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Canonical representation


These lecture notes provide an introduction to the theory of so-called canonical representations, a special type of (reducible) unitary representations. The simplest way to define them and to see their importance is done in the context of the group Sl(2, R), the group of 2x2 matrices of determinant one, see [16]. We have chosen a class of groups, namely G=SU(1, n), n>/1. Notice that SL(2, R) is isomorphic to SU (1, 1). Canonical representations can be seen, generally speaking, as the completion of L2 (G/K) with respect to a new G-invariant inner product, in the same spirit as the “complementary series” is obtained from the “principal series” for G. Here K=SU(n). But this is only one (but important) point of view, see section 5. Canonical representations occur also when studying tensor products of holomorphic and anti-holomorphic discrete series representations. This is explained in section 4. The connection with quantization in the sense of Berezin is not treated in these notes because we will emphasize the representation theory. On the other hand, Berezin has made a large contribution to the understanding of canonical representations. The main problem is to decompose the canonical representations into irreducible constituents. This is not an easy task. It has been done by Berezin [1] and, later, by Upmeier and Unterberger [15]. There are however, in both treatments, conditions on the set of parameters of the representations: only large parameters are allowed. For small parameters (see [3]) an interesting new phenomenon occurs: finitely many complementary series representations take part in the decomposition. We shall treat the case G=SU(1, n) in detail in these notes and try to illustrate all aspects of the theory of canonical representations we have mentioned.

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