Suppose that G is a simple adjoint reductive group over Q, with an exceptional Dynkin type, and with G(R) quaternionic (in the sense of Gross-Wallach). Then there is a notion of modular forms for G, anchored on the so-called quaternionic discrete series representations of G(R). The purpose of this paper is to give an explicit form of the Fourier expansion of modular forms on G, along the unipotent radical N of the Heisenberg parabolic P = M N of G. 1 forms is lacking if G is of this type. However, for certain groups G, Gross and Wallach [GW96] have singled out a special class of quaternionic discrete series representations that can take the place of the holomorphic discrete series representations above. More precisely, if G is a simple reductive group over R, then G(R) possesses quaternionic discrete series [GW96] if G isogenous to one of Sp(n, 1; H), SU(n, 2), SO(n, 4), the split exceptional group G 2 , or a form of the exceptional groups F 4 , E 6 , E 7 , E 8 with real rank four. Here H is Hamilton's quaternions.Throughout the paper, G will be a reductive adjoint group, over Q or over R, with G(R) isogenous to SO(n, 4) with n ≥ 3 or the quaternionic exceptional groups G 2 , F 4 , E 6 , E 7 , E 8 . The symmetric spaces associated to these groups do not have Hermitian structure. (The forms of E 6 and E 7 with quaternionic discrete series are not the real forms that have a Hermitian symmetric space.) Nevertheless, work of Gross-Wallach [GW94, GW96] on quaternionic discrete series representations and their continuations, Wallach [Wal03] on generalized Whittaker vectors, Gan [Gan00] on Siegel-Weil formulas, and Gan-Gross-Savin [GGS02] on modular forms on G 2 , all suggest that the group G might have a good theory of modular forms, anchored upon the quaternionic discrete series representations. Furthermore, these works suggest that said modular forms might have a refined Fourier expansion, similar to the Siegel modular forms on GSp 2n .The purpose of this paper is to give the explicit Fourier expansion for modular forms on G. To define the modular forms on G and discuss their Fourier expansions, we first recall some features of these groups. Denote by K a maximal compact subgroup of G(R) 0 , the connected component of the identity of G(R). Then K ≃ (SU(2)×L 0 )/µ 2 for a certain group L 0 , with µ 2 = {±1} embedded diagonally. If 2n ≥ 1 2 dim G(R)/K, then there is a quaternionic discrete series representation π n , whose minimal K-type is of the form Sym 2n (V 2 )⊠1 = V n [GW96], as a representation of SU(2)×L 0 . Here we write V 2 for the defining two-dimensional representation of SU(2) and V n = Sym 2n (V 2 )⊠1.Assume G is an adjoint reductive Q-group of type G 2 , F 4 , E 6 , E 7 or E 8 , with G(R) quaternionic as above. As in [GGS02] and [Wei06], one can define a modular form of weight n for G to be a homomorphism ϕ ∈ Hom G(R) (π n , A(G(Q)\G(A))), from π n to the space of automorphic forms on G(A), where π n is the quaternionic discrete series representation G(R) just discussed. We make a slightly broader defini...

We give two global integrals that unfold to a non-unique model and represent the partial Spin L-function on PGSp 6 . We deduce that for a wide class of cuspidal automorphic representations π, the partial Spin L-function is holomorphic except for a possible simple pole at s = 1, and that the presence of such a pole indicates that π is an exceptional theta lift from G2. These results utilize and extend previous work of Gan and Gurevich, who introduced one of the global integrals and proved these facts for a special subclass of these π upon which the aforementioned model becomes unique. The other integral can be regarded as a higher rank analogue of the integral of Kohnen-Skoruppa on GSp 4 .

Abstract. We give a Rankin-Selberg integral representation for the Spin (degree eight) L-function on PGSp 6 that applies to the cuspidal automorphic representations associated to Siegel modular forms. If π corresponds to a level one Siegel modular form f of even weight, and if f has a nonvanishing maximal Fourier coefficient (defined below), then we deduce the functional equation and finiteness of poles of the completed Spin L-function Λ(π, Spin, s) of π.

The Rankin-Selberg integral of Kohnen and Skoruppa produces the Spin L-function for holomorphic Siegel modular forms of genus two. In this paper, we reinterpret and extend their integral to apply to arbitrary cuspidal automorphic representations of PGSp 4 . We show that the integral is related to a non-unique model and analyze it using the approach of Piatetski-Shapiro and Rallis.

By applying the residue method for period integrals and Langlands-Shahidi's theory for residues of Eisenstein series, we study the period integrals for six spherical varieties. For each spherical variety, we prove a relation between the period integrals and certain automorphic Lfunctions. In some cases, we also study the local multiplicity of the spherical varieties.