On a strange invariant bilinear form on the space of automorphic forms

Abstract

Let F be a global field and \(G:=SL(2)\). We study the bilinear form \({{\mathcal {B}}}\) on the space of K-finite smooth compactly supported functions on \(G({\mathbb {A}})/G(F)\) defined by $$\begin{aligned} {{\mathcal {B}}}(f_1,f_2):={{\mathcal {B}}}_{\mathrm {naive}}(f_1,f_2)-\langle M^{-1}{{\mathrm{{CT}}}}(f_1)\, ,{{\mathrm{{CT}}}}(f_2)\rangle , \end{aligned}$$ where \({{\mathcal {B}}}_{\mathrm {naive}}\) is the usual scalar product, \({{\mathrm{{CT}}}}\) is the constant term operator, and M is the standard intertwiner. This form is natural from the viewpoint of the geometric Langlands program. To justify this claim, we provide a dictionary between the classical and ‘geometric’ theory of automorphic forms. We also show that the form \({{\mathcal {B}}}\) is related to S. Schieder’s Picard–Lefschetz oscillators.