Compositio Mathematica

Research Article

Geometric Waldspurger periods

Sergey Lysenkoa1

a1 Université Paris 6, Institut de Mathématiques, Analyse Algébrique, 175 rue du Chevaleret, F-75013 Paris, France (email:


Let X be a smooth projective curve. We consider the dual reductive pair $H=\mathrm {G\mathbb {O}}_{2m}$, $G=\mathrm {G\mathbb {S}p}_{2n}$ over X, where H splits on an étale two-sheeted covering $\pi :\tilde X\to X$. Let BunG (respectively, BunH) be the stack of G-torsors (respectively, H-torsors) on X. We study the functors FG and FH between the derived categories D(BunG) and D(BunH), which are analogs of the classical theta-lifting operators in the framework of the geometric Langlands program. Assume n=m=1 and H nonsplit, that is, $H=\pi _*{\mathbb {G}_m}$ with $\tilde X$ connected. We establish the geometric Langlands functoriality for this pair. Namely, we show that FG :D(BunH)→D(BunG) commutes with Hecke operators with respect to the corresponding map of Langlands L-groups LHLG. As an application, we calculate Waldspurger periods of cuspidal automorphic sheaves on BunGL2 and Bessel periods of theta-lifts from $\mathrm {Bun}_{\mathrm {G\mathbb {O}}_4}$ to $\mathrm {Bun}_{\mathrm {G\mathbb {S}p}_4}$. Based on these calculations, we give three conjectural constructions of certain automorphic sheaves on $\mathrm {Bun}_{\mathrm {G\mathbb {S}p}_4}$ (one of them makes sense for ${\mathcal D}$-modules only).

(Received March 17 2006)

(Accepted May 21 2007)

(Online publication March 14 2008)


  • geometric Langlands program;
  • Howe correspondence

2000 Mathematics subject classification

  • 11R39;
  • 14H60