The Ramanujan Journal

, Volume 17, Issue 2, pp 219–244 | Cite as

Inverting the Satake map for Sp n and applications to Hecke operators

  • Nathan C. Ryan
  • Thomas R. Shemanske


By compatibly grading the p-part of the Hecke algebra associated to Sp n (ℤ) and the subring of ℚ[x 0 ±1 ,…,x n ±1 ] invariant under the associated Weyl group, we produce a matrix representation of the Satake isomorphism restricted to the corresponding finite dimensional components. In particular, using the elementary divisor theory of integral matrices, we show how to determine the entries of this matrix representation restricted to double cosets of a fixed similitude. The matrix representation is upper-triangular, and can be explicitly inverted.

To address the specific question of characterizing families of Hecke operators whose generating series have “Euler” products, we define (n+1) families of polynomial Hecke operators t k n (p ) (in ℚ[x 0 ±1 ,…,x n ±1 ]) for Sp n whose generating series ∑t k n (p )v are rational functions of the form q k (v)−1, where q k is a polynomial in ℚ[x 0 ±1 ,…,x n ±1 ][v] of degree \(2^{k}{n\choose k}\) (2 n if k=0). For k=0 and k=1 the form of the polynomial is essentially that of the local factors in the spinor and standard zeta functions. For k>1, these appear to be new expressions. Taking advantage of the generating series and our ability to explicitly invert the Satake isomorphism, we explicitly compute the classical operators with the analogous properties in the case of genus 2. It is of interest to note that these operators lie in the full, but not generally the integral, Hecke algebra.


Satake map Hecke operators Siegel modular forms Symplectic group 

Mathematics Subject Classification (2000)

11F60 11F46 


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  1. 1.
    Andrianov, A.N.: Quadratic Forms and Hecke Operators. Springer, Berlin (1987) MATHGoogle Scholar
  2. 2.
    Breulmann, S., Kuss, M.: On a conjecture of Duke-Imamoḡlu. Proc. Am. Math. Soc. 128(6), 1595–1604 (2000) MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Cartier, P.: Representations of p-adic groups: a survey. In: Automorphic Forms, Representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1, pp. 111–155. Am. Math. Soc., Providence (1979) Google Scholar
  4. 4.
    Hafner, J., Walling, L.: Siegel modular forms and Hecke operators in degree 2. In: Millennial Conference on Number Theory, A.K. Peters (2002) Google Scholar
  5. 5.
    Krieg, A.: Das Vertauschungsgesetz zwischen Hecke-Operatoren und dem Siegelschen φ-Operator. Arch. Math. (Basel) 46(4), 323–329 (1986) MATHMathSciNetGoogle Scholar
  6. 6.
    Newman, M.: Integral Matrices. Pure and Applied Mathematics, vol. 45. Academic, New York (1972) MATHGoogle Scholar
  7. 7.
    Rhodes, J.A., Shemanske, T.R.: Rationality theorems for Hecke operators on GL n. J. Number Theory 102, 278–297 (2003) MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Satake, I.: Theory of spherical functions on reductive algebraic groups over \(\mathfrak{p}\) -adic fields. Inst. Hautes Etudes Sci. Publ. Math. 18, 1–69 (1963) CrossRefMathSciNetGoogle Scholar
  9. 9.
    Shimura, G.: Introduction to the Arithmetic Theory of Automorphic Functions. Publications of the Mathematical Society of Japan, No. 11. Iwanami Shoten, Tokyo (1971), Kanô Memorial Lectures, No. 1 MATHGoogle Scholar
  10. 10.
    Veenstra, T.B.: Siegel modular forms, L-functions, and Satake parameters. J. Number Theory 87(1), 15–30 (2001) MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Department of MathematicsBucknell UniversityLewisburgUSA
  2. 2.Department of MathematicsDartmouth CollegeHanoverUSA

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