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\(R\)-groups and elliptic representations for unitary groups. (English) Zbl 0856.22023
Let \(F\) be a locally compact, non-discrete, non-Archimedean local field of characteristic zero. Let \(G\) be the set of \(F\)-rational points of a connected reductive quasi-split algebraic group defined over \(F\). Let \(G'\) be the set of regular points of \(G\). An element \(x\) of \(G\) is said to be elliptic if the centralizer of \(x\) is compact, modulo the centre of \(G\). Denote by \(G^e\) the set of regular elliptic elements of \(G\). Let \({\mathcal E}_2(G)\) denote the set of equivalent classes of irreducible square-integrable representations of \(G\), and \({\mathcal E}_t(G)\) the set of equivalent classes of tempered representations of \(G\); then \({\mathcal E}_2(G)\subset {\mathcal E}_t(G)\). If \(\pi\in {\mathcal E}_t(G)\), let \(\Theta_\pi\) be the character of \(\pi\). Denote by \(\Theta^e_\pi\) its restriction to \(G^e\). \(\pi\) is said to be elliptic if \(\Theta^e_\pi\neq 0\).
In Section 1, the author gives a lucid summary of the basic results of (i) Harish-Chandra on discrete series and Plancherel measure, (ii) Knapp-Stein and Silberger on intertwining operators and \(R\)-groups and (iii) Arthur and Herb on elliptic representations. The author had earlier computed \(R\)-groups and elliptic representations for \(\text{SL}(n)\) [Pac. J. Math. 165, 77-92 (1994)] and \(\text{Sp}(2n)\) and \(\text{SO}(n)\) [Am. J. Math. 116, 1101-1151 (1994)].
In Section 2, the author studies the unitary groups \(U(n, n)\) and \(U(n, n+ 1)\) and determines the parabolic subgroups (Section 3) and their elliptic representations (Section 4).

22E50 Representations of Lie and linear algebraic groups over local fields
22E35 Analysis on \(p\)-adic Lie groups
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