MATHEMATICS

R. S. ISMAGILOV

DESCRIPTION OF UNITARY REPRESENTATIONS OF THE LORENTZ GROUP IN A SPACE WITH INDEFINITE METRIC

(Presented by Academician L. S. Pontryagin on 13 IV 1964)

1. In the present note we give a description of an arbitrary representation of the group \(G\) of complex unimodular matrices of the second order in a space with indefinite metric (the theory of such spaces is set forth in \((^{1,2})\)). The problem of studying such representations was posed by M. A. Naimark; he also obtained a number of results on unitary representations in \(\Pi\)-spaces \((^{4,5})\).

Irreducible representations of the group \(G\) that are unitary in a \(\Pi_n\)-space are described in \((^3)\). The study of an arbitrary unitary representation in a \(\Pi_n\)-space is complicated by the fact that for such representations there is no theorem on complete reducibility. We shall describe the “simplest” (after the irreducible ones) representations of the group \(G\) that are unitary in a \(\Pi_n\)-metric (representations of type \(T^k\) and \(S\)); any unitary representation turns out to be a direct orthogonal sum of a finite number of irreducible representations, a unitary (in the usual sense) representation, and a finite number of representations of type \(T^k\) and \(S'\) (some of these components may, of course, be absent).

We set forth the scheme for describing the representations. Let \(T_g\) be a representation of the group \(G\) that is unitary in the \(\Pi_n\)-space \(H\). It turns out that it is sufficient to consider only those representations for which there exists in \(H\) an \(n\)-dimensional null (i.e. with identically zero scalar product) invariant subspace. More precisely, the following holds.

Theorem 1. Let \(D\) be the maximal null subspace in \(H\) invariant with respect to \(T_g\) (if there are no such subspaces, we put \(D=0\)).

Then:

1) if \(D=0\), then there is a decomposition
\[ T_g=T_g^{i_1}\oplus T_g^{i_2}\oplus\cdots\oplus T_g^{i_s}\oplus U_g,\qquad i_k>0, \]
where \(T_g^{i_k}\) is an irreducible representation unitary in the \(\Pi_{i_k}\)-metric, and \(U_g\) is a unitary representation (in the usual sense);

2) if \(\dim D=k<n\), then
\[ T_g=T_g^{j_1}\oplus\cdots\oplus T_g^{j_m}\oplus T_g^*, \]
where the representations \(T_g^{j_k}\) are irreducible and unitary in the \(\Pi_{j_k}\)-metric \((j_k>0)\), while the representation \(T_g^*\) is unitary in the \(\Pi_k\)-metric; in addition, the subspace \(H^*\), in which \(T_g^*\) acts, contains \(D\), and the subspace \(D\) is a maximal null invariant subspace in \(H^*\).

Thus, in what follows we may restrict ourselves to representations \(T_g\) satisfying the following condition.

A. In the \(\Pi_n\)-space \(H\), where the representation \(T_g\) acts, there exists an \(n\)-dimensional null invariant subspace \(D\).

Introduce two more subspaces in \(H\): the subspace \(M\) of vectors \(x\in H\) orthogonal to \(D\), and a subspace \(R\subset H\) such that \(R\dotplus D=M\); it is obvious that the metric in \(R\) is positive.

Obviously, \(T_g M=M\); denote by \(T_g^M\) the restriction of \(T_g\) to the subspace \(M\). It turns out that the study of the representation \(T_g\) reduces to the study of the representation \(T_g^M\); more precisely, the following Theorems 2 and 3 hold.

Theorem 2. Let \(T'_g\) and \(T''_g\) be two representations satisfying condition A, and let \(D\) be the maximal \(n\)-dimensional null invariant subspace for \(T'_g\) and \(T''_g\). Suppose, furthermore, that \(T_g^{\prime M}=T_g^{\prime\prime M}\) (i.e. \(T'_g x=T''_g x\) for all \(x\in M\)). Then the representations \(T'_g\) and \(T''_g\) are unitarily equivalent.

Theorem 3. Let \(T_g\) satisfy condition A. Let
\[ M=M_1 \dotplus M_2, \]
where \(M_1\) and \(M_2\) are orthogonal in the sense of the metric \((x,y)\) and invariant with respect to \(T_g\). Then there exists a decomposition
\[ H=H_1\oplus H_2, \]
such that:
\[ \begin{aligned} &1)\quad T_g H_i=H_i \quad (i=1,2),\\ &2)\quad M_i\subset H_i \quad (i=1,2). \end{aligned} \]

The representation \(T_g^M\) is a coupling of an \(n\)-dimensional representation with a unitary one. Let us study such representations in greater detail.

1. Let \(M\) be a Hilbert space with scalar product \(\{x,y\}\), \(M=D\oplus R\), \(\dim D=n<\infty\). Let \((x,y)\) denote the degenerate scalar product in \(M\) defined as follows: \((x,y)=\{x,y\}\) if \(x,y\in R\); \((x,y)=0\) if \(x\in D\) or \(y\in D\). Consider a representation \(T_g\) of the group \(G\) in \(M\), preserving the scalar product \((x,y)\), (i.e. \((T_g x,T_g y)=(x,y)\), \(x,y\in M\)); obviously, \(T_g D=D\), and for \(x\in R\)
   \[ T_g x=V_g x+A_g x, \tag{1} \]
   where \(V_g\) is a unitary representation of \(G\) in \(R\), while \(A_g\) maps \(R\) into \(D\). Let us consider one special class of such representations.

Definition 1. We shall say that a representation \(T_g\), acting in \(M\) and possessing the properties described above, is a representation of type \(T_k\), if: 1) the restriction \(S_g\) of the representation \(T_g\) to \(D\) is a multiple of the irreducible (finite-dimensional) representation \(S_{n,n}\), while \(V_g\) is a multiple* of the unitary irreducible representation \(T_{n,-n}\), which is the nearest relative of the representation \(S_{n,n}\) ([3], p. 212); 2) the space \(M\) cannot be decomposed into a direct sum of two subspaces invariant with respect to \(T_g\) and orthogonal in the sense of the bilinear form \((x,y)\).

From condition 2) it follows, in particular, that \(T_g\) contains no unitary parts, i.e. in \(M\) there do not exist invariant subspaces on which the form \((x,y)\) is nondegenerate (and hence positive definite).

Definition 1′. A representation \(T_g\), acting in \(M\), will be called a representation of type \(S\), if: 1) \(S_g=E\) (i.e. \(T_g x=x\), \(x\in D\)); 2) \(V_g=V'_g\oplus V''_g\), where \(V'_g\) is a multiple of the representation \(T_{1,-1}\) (the nearest relative of the identity representation), and the representation \(V''_g\) decomposes only into representations of the supplementary series; 3) the restriction of the representation \(T_g\) to \(H(V'_g)\oplus D\) (here \(H(V'_g)\) denotes that subspace of the space \(R\) on which \(V'_g\) acts) is a representation of type \(T_1\).

It turns out that from unitary (in the usual sense) representations, finite-dimensional representations, and representations of types \(T_k\) and \(S\) (see the definition), one can compose any representation possessing the properties listed at the beginning of this section. More precisely, the decomposition
\[ T_g=U_g\dotplus L_g\dotplus \sum_1^s T_g^k\dotplus T'_g, \]
holds

* The multiplicity may be infinite.

where \(U_g\) is a unitary representation, \(L_g\) is a finite-dimensional representation; the representation \(T_g^k\) belongs to type \(T_k\), and the representation \(T'_g\) to type \(S_1\); moreover, the subspaces on which these components of the representation \(T_g\) act are mutually orthogonal in the sense of the form \((x,y)\). We note that \(L_g\) includes, in particular, all those finite-dimensional subrepresentations that are not relatives of any unitary representations.

1. It is now easy to describe the representations that are unitary in the \(\Pi_n\)-metric.

Definition 2. Let \(T_g\) be a representation of a group \(G\) that is unitary in the \(\Pi_n\)-metric. We shall call \(T_g\) a representation of type \(T^k\) if in the representation space there exists an \(n\)-dimensional null invariant subspace \(D\), and the restriction \(T_g^M\) of the representation \(T_g\) to the subspace \(M=\{x: x\perp D\}\) is a representation of type \(T_k\). A representation of type \(S'\) is defined analogously. We now formulate the main theorem.

Theorem 4. A representation \(T_g\) of a group \(G\) that is unitary in the \(\Pi_n\)-metric admits a decomposition into a direct orthogonal sum

\[ T_g=L_g\oplus U_g\oplus \sum_{i=1}^{s}\oplus T_g^{k_i}\oplus \sum_{j=1}^{k}\oplus \widetilde{T}_g^{p_j}\oplus \sum_{s=1}^{l}\oplus T_g^{q_s}, \]

where \(L_g\) is finite-dimensional, \(U_g\) is a unitary (in the ordinary sense) representation; \(T_g^{k_i}\) belongs to type \(T^{k_i}\), \(\widetilde{T}_g^{p_j}\) to type \(S'\); \(T_g^{q_s}\) is irreducible and unitary in the \(\Pi_{q_s}\)-metric \((q_s>0)\).

1. Theorem 2 asserts that a representation \(T_g\) that is unitary in the \(\Pi_n\)-metric is determined, up to unitary equivalence, by its part \(T_g^M\); the following question arises: let \(D\) be an \(n\)-dimensional null subspace of a \(\Pi_n\)-space \(H\), \(M=\{x: x\perp D\}\), and let \(T_g^M\) be a representation of the group \(G\) in \(M\) preserving the bilinear form \((x,y)\) (i.e. \((T_gx,T_gy)=(x,y)\), \(x,y\in M\)). Can the representation \(T_g^M\) be extended to a unitary representation \(T_g\) in the whole space \(H\)?

The answer to this question is positive if the representation \(V_g\) that was constructed by formula (1) decomposes into a finite number of irreducible representations. The general case remains unclear.

In conclusion I express my gratitude to M. A. Naimark for his advice.

Voronezh State University

Received
2 IV 1964

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