An explicit construction of non-tempered cusp forms on $$O(1,8n+1)$$ O ( 1
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An explicit construction of non-tempered cusp forms on O(1, 8n + 1) Yingkun Li1 · Hiro-aki Narita2
· Ameya Pitale3
Received: 25 June 2019 / Accepted: 12 August 2019 © Fondation Carl-Herz and Springer Nature Switzerland AG 2019
Abstract We explicitly construct non-holomorphic cusp forms on the orthogonal group of signature (1, 8n + 1) for an arbitrary natural number n as liftings from Maass cusp forms of level one. In our previous works [31] and [24] the fundamental tool to show the automorphy of the lifting was the converse theorem by Maass. In this paper, we use the Fourier expansion of the theta lifts by Borcherds [4] instead. We also study cuspidal representations generated by such cusp forms and show that they are irreducible and that all of their non-archimedean local components are non-tempered while the archimedean component is tempered, if the Maass cusp forms are Hecke eigenforms. Our non-archimedean local theory relates Sugano’s local theory [39] to non-tempered automorphic forms or representations of a general orthogonal group in a transparent manner. Keywords Lifting from Maass cusp forms · Non-tempered cusp forms · Orthogonal group of signature (1, 8n) · Theta lifting · Special Bessel models Mathematics Subject Classification Primary 11F27 · 11F55 · 11F70 Résumé Cet article décrit une construction explicite de formes paraboliques non-holomorphes sur le groupe orthogonal de signature (1, 8n + 1), pour un entier n arbitraire, à partir de relèvements de formes de Maass paraboliques de niveau un. Cela fait suite aux travaux [31] et [24], où l’outil fondamental pour démontrer l’automorphie du relèvement est le “converse theorem” de Maass. Cet article utilise les séries de Fourier associées aux relèvements theta
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Hiro-aki Narita [email protected] Yingkun Li [email protected] Ameya Pitale [email protected]
1
Fachbereich Mathematik, Technische Universität Darmstadt, Schlossgartenstr. 7, 64289 Darmstadt, Germany
2
Department of Mathematics, Faculty of Science and Engineering, Waseda University, 3-4-1 Ohkubo, Shinjuku, Tokyo 169-8555, Japan
3
Department of Mathematics, University of Oklahoma, Norman, OK, USA
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de Borcherds [4]. Nous étudions aussi les représentations cuspidales engendrées par de telles formes paraboliques et montrons qu’elles sont irréductibles et que toutes leurs composantes locales non-archimédiennes sont non-tempérées, alors que la composante archimédienne est tempérée, lorsque la forme de Maass de départ est vecteur propre pour les opérateurs de Hecke. Notre théorie locale non-archimédenne relie de maniére plus transparente la théorie locale de Sugano [39] aux formes et aux représentations automorphes non-tempérées d’un groupe orthogonal général.
1 Introduction A unique feature of automorphic forms or representations of reductive groups of higher rank (or of larger matrix size) is the existence of non-tempered cusp forms or cuspidal representations, namely cuspidal representations which have a non-tempered local component. Due to such existence the Rama
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