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This chapter shows how the book's main theory applies to computing the periods of Eisenstein and Poincaré series. Our Poincaré series are eigenfunctions of the Laplacian, and they arise naturally in the theory. In particular, the periods of the Poincaré series are Kloosterman sums.

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.

This chapter introduces the Tokuyama's Theorem, first by writing the Weyl character formula and restating Schur polynomials, the values of the Whittaker function multiplied by the normalization constant. The λ‎-parts of Whittaker coefficients of Eisenstein series can be profitably regarded as a deformation of the numerator in the Weyl character formula. This leads to deformations of the Weyl character formula. Tokuyama gave such a deformation. It is an expression of ssubscript Greek small letter lamda(z) as a ratio of a numerator to a denominator. The denominator is a deformation of the Weyl denominator, and the numerator is a sum over Gelfand-Tsetlin patterns with top row λ‎ + ρ‎.

This chapter proves the theorem that asserts the modularity of the generating series and the theorem dealing with abelian varieties parametrized by Shimura curves. Before presenting the proofs, the chapter considers the new space of Schwartz functions and constructs theta series and Eisenstein series from such functions. It proceeds by discussing discrete series at infinite places, modularity of the generating series, degree of the generating series, and the trace identity. It also presents the pull-back formula for the compact and non-compact cases. In particular, it describes CM cycles on the Shimura curve, pull-back as cycles, degree of the pull-back, and some coset identities.

This chapter reviews some basic results on Weil representations, theta liftings and Eisenstein series. In particular, it introduces a proof of the Waldspurger formula. The theory of Weil representation is applied to an integral representation of the Rankin–Selberg L-function and to a proof of Waldspurger's central value formula. The chapter mostly follows Waldspurger's treatment with some modifications including Kudla's construction of incoherent Eisenstein series. It first describes the classical theory of Weil representation for an orthogonal space over a local field before discussing theta functions, the Siegel–Weil formula, and normalized local Shimizu lifting. The main result is an integral formula for the L-series using a kernel function. The Waldspurger formula is a direct consequence of the Siegel–Weil formula. After presenting the proof of Waldspurger formula, the chapter lists some computational results on three types of incoherent Eisenstein series.

This chapter provides the Eisenstein cohomology. It begins with the Poincaré duality and maximal isotropic subspace of boundary cohomology. Here, the chapter considers the compatibility of duality isomorphisms with the connecting homomorphism. It then states and proves the main result on rank-one Eisenstein cohomology. Thereafter, the chapter presents a theorem of Langlands: the constant term of an Eisenstein series. It draws some details from the Langlands–Shahidi method in this context. Induced representations are examined, as are standard intertwining operators. The chapter finally illustrates the Eisenstein series, the constant term of an Eisenstein series, and the holomorphy of the Eisenstein series at the point of evaluation.