Exceptional theta correspondence F₄\timesPGL₂ for level one automorphic representations

2025/01/31 by Shan, Yi
#11F #11F27 #11F55 #11F67 #11F70 #20G41 #FOS: Mathematics #Number Theory (math.NT) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2501.19101

Abstract

Let F₄ be the unique (up to isomorphism) connected semisimple algebraic group over ℚ of type F₄, with compact real points and split over ℚ_p for all primes p. A conjectural computation by the author in arxiv:2407.05859 predicts the existence of a family of level one automorphic representations of F₄, which are expected to be functorial lifts of cuspidal representations of PGL₂ associated with Hecke eigenforms. In this paper, we study the exceptional theta correspondence for F₄\timesPGL₂, and establish the existence of such a family of automorphic representations for F₄. Motivated by the work of Pollack, our main tool is a notion of "exceptional theta series" on PGL₂, arising from certain automorphic representations of F₄. These theta series are holomorphic modular forms on SL₂(ℤ), with explicit Fourier expansions, and span the entire space of level one cusp forms.

Related

• The Theta Correspondence, Periods of Automorphic Forms and Special Values of Standard Automorphic L-Functions
• Exceptional Siegel-Weil theorems for compact Spin₈
• Theta correspondences for GSp(4)
• Level one automorphic representations of an anisotropic exceptional group over ℚ of type F₄
• Howe duality and dichotomy for exceptional theta correspondences