NONSINGULAR MATRICES OF THIRD ORDER

(Presented by Academician P. S. Aleksandrov, 11 XII 1965)

In this note all irreducible unitary representations of the group \(GL(3,R)\) are listed.

Consider the subgroup \(K \subset GL(3,R)\) of matrices \(\|a_{ij}\|\) \((i=1,2,3)\) with the condition \(a_{31}=a_{32}=0\). For a matrix \(k \in K\), denote by \(A(k)\) the diagonal block of size \(2\times 2\) situated in the left upper corner. Suppose that a representation of the subgroup \(K\) has the form

\[ k\to |a_{33}|^{i\rho}\operatorname{sgn}^{\varepsilon}(a_{33})T_{A(k)}, \tag{1} \]

where \(k\in K\); \(\rho\) is a real number; the index \(\varepsilon=0,1\); \(T_{A(k)}\) is any irreducible unitary representation of the group \(GL(3,R)\).

Theorem 1. Every irreducible unitary representation of the group \(GL(3,R)\) is either one-dimensional or equivalent to a representation induced from the stationary subgroup \(K\) and an inducing representation of the form (1).

For the classification of all representations of the group \(GL(3,R)\), it remains to indicate which of the representations appearing in Theorem 1 are equivalent. We first list all irreducible unitary representations of the group \(GL(2,R)\). This is easy to do, knowing all representations of the unimodular group \(SL(2,R)\), found by Bargmann \((^1)\). It turns out that there exist 4 series of irreducible unitary representations of the group \(GL(2,R)\).

I. The continuous series is realized in \(L^2(-\infty,\infty)\):

\[ T_{\left(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix}\right)}f(x) = f\left(\frac{ax+c}{bx+d}\right) \left|\frac{bx+d}{ad-bc}\right|^{-1+i\rho_1} \operatorname{sgn}^{\varepsilon_1}\left(\frac{bx+d}{ad-bc}\right) |ad-bc|^{i\rho_2}\operatorname{sgn}^{\varepsilon_2}(ad-bc). \]

II. The discrete series is realized in the space \(\mathcal H_s\) \((s=1,2,3,\ldots)\) of functions analytic in the upper and lower half-planes separately, with the condition \(\varphi\in\mathcal H_s\) if

\[ \int |\varphi(z)|^2 |\operatorname{Im} z|^{s-1}\,dz\,d\bar z<\infty, \]

\[ T_{\left(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix}\right)}f(z) = f\left(\frac{az+c}{bz+d}\right) \left(\frac{bz+d}{ad-bc}\right)^{-s-1} |ad-bc|^{i\rho}. \]

III. The supplementary series is realized in the space \(L_s\) \((0<s<1)\) of functions on the line with scalar product defined by the kernel \(|x_1-x_2|^{s-1}\):

\[ T_{\left(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix}\right)}f(x) = f\left(\frac{ax+c}{bx+d}\right) \left(\frac{bx+d}{ad-bc}\right)^{-s-1} |ad-bc|^{i\rho}\operatorname{sgn}^{\varepsilon}(ad-bc). \]

IV. The degenerate series consists of one-dimensional representations:

\[ T_{\left(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix}\right)} = |ad-bc|^{i\rho}\operatorname{sgn}^{\varepsilon}(ad-bc). \]

Representations of the group \(GL(3,R)\) obtained from representations of types I–IV will be called, respectively, the continuous, discrete, supplementary, and degenerate series.

In the case of the continuous series, the representation of the subgroup \(K\) itself is induced by a unitary representation of the subgroup \(K_0 \subset K\), consisting of upper triangular matrices. For this reason the representation of the continuous series of the group \(GL(3,R)\) is also induced by a unitary representation of \(K_0\). The inducing representation will be one-dimensional, of the form

\[ k_0 \to \prod_{i=1}^{3} |a_{ii}|^{\rho_i}\operatorname{sgn}^{\varepsilon_i}(a_{ii}), \qquad k_0 \in K_0 . \tag{2} \]

Theorem 2. Among the induced representations of the group \(GL(3,R)\) listed above, equivalence occurs only among representations of the continuous series. Let two representations of the subgroup \(K_0\) be given by the sets \((\rho_i)(\varepsilon_i)\) and \((\rho_i')(\varepsilon_i')\), respectively \((i=1,2,3;\) see (2)). Then the representations of the group \(GL(3,R)\) induced by them are equivalent if and only if the sets \((\rho_i')\) and \((\varepsilon_i')\) are obtained from the sets \((\rho_i)\) and \((\varepsilon_i)\), respectively, by one and the same permutation.

The method we use for finding all representations is based on a theorem of A. A. Kirillov \((^2)\), according to which the restriction of an irreducible unitary representation of the group \(GL(3,R)\) to the subgroup \(K\) remains irreducible. All representations of the subgroup \(K\) and the corresponding representations of the Lie algebra \(K\) are considered. From the commutation relations one finds the Lie operators of the missing one-parameter subgroups. The main difficulty here lies in the fact that, a priori, the Gårding space or any other natural domain of definition of the Lie operators for the group \(GL(3,R)\) is not known in advance. One device that makes it possible to overcome this difficulty consists in studying, among the missing Lie operators, the one that corresponds to a compact one-parameter subgroup. For this operator it turned out to be possible to find all self-adjoint extensions to which a representation of \(GL(3,R)\) may correspond.

We note that only the continuous and discrete series enter the Plancherel formula found by B. D. Romm \((^3)\). (In his work these series are given for the unimodular group and are denoted by \(d_0\) and \(d_1\), respectively; moreover, for the cases \(d_1\) with \(\varepsilon=0\) and \(\varepsilon=1\) (formula (1) of \((^3)\)) the representations are equivalent.) These basic series were first indicated by I. M. Gelfand and M. I. Graev in \((^4)\).

In conclusion I express my gratitude to Prof. A. A. Kirillov for his constant attention and advice in solving this problem.

Received
3 XII 1965

CITED LITERATURE

1. V. Bargmann, Ann. Math., 48, No. 3, 568 (1947).
2. A. A. Kirillov, DAN, 144, No. 1, 37 (1962).
3. B. D. Romm, DAN, 160, No. 6, 1269 (1965).
4. I. M. Gelfand, M. I. Graev, Izv. AN SSSR, 17, No. 3, 189 (1953).