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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).

##### MSC:
 2.2e+51 Representations of Lie and linear algebraic groups over local fields 2.2e+36 Analysis on $$p$$-adic Lie groups
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