This chapter and the next chapter are the technical heart of this book. In this chapter, we estimate the Eisenstein series on GL(n) for the Borel subgroup B. This reduces to the estimate of Whittaker functions on GL(n). The estimate of the Whittaker functions consists of two problems. One is at finite places, and the other is at infinite places. We prove the explicit formula for the p-adic Whittaker functions in §2.3 following Shintani [66]. Even though some estimate of Whittaker functions are known, the uniformity of the estimate is a big issue in this book, because we intend to consider contour integrals. Therefore, it is necessary to prove an estimate which is of polynomial growth with respect to the parameter of the Lie algebra. For this purpose, we generalize Shintani's approach in [64], which is the naive use of integration by parts. As far as GL(3) is concerned, Bump's estimate [5] is the best possible estimate. Our estimate, even though it is enough for our purposes for the moment, may not be the optimum estimate, and we may have to prove a better estimate in the future to handle more prehomogeneous vector spaces.

The Fourier expansion of automorphic forms on GL(n)

In this section, we review the notion of the Fourier expansion of automorphic functions on GL(n) following Piatecki–Shapiro in [51].