Department of Mathematics, Yale University, New Haven, Connecticut 06520.
Proc Natl Acad Sci U S A. 1982 Dec;79(24):7955-7. doi: 10.1073/pnas.79.24.7955.
E. Hecke initiated the application of representation theory to the study of cusp forms. He showed that, for p a prime congruent to 3 mod 4, the difference of multiplicities of certain conjugate representations of SL(F(p)) on cusp forms of degree 1, level p, and weight >/=2 is given by the class number h(-p) of the field Q( radical-p). We apply the holomorphic Lefschetz theorem to actions on the Igusa compactification of the Siegel moduli space of degree 2 to compute the values of characters of the representations of Sp(4)(F(p)) on certain spaces of cusp forms of degree 2 and level p at parabolic elements of this group. Our results imply that here too, the difference in multiplicities of conjugate representations of Sp(4)(F(p)) is a multiple of h(-p).
E. Hecke 开创了将表示论应用于研究尖形式的先河。他表明，对于 p 是满足模 4 余 3 的素数，SL(F(p)) 在模形式上的某些共轭表示的多重度之差，这些模形式的次数为 1，水平为 p，权大于等于 2，由 Q( radical-p) 的类数 h(-p)给出。我们将全纯 Lefschetz 定理应用于作用在 Igusa 紧化的二次型的 Siegel 模空间上，以计算 Sp(4)(F(p))在这个群的某些二次型和水平 p 的尖形式空间上的表示的特征值。我们的结果表明，在这里，Sp(4)(F(p))的共轭表示的多重度之差也是 h(-p)的倍数。
