Residual spectrum of \(\mathrm{GL}_{2n}\) distinguished by the symplectic group. (English) Zbl 1220.11072
From the introduction: Let \(F\) be a number field and let \(\mathbb A\) be the ring of adèles of \(F\). For
any positive integer \(r\) we denote by \(G_r\) the group \(\mathrm{GL}_r\) viewed as an algebraic group over \(F\). We fix an integer \(n\) and denote \(G = G_{2n}\). Let \(K\) be the standard maximal compact of \(G(\mathbb A)\). For any algebraic group \(Q\) defined over \(F\), denote \(Q(\mathbb A)^1=\cap_\chi\text{ker}\,|\chi|\) where \(\chi\) ranges over the algebraic characters of \(Q\). There is a direct sum decomposition
\[
L^2\left(G(F)\backslash G(\mathbb A)^1\right) = L^2_{\text{disc}}(G)\oplus L^2_{\text{cont}}(G)
\]
to a discrete and a continuous part. The discrete part \(L^2_{\text{disc}}(G)\) decomposes into a direct sum of irreducible representations. By an irreducible, discrete spectrum representation of \(G(\mathbb A)^1\) we mean an irreducible summand of \(L^2_{\text{disc}}(G)\).
In this work we determine the irreducible, discrete spectrum representations of \(G(\mathbb A)^1\), that have a nonvanishing symplectic period. This completes the work of H. Jacquet and S. Rallis [J. Reine Angew. Math. 423, 175–197 (1991; Zbl 0734.11035)].
In this work we determine the irreducible, discrete spectrum representations of \(G(\mathbb A)^1\), that have a nonvanishing symplectic period. This completes the work of H. Jacquet and S. Rallis [J. Reine Angew. Math. 423, 175–197 (1991; Zbl 0734.11035)].
Reviewer: Olaf Ninnemann (Uffing am Staffelsee)
MSC:
| 11F70 | Representation-theoretic methods; automorphic representations over local and global fields |
| 11F67 | Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols |
| 22E55 | Representations of Lie and linear algebraic groups over global fields and adèle rings |
Citations:
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