FORMAL ANALYTIC CONTINUATION OF GELFAND'S FINITE DIMENSIONAL REPRESENTATIONS OF gl(n,C)
Sep, 197827 pages
Published in:
- J.Math.Phys. 20 (1979) 820
DOI:
Report number:
- CRM-818
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Abstract: (AIP)
The article contains three results: I. It is shown that among the 2−n n! (n+1) ! discrete series of representations of the Lie algebra gl(n,C) of complex n×n matrices described in the literature, the majority are not representations at all. Thus for n=3 and 4 one has respectively 12 and 45 series of representations instead of 18 and 180. II. In addition to the p+1 discrete unitary series of representations of u(p,q) [the Lie algebra of the group U(p,q), p?q, and p+q=n] there exist other discrete series of gl(n,C) which become unitary when restricted to its real subalgebra u(p,q). For n=3 there are four such series all corresponding to the chain u(2,1) ⊆u(1,1) ⊆u(1); for n=4 there exist six such series for u(3,1) and four series for u(2,2). Furthermore, some of the gl(n,C) series whose restriction to the real case do not provide unitary representations in general, do contain (infinitely many) particular representations which are unitary. Such unitary representations are contained inside of two of the four series for n=3 and inside of seven of the 27 series for n=4. III. Some properties of indecomposable representations of the Lie algebras for the groups of inhomogeneous transformations are shown using the discrete series of gl(n,C).References(6)
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