Some results on L-indistinguishability for SL(r). (English) Zbl 0553.10024

Shahidi, Freydoon

Can. J. Math. 35, 1075-1109 (1983).

Let F be a number field, \({\mathbb{A}}={\mathbb{A}}_ F\) its adèle ring. Given a parabolic subgroup \(P=MN\) of SL(r,F), one can consider the terms in the trace formula for SL(r,\({\mathbb{A}})\) coming from the Eisenstein series with P. let \(W=W(M)\) be the Weyl group for M. J. Arthur has shown that among the terms appearing in the trace formula are those that are (up to a constant multiple) \((*)\quad Tr(M_{P| P}(w,0)I(\sigma,P,f)),\) where \(f\in C_ c^{\infty}(SL(r,{\mathbb{A}})),\quad w\in W(M)_{reg},\) and \(\sigma\) is a cusp form on M(\({\mathbb{A}})\) with \(w\sigma\) \(\cong \sigma.\)
There are cusp forms on a certain endoscopic group \(H\subseteq SL(r,{\mathbb{A}})\) which should not lift to cusp forms on SL(r,\({\mathbb{A}})\); here, H is the group of elements in \(GL(p,{\mathbb{A}}_ E)\) whose determinants have norm 1, E is a cyclic extension of F of degree m, and \(mp=r\). Hence their traces should not contribute to the trace formula, and they should cancel off certain terms of the form (*). Results of this sort have been proved, using the conjectured transfer of orbital integrals. The author proves the corresponding local results.

Reviewer: L.Corwin

MSC:

11F70	Representation-theoretic methods; automorphic representations over local and global fields
11S37	Langlands-Weil conjectures, nonabelian class field theory
22E50	Representations of Lie and linear algebraic groups over local fields
11F85	\(p\)-adic theory, local fields

Keywords:

adelic group; trace formula; Eisenstein series; cusp form; endoscopic group