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On the transfer of orbital integrals for the bilinear groups (\(p\)-adic case). (Sur le transfert des intégrales orbitales pour les groupes linéaires (cas \(p\)-adique).) (French) Zbl 0893.22009

D. Kazhdan [On lifting, Lect. Notes Math. 1401, 209-249 (1984; Zbl 0538.20014)] lifted (automorphic and) admissible representations of \(GL(m,E)\) to those of \(GL(n,F)\), where \(E\) is a cyclic extension of \(F\) of degree \(d=n/m\), for \(m=1\), by using the trace formula and transferring orbital integrals of spherical and general functions. Y.-L. Waldspurger [Can. J. Math. 43, 852-896 (1991; Zbl 0760.22026)] transfered the orbital integrals of spherical functions for any \(m\) (dividing \(n\)). Using the trace formula Y. Flicker [Compos. Math. 67, 271-300 (1988; Zbl 0661.22010)] deduced the lifting, and showed that transfer of orbital integrals of general functions then follows from the trace Paley-Wiener theorem of J. Bernstein, P. Deligne and D. Kazhdan [J. Anal. Math. 47, 180-192 (1986; Zbl 0634.22011)] (for a local proof see Y. Flicker [Proc. Symp. Pure Math. 58, Part II, 171-196 (1995; Zbl 0840.22031)]), and that Waldspurger’s fundamental lemma would follow from the computation of the twisted (as in Kazhdan’s article) character of the trivial or alternatively the Steinberg representation. But the key point is that transfer of orbital integrals for general functions follows in this relatively simple case of \(GL(n)\), where multiplicity one and rigidity are available, from the fundamental lemma.
The present work studies the transfer of orbital integrals for general functions directly locally, using the germ of the twisted character of the Steinberg representation, being led to a lengthy computation.

MSC:

22E50 Representations of Lie and linear algebraic groups over local fields
11F70 Representation-theoretic methods; automorphic representations over local and global fields
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