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

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36. [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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38. [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. 
39. [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]. 
40. [13] Here and elsewhere we use brackets for indices defined as follows: 
41. [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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