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Unitary representations of the SL (2, C) group in horospheric basis
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23.At first glance one would think that Eq. (1.20) for the matrix elements can be put down at once by means of the basis functions and thus, the way in which Eq. (1.20) has been óbtained is superfluously complicated. Actually, in the present case (and in general, in the case of continuous bases) the notion of matrix element of a representation has a slightly restricted meaning. Operators of representations act on functions as indicated in Eq. (1.19) and interpretation of the matrix elements as the kernel of the integral transformation (1.20) is possible only if the interchange of the order of integrations in (1.19) is legitimate. However, a detailed investigation shows that at integrations in Eq. (1.19) cannot be interchanged. If in the explicit form of as given by Eq. (2.16) one still tries to take it turns out to be nonexisting (an oscillating undetermined expression). Due to the symmetry of the functions a similar statement can be made on the limit too.
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