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A Fourier summation formula for the symmetric space \(GL(n)/GL(n-1)\). (English) Zbl 0789.11035
This paper is devoted to a discussion of the meaning of a certain type of ‘Fourier summation formula’ in the theory of automorphic representations and to its proof in a special case. The underlying problem concerns the restriction of automorphic representations to a subgroup. In this context one considers a reductive algebraic group \(G\) over a global field \(k\) and a kernel \(K_ f(g_ 1,g_ 2)=\sum_{\gamma \in G(k)} f(g^{-1}_ 1 \gamma g_ 2)\) of the type one considers in the theory of the Selberg trace formula. Let \(H\) be a reductive subgroup of \(G\). In this paper \(G\) is mainly \(GL(n)\) and \(H\) then \(GL(n-2)\). For a one-dimensional representation \(\xi\) of \(H\) and a one-dimensional nondegenerate representation \(\psi\) of a certain unipotent subgroup times the centre, the author considers the Fourier decomposition of \(K_ f (g_ 1,g_ 2)\) with respect to \(\xi\) in \(g_ 1\) and to \(\psi\) in \(g_ 2\) and this makes use of the spectral decomposition, due to Langlands of \(K_ f\). The argument is complete only when \(n=3\). The resulting formula, like the trace formula, has a ‘geometric or group-theoretic’ side and a ‘spectral’ side. The main result is that the spectral side is supported on induced unitary representations. The proof of the formula is intricate and makes use of the theory of Eisenstein series and of the Bernstein centre.

MSC:
11F72 Spectral theory; trace formulas (e.g., that of Selberg)
22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings
11F70 Representation-theoretic methods; automorphic representations over local and global fields
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