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• W2334807767 abstract "Let G be a separable locally compact group, Z a closed subgroup contained in the center of G such that center (G)/Z is compact. An irreducible unitary representation X of G is said to be square integrable mod Z (or a member of the discrete series of G) if there exist non-zero vectors qA, / in the representation space g(w) such that G 1(wr(g)(p: V)12 dg < oo, where dg is the right Haar measure of G/Z. The problem of classifying semi-simple Lie groups and connected, simply connected nilpotent Lie groups having discrete series has been solved by Harish-Chandra, and J. Wolf and C. C. Moore. In this paper we shall consider the problem for another class of Lie groups whose radicals are connected, simply connected nilpotent Lie groups (these are called U-groups). Thus let G be a connected U-group, N its radical, and S a maximal connected semi-simple subgroup of G. Apart from some technical requirements on S it will be proved that G has discrete series if and only if: (A) The center of N is the connected component of the identity in the center of G, and (B) Both N and S have discrete series. Furthermore every member of the discrete series of G may be written as the tensor product of a member of the discrete series of S extended trivially to N with an irreducible representation of G whose restriction to N is a member of the discrete series of N. Finally an algorithm to determine U-groups having discrete series as successive extensions of simpler U-groups is also described. Roughly speaking, Mackey's theory of group extensions ([10]) will be applied to solve the problem by induction. Thus let H be a closed normal" @default.
• W2334807767 created "2016-06-24" @default.
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• W2334807767 date "1976-11-01" @default.
• W2334807767 modified "2023-10-18" @default.
• W2334807767 title "Lie Groups With Square Integrable Representations" @default.
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• W2334807767 doi "https://doi.org/10.2307/1970965" @default.
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