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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.

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.

Reviewer: Stephen Gelbart (Rehovot)

### 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 |