On the distinguished spectrum of \(\mathrm{Sp}_{2n}\) with respect to \(\mathrm{Sp}_n \times \mathrm{Sp}_n\). (English) Zbl 1448.11098
Let \(G\) denote a reductive group over a number field \(F\) and let \(H\) stand for a closed subgroup of \(G\) defined over \(F\). Let \(\mathbb{A}\) denote the ring of adéles over \(F\).
In the paper under the review, the authors introduce a the notion of the \(H\)-distinguished automorphic spectrum of \(G\), \(L^2_{H\text{-dist}}(G(F) \setminus G(\mathbb{A}))\), as the orthogonal complement in \(L^2 (G(F) \setminus G(\mathbb{A}))\) of the space of pseudo-Eisenstein series \(\varphi\) on \(G(F) \setminus G(\mathbb{A})\) such that \(\int_{H(F) \setminus H(\mathbb{A})} \varphi(hg)dh= 0\) for all \(g \in G(\mathbb{A})\).
In the case when \(G = \mathrm{GL}_{2n}\) and \(H = \mathrm{Sp}_n\), the authors obtain a complete description of \(L^2_{H\text{-dist}}(G(F) \setminus G(\mathbb{A}))\) and its discrete part. The results in this case suggest a close relation between the distinguished spectrum and the automorphic spectrum of \(\mathrm{GL}_n\) through functoriality.
The authors also consider the case when \(G= \mathrm{Sp}_{2n}\) and \(H = \mathrm{Sp}_n \times \mathrm{Sp}_n\), and obtain upper and lower bounds for the considered case. Again, the results suggest a close relation between the distinguished spectrum for the pair \((\mathrm{Sp}_{2n}, \mathrm{Sp}_n \times \mathrm{Sp}_n)\) and that of the pair \((\mathrm{GL}_{2n}, \mathrm{GL}_n \times \mathrm{GL}_n)\) through functoriality.
In the paper under the review, the authors introduce a the notion of the \(H\)-distinguished automorphic spectrum of \(G\), \(L^2_{H\text{-dist}}(G(F) \setminus G(\mathbb{A}))\), as the orthogonal complement in \(L^2 (G(F) \setminus G(\mathbb{A}))\) of the space of pseudo-Eisenstein series \(\varphi\) on \(G(F) \setminus G(\mathbb{A})\) such that \(\int_{H(F) \setminus H(\mathbb{A})} \varphi(hg)dh= 0\) for all \(g \in G(\mathbb{A})\).
In the case when \(G = \mathrm{GL}_{2n}\) and \(H = \mathrm{Sp}_n\), the authors obtain a complete description of \(L^2_{H\text{-dist}}(G(F) \setminus G(\mathbb{A}))\) and its discrete part. The results in this case suggest a close relation between the distinguished spectrum and the automorphic spectrum of \(\mathrm{GL}_n\) through functoriality.
The authors also consider the case when \(G= \mathrm{Sp}_{2n}\) and \(H = \mathrm{Sp}_n \times \mathrm{Sp}_n\), and obtain upper and lower bounds for the considered case. Again, the results suggest a close relation between the distinguished spectrum for the pair \((\mathrm{Sp}_{2n}, \mathrm{Sp}_n \times \mathrm{Sp}_n)\) and that of the pair \((\mathrm{GL}_{2n}, \mathrm{GL}_n \times \mathrm{GL}_n)\) through functoriality.
Reviewer: Ivan Matić (Osijek)
MSC:
| 11F70 | Representation-theoretic methods; automorphic representations over local and global fields |
| 22E35 | Analysis on \(p\)-adic Lie groups |
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