MATHEMATICS

M. A. NAIMARK

ON UNITARY REPRESENTATIONS OF THE LORENTZ GROUP IN THE SPACE \(\Pi_k\)

(Presented by Academician L. S. Pontryagin, 4 V 1963)

1. Let \(\Pi\) be a space with an indefinite metric and finite rank of indefiniteness \(k\) (see \((^1)\)), let \(G\) be a locally compact group, and let \(g \to U_g\) be a unitary representation of the group \(G\) in the space \(\Pi_k\); here unitarity is understood in the sense of the indefinite scalar product \((\xi,\eta)\) in \(\Pi\), so that \((U_g\xi,U_g\eta)=(\xi,\eta)\) for all \(\xi,\eta\in\Pi_k,\ g\in G\). We shall say that the representation \(g\to U_g\) of the group \(G\) discretely contains the representation \(g\to V_g\), if in the space \(\Pi_k\) there exists a closed subspace \(\mathfrak M\), invariant with respect to all operators \(U_g\), and on which the restriction of the representation \(g\to U_g\) is equivalent to the representation \(g\to V_g\).

It is not difficult to show that (in contrast to ordinary unitary representations) every unitary representation in the space \(\Pi_k\) discretely contains an irreducible representation* (coinciding with the original representation if it itself is irreducible).

A representation \(g\to W_g\) in a space \(R\) is called nonunitary if in \(R\) there is no nondegenerate scalar product \((\xi,\eta)\) (definite or indefinite), invariant with respect to all operators of the representation. A unitary representation may discretely contain irreducible nonunitary representations. It turns out, however, that in the case of a complex semisimple Lie group the direct sum of all such representations is normally split off; thus the study of general unitary representations of a complex semisimple Lie group is reduced to the study of its unitary representations that do not discretely contain nonunitary representations.

For simplicity of exposition, we shall set out here (see item 4) a brief proof of this result for the case \(G=\mathfrak A\), where \(\mathfrak A\) is the complex unimodular group of second order (locally isomorphic to the Lorentz group). We first give a number of auxiliary propositions, some of which are of independent interest.

2. Let \(G\) be an arbitrary locally compact group and let \(g\to U_g\) be a unitary representation of the group \(G\) in the space \(\Pi_k\).

I\(**\). If \(\mathfrak M\) is a closed subspace in \(\Pi_k\), invariant with respect to \(g\to U_g\), and on which the restriction \(g\to V_g\) of the representation \(g\to U_g\) is irreducible and nonunitary, then \(\mathfrak M\) is a null subspace in \(\Pi_k\); consequently ***, \(\dim \mathfrak M\le k\), and the representation \(g\to V_g\) is finite-dimensional.

We shall denote by \(g\to \hat V_g\) the representation conjugate to \(g\to V_g\), so that \(\hat V_g=(V_g^{-1})'\), where \(A'\) denotes the operator adjoint to \(A\).

II. Let the representation \(g\to V_g\) in a finite-dimensional space \(R\) be the direct sum of two irreducible nonunitary representations \(g\to V_g^{(1)}\) and \(g\to V_g^{(2)}\) in the spaces \(R_1\) and \(R_2\), so that \(R=R_1 \dotplus R_2\) and \(V_g(\xi_1+\xi_2)=V_g^{(1)}\xi_1+V_g^{(2)}\xi_2\) for \(\xi_1\in R_1,\ \xi_2\in R_2\). If in \(R\) one can introduce a nondegenerate scalar product in which the representation \(g\to V_g\) is unitary, then the representations \(g\to V_g^{(1)}\) and \(g\to \hat V_g^{(2)}\) are equivalent.

* This result was also obtained independently by R. S. Ismagilov.
* Propositions I–V partially overlap with results of Shlider \((^2)\).
** See \((^1)\), Lemma 1.2.

III. An irreducible finite-dimensional representation \(g \to V_g\) is unitary if and only if it is equivalent to the conjugate representation \(g \to \hat V_g\).

From II and III we conclude:

IV. The direct sum of two irreducible nonunitary finite-dimensional representations equivalent to one another is a nonunitary representation.

We shall say that the representation \(g \to V_g\) is contained \(p\)-fold in the representation \(g \to U_g\) if: 1) in the space \(\Pi_k\) there exist closed subspaces \(\mathfrak M_1,\ldots,\mathfrak M_p\) such that

\[ U_g\xi_j=U_{1j}(g)\xi_j+U_{2j}(g)\xi_j+\cdots+U_{lj}(g)\xi_l \quad \text{for } \xi_j\in\mathfrak M_j,\ j=1,\ldots,p, \]

where \(U_{lj}(g)\), \(l\le j\), is an operator from \(\mathfrak M_j\) to \(\mathfrak M_l\), and \(g\to U_{jj}(g)\) is a representation of the group \(G\) in the space \(\mathfrak M_j\), equivalent to the representation \(g\to V_g\); 2) \(p\) is the maximal number satisfying condition 1). The subspace \(\mathfrak M=\mathfrak M_1+\cdots+\mathfrak M_p\) will then be called the carrier of the representation \(g\to V_g\) in the representation \(g\to U_g\).

V. Every nonunitary irreducible representation \(g\to V_g\) can be contained in a unitary representation \(g\to U_g\) only with finite multiplicity, and the carrier \(\mathfrak M\) of the representation \(g\to V_g\) in the representation \(g\to U_g\) is a null subspace; consequently, \(\dim \mathfrak M\le k\).

VI. If in a unitary representation \(g\to U_g\) two nonunitary irreducible representations \(g\to V_g^{(1)}\) and \(g\to V_g^{(2)}\) are contained with multiplicity, and if the representations \(g\to V_g^{(1)}\) and \(g\to \hat V_g^{(2)}\) are not equivalent, then the carriers of the representations \(g\to V_g^{(1)}\) and \(g\to V_g^{(2)}\) are mutually orthogonal.

Combining these propositions, we arrive at the following result:

Theorem 1. A unitary representation in the space \(\Pi_k\) can discretely contain only a finite number of irreducible nonunitary representations and only with finite multiplicities, and the direct sum of the carriers of all these representations is a subspace of dimension \(\le 2k\).

1. Theorem 2. Let \(\mathcal H\) be a family of permutable bounded Hermitian operators in the space \(\Pi_k\), and let \(\xi_0\) be a nonzero nonnegative vector from \(\Pi_k\) satisfying the conditions:

1) \(H\xi_0=\lambda(H)\xi_0\) for all \(H\in\mathcal H\), where \(\lambda(H)\) is a numerical function on \(\mathcal H\);

2) there exist \(H\in\mathcal H\) such that \(\lambda(H)\) is not real.

Then in \(\Pi_k\) there exist two skew-conjugate null subspaces \(\mathfrak N,\mathfrak N'\), and in them bases \(\xi_0,\xi_1,\ldots,\xi_r;\ \xi'_0,\xi'_1,\ldots,\xi'_r\), possessing the following properties

\[ (\xi_\nu,\xi'_l)= \begin{cases} 0 & \text{for } \nu\ne l,\\ 1 & \text{for } \nu=l, \end{cases} \quad \nu,l=0,1,\ldots,r; \tag{1} \]

\[ H\xi_0=\lambda(H)\xi_0,\qquad H\xi_\nu=\lambda_{\nu 0}(H)\xi_0+\cdots+\lambda_{\nu,\nu-1}(H)\xi_{\nu-1}+\lambda(H)\xi_\nu, \]

\[ \nu=1,\ldots,r; \tag{2} \]

\[ H\xi'_r=\overline{\lambda(H)}\xi'_r,\qquad H\xi'_\nu=\overline{\lambda(H)}\xi'_\nu+\overline{\lambda_{\nu+1,\nu}(H)}\xi'_{\nu+1}+\cdots+\overline{\lambda_{r,\nu}(H)}\xi'_r, \]

\[ \nu=0,1,\ldots,r-1, \tag{3} \]

for all \(H\in\mathcal H\), where \(\lambda_{\nu\rho}(H)\) are numerical functions on \(\mathcal H\).

Remark 1. Let \(\mathcal H\) be a family of permutable bounded Hermitian operators in \(\Pi_k\), and suppose that one of the operators \(H_0\in\mathcal H\) has a nonreal eigenvalue \(\lambda_0\). Then there exist a nonnegative vector \(\xi_0\ne0\) and a numerical function \(\lambda(H)\) on \(\mathcal H\) such that \(H\xi_0=\lambda(H)\xi_0\) for all \(H\in\mathcal H\), \(\lambda(H_0)=\lambda_0\).

Remark 2. Theorem 2 and Remark 1 carry over, with the appropriate changes, to families of permutable unitary operators in \(\Pi_k\).

1. We can now prove the following main theorem:

Theorem 3. Let \(a \to U_a\) be a unitary representation of the group \(\mathfrak A\) in the space \(\Pi_k\), and suppose that this representation discretely contains a nonunitary irreducible representation \(a \to V_a\), acting in the invariant subspace \(\mathfrak M_0 \subset \Pi_k\).

Then:

1) The representation \(a \to U_a\) also discretely contains the conjugate representation \(a \to \hat V_a\), acting in the invariant subspace \(\mathfrak M'_0 \subset \Pi_k\), skew-linked with \(\mathfrak M_0\).

2) There exists a finite-dimensional subspace \(\Pi_{k'} \subset \Pi_k\), \(k' \leq k\), invariant with respect to the representation \(a \to U_a\) and possessing the following properties: a) the restriction of the representation \(a \to U_a\) to \(\Pi_k \ominus \Pi_{k'}\) contains no discrete nonunitary irreducible representations; b) the restriction of the representation \(a \to U_a\) to \(\Pi_{k'}\) is an orthogonal sum of a finite number (possibly one) of representations, each of which is a direct sum of two mutually conjugate nonunitary representations acting in two skew-linked null subspaces.

Proof. Consider the representations \(x \to U_x\), \(x \to V_x\) of the group ring \({}^{*}X\) of the group \(\mathfrak A\), corresponding to the representations \(a \to U_a\), \(a \to V_a\). Let the indices \(j,q\) be chosen so that \(\mathfrak M_j^q = E_{jj}^q \mathfrak M_0 \ne (0)\); then \(\mathfrak M_j^q\) is one-dimensional. Let \(\xi_0 \in \mathfrak M_j^q\), \(\xi_0 \ne 0\); then

\[ U_x \xi_0 = V_x \xi_0 = \lambda(x)\xi_0 \quad \text{for } x \in X_j^q, \tag{4} \]

where \(\lambda(x)\) is a multiplicative linear functional on \(X_j^q\). Since \(X_j^q\) is a commutative ring with involution, the operators \(U_x\), \(x=x^* \in X_j^q\), form a family \(\mathcal H\) of commuting bounded Hermitian operators; moreover, because of the nonunitarity of the representation \(a \to V_a\), the function \(\lambda(x)\) takes non-real values for some \(x=x^* \in X_j^q\). Consequently, Theorem 2 is applicable to this family \(\mathcal H\). Let \(\xi_0,\ldots,\xi_r\) and \(\xi'_0,\ldots,\xi'_r\) be basis vectors satisfying conditions (1)—(3); put \(\mathfrak M_\nu=\{U_x\xi_\nu,\ x\in X\}\), \(\mathfrak M'_\nu=\{U_x\xi'_\nu,\ x\in X\}\), \(\nu=0,1,\ldots,r\). It is easy to show that \(E_{jj}^q\xi_\nu=\xi_\nu\), \(E_{jj}^q\xi'_\nu=\xi'_\nu\). Hence, for \(x,y\in X\),

\[ (U_x\xi_\nu, U_y\xi_\nu)=(U_{y^*x}\xi_\nu,\xi_\nu)=(U_{y^*x}E_{jj}^q\xi_\nu,E_{jj}^q\xi_\nu)= \]

\[ =(U_{x'}\xi_\nu,\xi_\nu)=\lambda(x')(\xi_\nu,\xi_\nu)=0, \]

where \(x'=e_{jj}^q y^* x e_{ij}^q \in X_j^q\) (recall that \((\xi_\nu,\xi_\nu)=0\)); consequently, \(\mathfrak M_\nu\) is a null subspace and, analogously, \(\mathfrak M'_\nu\) is a null subspace. Therefore \(\dim \mathfrak M_\nu \leq k\), \(\dim \mathfrak M'_\nu \leq k\). On the other hand, every irreducible finite-dimensional representation of the group \(\mathfrak A\) is completely reducible. Using this fact, the vectors \(\xi_0,\ldots,\xi_r\) and \(\xi'_0,\ldots,\xi'_r\) may be chosen so that: \(\alpha)\) \(U_x\xi_\nu=\lambda(x)\xi_\nu\), \(U_x\xi'_\nu=\overline{\lambda}(x)\xi'_\nu\) for \(\nu=0,1,\ldots,r\); \(\beta)\) the restriction of the representation \(a \to U_a\) to \(\mathfrak M_\nu\) and to \(\mathfrak M'_\nu\) is irreducible. By \(\alpha)\), on \(\mathfrak M_\nu\) it is equivalent to the representation \(a \to V_a\), and on \(\mathfrak M'_\nu\) to the representation \(a \to \hat V_a\). Let us consider, in particular, \(\mathfrak M_0\) and \(\mathfrak M'_0\); they are skew-linked. Indeed, the set \(\mathfrak N_0\) of all \(\xi\) in \(\mathfrak M_0\) orthogonal to \(\mathfrak M'_0\) forms a subspace in \(\mathfrak M_0\) invariant with respect to all \(V_a\), different from \(\mathfrak M_0\), since \((\xi_0,\xi'_0)=1\); consequently, by irreducibility of the representation \(a \to V_a\), \(\mathfrak N_0=(0)\). Similarly \(\mathfrak N'_0=(0)\), where \(\mathfrak N'_0\) is the set of all \(\xi'\) in \(\mathfrak M'_0\) orthogonal to

* Here and below we use the notation and terminology of [3], chiefly § 15 in [3]; the operator \(E_{ii}^q\) corresponds to the representation \(a \to U_a\).

to $\mathfrak M_0$, and this means that $\mathfrak M_0$ and $\mathfrak M'_0$ are skew-associated. Consequently, for $\mathfrak M_0$ and $\mathfrak M'_0$ assertion 1) of Theorem 3 holds.

Let us now prove assertion 2). $\mathfrak M_0 \dot{+} \mathfrak M'_0$ is a finite-dimensional space $\Pi_{k_1}$, $k_1 \leq k$, invariant with respect to all $U_a$, so that*

\[ \Pi_k = (\mathfrak M_0 \dot{+} \mathfrak M'_0)\oplus \Pi_{k-k_1}, \]

where

\[ \Pi_{k-k_1}=(\mathfrak M_0 \dot{+} \mathfrak M'_0)^\perp \]

is also invariant with respect to all $U_a$. If the representation $a \to U_a$ does not contain discrete nonunitary representations in $\Pi_{k-k_1}$, then for $\Pi_{k'}=\Pi_{k_1}$ assertion 2) of the theorem holds. Otherwise, applying the preceding argument to the restriction of the representation $a\to U_a$ to $\Pi_{k-k_1}$, we obtain that

\[ \Pi_{k-k_1}=(\mathfrak M_0^{(1)} \dot{+} \mathfrak M_0^{(1)'})\oplus \Pi_{k-k_1-k_2}, \]

where $\mathfrak M_0^{(1)}$, $\mathfrak M_0^{(1)'}$ are skew-associated, and on them the restrictions of the representation $a\to U_a$ are irreducible, nonunitary, and mutually conjugate. Repeating this argument and taking into account that $k_1+k_2+\cdots\leq k$, after a finite number of steps we obtain a subspace of the form

\[ \Pi_{k'}=(\mathfrak M_0\dot{+}\mathfrak M'_0)\oplus\cdots\oplus(\mathfrak M_0^{(\nu)}\dot{+}\mathfrak M_0^{(\nu)'}), \]

possessing properties a) and b).

Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR

Received
25 IV 1963

CITED LITERATURE

¹ I. S. Iokhvidov, M. G. Krein, Tr. Mosk. matem. obshch., 5, 367 (1956); 8, 413 (1959). ² S. Schlieder, Zs. Naturforsch., A 15a, 448 (1960). ³ M. A. Naimark, Linear Representations of the Lorentz Group, Moscow, 1958.

* $\Pi_0$ denotes the ordinary Hilbert space.