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Fourier coefficients of $\operatorname{Sp}(4)$ Eisenstein Series

Siu Hang Man

Abstract

We compute explicit formulae for the constant terms and Fourier coefficients for Eisenstein series on $\operatorname{Sp}(4,\mathbb{R})$, in terms of zeta functions and Whittaker functions. We also develop a generalisation of Ramanujan sums to $\operatorname{Sp}(4,\mathbb{Z})$, which appears as coefficients in the Fourier expansion for the minimal Eisenstein series.

Fourier coefficients of $\operatorname{Sp}(4)$ Eisenstein Series

Abstract

We compute explicit formulae for the constant terms and Fourier coefficients for Eisenstein series on , in terms of zeta functions and Whittaker functions. We also develop a generalisation of Ramanujan sums to , which appears as coefficients in the Fourier expansion for the minimal Eisenstein series.

Paper Structure

This paper contains 18 sections, 23 theorems, 152 equations.

Table of Contents

  1. Introduction
  2. Group decompositions
  3. Coset representatives
  4. Plücker coordinates
  5. Bruhat decomposition
  6. Eisenstein series
  7. Constant terms
  8. Constant term along P0
  9. Constant term along Pa
  10. Constant term along Pb
  11. $\operatorname{Sp}(4)$ Ramanujan sums
  12. Fourier coefficients of eisenstein series
  13. Invariant differential operators
  14. Jacquet's Whittaker functions
  15. Fourier Coefficients
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $\chi = \chi_{m_1,m_2}$ be a character of $U(\mathbb{Z})\backslash U(\mathbb{R})$. Then the Fourier coefficients of the minimal Eisenstein series $E_0 (g,\nu)$ are given as follows. For $m_1=m_2=0$ we have For $m_1\ne 0$, $m_2=0$ we have For $m_1=0$, $m_2\ne 0$ we have For $m_1,m_2\ne 0$ we have Here $W_w(g,\nu,\chi)$ are Jacquet's Whittaker functions, defined in eq:wd, $\sigma_\nu(m) = \s

Theorems & Definitions (35)

  • Theorem 1.1
  • Proposition 3.1
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • proof
  • Theorem 4.1: Langlands Langlands1976
  • Proposition 4.2
  • Proposition 4.3
  • Proposition 4.4
  • ...and 25 more