An explicit construction of non-tempered cusp forms on O(1 , 8 n+ 1)

Yingkun Li, Hiro aki Narita*, Ameya Pitale

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)


We explicitly construct non-holomorphic cusp forms on the orthogonal group of signature (1 , 8 n+ 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.

Original languageEnglish
Pages (from-to)349-384
Number of pages36
JournalAnnales Mathematiques du Quebec
Issue number2
Publication statusPublished - 2020 Oct 1


  • Lifting from Maass cusp forms
  • Non-tempered cusp forms
  • Orthogonal group of signature (1, 8n)
  • Special Bessel models
  • Theta lifting

ASJC Scopus subject areas

  • General Mathematics


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