Hecke Eigenvalues of Ikeda Lifts

In this paper, we study the Hecke eigenvalues of Ikeda lifts. Using the spherical map for the Hecke algebra of the symplectic group, we obtain an explicit formula for the eigenvalues ${\lambda }_F(p^r)$. From this formula, we show that ${\lambda }_F(p^r)$ can be written as a polynomial in $p^{\pm 1\mathrm{/}2}$ with a positive leading term. Furthermore, we prove that the coefficients of this polynomial are bounded and, as a consequence, the Hecke eigenvalues ${\lambda }_F(p^r)$ are positive for all sufficiently large primes $p$.