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Representations of twisted spaces on a connected reductive \(p\)-adic group. (Représentations des espaces tordus sur un groupe réductif connexe \(p\)-adique.) (French) Zbl 1409.22013

Astérisque 386. Paris: Société Mathématique de France (SMF) (ISBN 978-2-85629-851-0/pbk). ix, 366 p. (2017).
Walking to a lecture at the Maryland conference “Lie Groups Representations II”, in 1982 or 1983, I followed Langlands and his disciples. One of them asked Langlands whether the rumors that Kazhdan intended to announce an extension of the GL(2) and its inner forms correspondence to \(\mathrm{GL}(n)\), as well as an extension of Langlands work on endoscopy from SL(2) to \(\mathrm{SL}(n)\), and whether Langlands could explain to us the implied striking ideas. I do not know whether this apparently not too PC question has anything to do with the fact that this very original young scholar, who got a professorship at a leading university, stopped publishing.
In any case, the relevant works of Kazhdan appeared in this conference proceedings: Springer Lecture Notes in Math. 1041, 1984, in particular the second work alluded to above: “On lifting”, on pages 209–249. Here, Kazhdan observed that Langlands in fact studied representations \(\pi\) invariant under twisting by a character \(\omega\) of the determinant, thus \(\pi(g)=\pi(g)\otimes\omega(\mathrm{det}(g))\) for \(g\) in \(\mathrm{GL}(n,F)\), he introduced a twisted by \(\omega\) trace formula on \(\mathrm{GL}(n)\), and proceeded to milk it using a local-to-global machinery. These striking ideas opened the way to extend Langlands early work from rank one to general rank. Kazhdan later modestly stated that the idea of using the local-to-global machine already appear in Jacquet-Langlands.
So to carry out the analysis Kazhdan needed local information, some are expected extensions of Harish-Chandra’s deep results on the local behaviour of characters on the regular set, and their local integrability, also in this new, twisted case, as well as the trace Paley-Wiener theorem, which describes the image and kernel of the “Fourier” transform \(f\mapsto\text{trace}\,\pi(f)\), also in the new, twisted case. These local technicalities were studied in many papers, some by this reviewer (who first combined the twisting by the character \(\omega\) and the Galois twisting, say \(\theta\), that appeared in Langlands’ base change theory) and some by others. Perhaps what is missing is a readable (say by new graduate students) account.
The text under review, of more than 350 pages, consists of two articles. I copy the English abstract of the first one: “Twisted characters of admissible representations of twisted spaces on a connected reductive \(\mathfrak p\)-adic group. — Let \(F\) be a non-Archimedean locally compact field (\(\mathrm{car}(F)\geq 0\)) (sic. I suppose this explains why this long text is in French), \(\mathrm{G}\) a connected reductive group defined over \(F\), \(\theta\) be an \(F\)-automorphism of \(\mathrm{G}\), and \(\omega\) be a character of \(\mathrm{G}(F)\). We fix a Haar measure \(dg\) on \(\mathrm{G}(F)\). For a smooth irreducible \((\theta,\omega)\)-stable complex representation \(\pi\) of \(\mathrm{G}(F)\), that is such that \(\pi\circ\theta\simeq\pi\otimes\omega\), the choice of an isomorphism \(A\) from \(\pi\otimes\omega\) to \(\pi\circ\theta\) defines a distribution \(\Theta^A_\pi\), called the \(\langle\langle\) \((A-)\)twisted character of \(\pi\) \(\rangle\rangle\): for a compactly supported locally constant function \(f\) on \(\mathrm{G}(F)\), we put \(\Theta^A_\pi(f)=\text{trace}(\pi(fdg)\circ A)\). In this paper, we study these distributions \(\Theta^A_\pi\), without any restrictive hypothesis on \(F\), \(\mathrm{G}\) or \(\theta\). We prove in particular that the restriction of \(\mathrm{G}\) on the open dense subset of \(\mathrm{G}(F)\) formed of those elements which are \(\theta\)-quasi-regular is given by a locally constant function, and we describe how this function behaves with respect to parabolic induction and and Jacquet restriction. This leads us to take up again the Steinberg’s (sic) theory of automorphisms of an algebraic group, from a rational point of view.”
Well, looking at twisted characters leads to functions of the coset \(\mathrm{G}\theta(F)=\mathrm{G}(F)\theta\) of the semidirect product \(\mathrm{G}(F)\rtimes\langle\theta\rangle\). The space \(\mathrm{G}(F)\theta\) is denoted in this work by \(\mathrm{G}(F)^\sharp\). I am not sure of the advantage in introducing this new notation. In any case, a more detailed description of the contents of the various chapters can be found in Subsection 1.10 on page 8.
The second article has the following English abstract: “The Fourier transform for twisted spaces on a connected reductive \(\mathfrak p\)-adic group. Let \(\mathrm{G}\) be a connected reductive group defined over a non-Archimedean local field \(F\). Put \(G=\mathrm{G}(F)\). Let \(\theta\) be an \(F\)-automorphism of \(\mathrm{G}\), and let \(\omega\) be a smooth character of \(G\). This paper is concerned with the smooth complex representations \(\pi\) of \(G\) such that \(\pi^{\theta}=\pi\circ\theta\) is isomorphic to \(\omega\pi=\omega\circ\pi\). If \(\pi\) is admissible, in particular irreducible, the choice of an isomorphism \(A\) from \(\omega\pi\) to \(\pi^\theta\) (and of a Haar measure on \(G\)) defines a distribution \(\Theta^A_\pi=\mathrm{tr}(\pi\circ A)\) on \(G\). The twisted Fourier transform associates to a compactly supported locally constant function \(f\) on \(G\), the function \((\pi,A)\mapsto\Theta^A_\pi(f)\) on a suitable Grothendieck group. Here we describe its image (Paley-Wiener theorem) and its kernel (spectral density theorem).”
The Trace Paley-Wiener theorem was first established by J. Bernstein et al. [J. Anal. Math. 47, 180–192 (1986; Zbl 0634.22011)], and in the twisted case by a purely local technique (which is followed in the article under review) in the reviewer’s [Proc. Symp. Pure Math. 58, 171–196 (1995; Zbl 0840.22031)].

MSC:

22E50 Representations of Lie and linear algebraic groups over local fields
22-02 Research exposition (monographs, survey articles) pertaining to topological groups
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