Reduction of the most degenerate unitary irreductible representations of $S{O}_0(m,n)$ when restricted to a non-compact rotation subgroup

Limić, N.; Niederle, J.

Annales de l'institut Henri Poincaré. Section A, Physique Théorique, Tome 9 (1968) no. 4, pp. 327-355.

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[9] The group G is unimodular if | det AdG(g) | = 1 for all g ∈ G, where AdG(g) is the automorphism of the Lie algebra g of the group G defined by AdG(g): g ∋ X → AdG(g)X = dI(g)eX and I(g) is the inner isomorphism of G onto itself. The group SO0(r, s) is a semi-simple (in fact simple) Lie group, hence unimodular. For the group G = Tm+n-2 s SO0(m - 1, n - 1) we also have | det AdG(g) | = 1, as follows from the following argument. G is a connected group and therefore every neighbourhood U(e) of the identity element e ∈ G generates the whole group G. As G ∋ g → Ad(g) ∈ GL(g) is the homomorphism, it suffices to prove that | det Ad(g) | = 1 for a g ∈ U(e). We choose such U(e) for which a neighbourhood V(o) ∈ g exists such that V(0) ∋ X → exp X ∈ U(e) is the diffeomorphism. Then for every g = exp X ∈ U(e) we have | det Ad (exp X) | = exp { TradX }. In the basis of the Lie algebra of the considered group G, which is the union of the basis of Lie algebras of the groups Tm+n-2 and SO0(m - 1, n - 1), one easily calculates that Tr adX = 0.

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[11] The Hilbert space h(M) is a Hilbert space vector of which the equivalence classes of complex valued measurable functions f(p) on M such that and the scalar product is defined by Addition of vectors and multiplication of vectors by complex numbers is defined as the corresponding operations with the complex valued functions.

[12] The invariant C2 is a Casimir operator gμνXμXν where gμν is the Cartan metric tensor of the Lie algebra s0(m, n) in a basis X1, X2, ..., X[m+n 2].

[13] Here and elsewhere we use brackets for indices defined as follows:

[14] For instance all UI representations of the group SO0(m, n) m ≥ n ≥ 2 related with three homogeneous spaces M which are classified by the same real number λ ∈ (0, ∞) and the same eigenvalue of the operator P are equivalent.

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