Whittaker Functions

This chapter shows that Weyl group multiple Dirichlet series are expected to be Whittaker coefficients of metaplectic Eisenstein series. The fact that Whittaker coefficients of Eisenstein series reduce to the crystal description that was given in Chapter 2 is proved for Type A. On the adele group, the corresponding local computation reduces to the evaluation of a type of λ‎-adic integral. These were considered by McNamara, who reduced the integrals to sums over crystals by a very interesting method. A full treatment of this topic is outside the scope of this work, but it is introduced in this chapter by considering the case where n = 1. In this chapter, F is used to denote a nonarchimedean local field and F to denote a global field. The values of the Whittaker function are Schur polynomials multiplied by the normalization constant.

Integral moments of automorphic L-functions

We obtain second integral moments of automorphic L-functions on adele groups GL2 over arbitrary number fields, by a spectral decomposition using the structure and representation theory of adele groups GL1 and GL2. This requires reformulation of the notion of Poincaré series, replacing the collection of classical Poincaré series over GL2(ℚ) or GL2(ℚ(i)) with a single, coherent, global object that makes sense over a number field. This is the first expression of integral moments in adele-group terms, distinguishing global and local issues, and allowing uniform application to number fields. When specialized to the field of rational numbers ℚ, we recover the classical results on moments.

On the lifting of hermitian modular forms

Let K be an imaginary quadratic field with discriminant −D. We denote by 𝒪 the ring of integers of K. Let χ be the primitive Dirichlet character corresponding to K/ℚ. Let $\Gamma ^{(m)}_K=\mathrm {U} (m,m)({\mathbb Q})\cap \mathrm {GL}_{2m}({\cal O})$ be the hermitian modular group of degree m. We construct a lifting from S2k(SL2(ℤ)) to S2k+2n(ΓK(2n+1),det −k−n) and a lifting from S2k+1(Γ0(D),χ) to S2k+2n(ΓK(2n),det −k−n). We give an explicit Fourier coefficient formula of the lifting. This is a generalization of the Maass lift considered by Kojima, Krieg and Sugano. We also discuss its extension to the adele group of U(m,m).

Residual automorphic representations of Sp4

Let G = Sp4 be the symplectic group of degree two defined over an algebraic number field F and K the standard maximal compact subgroup of the adele group G (A). By the general theory of Eisenstein series ([14]), one knows that the Hilbert space L2(G(F)\G(A)) has an orthogonal decomposition of the formL2(G(F)\G(A)) = L2(G) ⊕ L2(B) ⊕ L2(P1) ⊕ L2(P1),where B is a Borel subgroup and Pi are standard maximal parabolic subgroups in G for i = 1,2. The purpose of this paper is to study the space L2d(B) associated to discrete spectrurns in L2(B).