On the theta functional of Weil representations for symplectic loop groups

Abstract

We prove the convergence of the theta functional for loop group Weil representation under certain natural conditions and discuss the modularity.

Theta liftings for $(\mathrm{GL}_n,\mathrm{GL}_n)$ type dual pairs of loop groups

“…In this section we briefly recall the Weil representations of symplectic loop groups over local and global fields, as well as the absolute convergence of theta functionals, following [Z] and [LZ1].…”

“…We also need Q t := Q((t))∩A t . Let (G, G ′ ) be a reductive dual pair over Q sitting inside Sp(W ), Following [LZ1], we have the adelic loop symplectic group Sp(W A t ), which is a central extension of Sp(W A t ) and can be written as a certain restricted product of local loop groups. Its Weil representation, denoted by ω, is realized on a certain space S ′ (V −,A ) of Bruhat-Schwartz functions, where V − := W [t −1 ]t −1 is a Larangian space of W t .…”

“…converges absolutely [Z,LZ1], and g → θ(ω(g)f ) defines an automorphic function on the arithmetic quotient Sp(W t )\ Sp(W A t ).…”

“…The organization of this paper is as follows. In Section 2, we briefly recall the local and global theory of Weil representations and theta functionals for loop symplectic groups following [Z,LZ1]. In Section 3, we construct a different model of the representation of GL n (A) and prove that is isomorphic to S ′ (V −,A ).…”

Theta functions and arithmetic quotients of loop groups

“…The most technical problem is the convergence of the limit defining ϑ( g). Using a variant of the lemmas in [4] and [12], together with some elementary Fourier analysis, in particular the Poisson summation formula, we establish in Section 3 the uniform convergence of ϑ( g) for g varying in certain Siegel subset of G. In the course of the proof, the use of Iwasawa decompositions for loop groups simplify our considerations. Motivated by the work of Y. Zhu [12] on Weil representations, we also extend the theta functions to loop symplectic groups in Section 4.…”