Theta lifts to certain cohomological representations of indefinite orthogonal groups

2023/07/06 by Takuya Miyazaki, Miyazaki, Takuya, Saito Yohei +1 · 1 citation
Mathematics · Chemistry · #Advanced Algebra and Geometry #Molecular spectroscopy and chirality #Algebraic Geometry and Number Theory

paper · pdf · doi:10.48550/arxiv.2307.02926

Abstract

Howe and Tan (1993) investigated a degenerate principal series representation of indefinite orthogonal groups O(V) and explicitly described its composition series. They showed that there exists a unique unitarizable irreducible submodule Π, which is isomorphic to a cohomological representation. In this paper we construct orthogonal automorphic forms belonging to Π as theta liftings of holomorphic Mp₂(ℝ) cusp forms by using the Borcherds' method. We will give a useful observation in computing their Fourier expansions that enables us to give very precise descriptions of the liftings specially belonging to Π. We will also determine the cuspidal supports of the liftings, then confirm the square integrability of those automorphic forms on orthogonal groups.

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