Automorphic Forms on Certain Affine Symmetric Spaces.

In this thesis, we consider automorphic periods associated to certain affine symmetric spaces such as the symmetric pairs &parl0;Sp4n,Res K/kSp2n&parr0; and&parl0;Gsp4n ,ResK/k GSp2n&parr0;, where k is a number field and K is an Etale algebra over k of dimension 2. We consider the period integral of a cusp forms of Sp4n( Ak ) against with an Eisenstein series of the symmetric subgroup Res K/kSp2. We expect to establish an identity between this period integrals and the special value of the spin L-function of the symplectic group.

In the local theory, using Aizenbud and Gourevitch's generalized Harish-Chandra method and traditional methods, i.e. the Gelfand-Kahzdan theorem, we can prove that these symmetric pairs are Gelfand pairs when Kv is a quadratic extension field over kv for any n, or Kv is isomorphic to k v x kv for n ≤ 2. Since (U(J2n , kv( t )), Sp2n(kv)) is a descendant of (Sp4n(k v), Sp2nk v) x Sp2n( kv)), we prove that it is a Gelfand pair for both archimedean and non-archimedean fields.

According to the Yu' construction of irreducible tame supercuspidal representations, we give a parametrization of the distinguished tame supercuspidal representation of symplectic groups in this thesis. Applying the dimension formula of the space HomH(pi, 1) given by Hakim and Murnaghan, we prove that if (G, H) is the symmetric pair (U(J2n, Kv), Sp2n( kv)) there is no H-distinguished tame supercuspidal representation, where Kv is a quadratic extension over kv. In addition, for the symmetric pair (Sp4n(kv), Sp2n(Kv)), we give the sufficient and necessary conditions of generic cuspidal data such that the corresponding tame supercuspidal representations are H-distinguished. Note that our case is the first case worked out with none of G and H being the general linear groups. Furthermore, motived by a sub-question, we also give an example for the distinguished representations of finite groups of Lie Type in a low rank case. In particular, we show that theta 10 is the unique SL2( Fq2 )-distinguished cuspidal representation of Sp4( Fq ).