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Representations of central simple $$p$$-adic algebras. (Représentations des algèbres centrales simples $$p$$-adiques.) (French) Zbl 0583.22009
Représentations des groupes réductifs sur un corps local, Travaux en Cours, 33-117 (1984).
The authors establish a bijection between the (isomorphism classes of) irreducible square-integrable representations of $$\mathrm{GL}(n,F)=:G$$ with those of the multiplicative group $$G'$$ of a central simple algebra over a local non-Archimedean field $$F$$. Here $$G'=\mathrm{GL}(m,D)$$, where $$D$$ is a division algebra of dimension $$d^ 2$$ (and central) over $$F$$, and $$n=md$$. The idea of the proof, which goes back to H. Jacquet and R. P. Langlands for the case of $$\mathrm{GL}(2,F)$$, is to embed this local problem into a global context and then apply the Selberg trace formula [cf. §16 of “Automorphic forms on $$\mathrm{GL}(2)$$”, Lect. Notes Math. 114 (1970; Zbl 0236.12010)].
For the general case, it is important that a simpler (more restrictive) form of the trace formula (due initially to Deligne) be used. The end result is a bijection $$\pi'\to\pi$$ of $$E^ 2(G')$$ onto $$E^ 2(G)$$ characterized by the following relation between the characters $$X_{\pi'}$$ and $$X_{\pi}$$: suppose $$g'$$ is a regular element in $$G'$$ whose conjugacy class corresponds to that of $$g$$ in $$G^{\text{reg}}$$; then $(-1)^ mX_{\pi'}(g')=(-1)^ nX_{\pi}(g).$ Such a result implies that the representation theory of $$G'$$ is reflected in that of $$G$$.
The proof involves not only much of what was previously known concerning the representations of $$p$$-adic groups, but also many new results as well: orthogonality relations for characters of square-integrable representations and their relation with Howe’s conjecture, the existence of “pseudo-coefficients” for $$G$$ and $$G'$$, the proof of Paley-Wiener theorems, etc. Because many of these important results were announced years ago without proofs, the appearance of this detailed and long-awaited manuscript is a significant and welcome event.
For the entire collection see Zbl 0544.00007.

##### MSC:
 22E50 Representations of Lie and linear algebraic groups over local fields 11F70 Representation-theoretic methods; automorphic representations over local and global fields 11F72 Spectral theory; trace formulas (e.g., that of Selberg) 11S45 Algebras and orders, and their zeta functions 11F85 $$p$$-adic theory, local fields
##### Citations:
Zbl 0544.00007; Zbl 0236.12010