Klingen Eisenstein series

In mathematics, a Klingen Eisenstein series is a Siegel modular form of weight k and degree g depending on another Siegel cusp form f of weight k and degree r g + r + 1 an even integer. The Klingen Eisenstein series is


 * $$ \sum_{\binom{ab}{cd}\isin P_r\setminus\Gamma_g} f\left(\frac{a\tau+b}{c\tau+d}\right)\det(c\tau+d)^{-k}.$$

It is a Siegel modular form of weight k and degree g. Here Pr is the integral points of a certain parabolic subgroup of the symplectic group, and Γr is the group of integral points of the degree g symplectic group. The variable τ is in the Siegel upper half plane of degree g. The function f is originally defined only for elements of the Siegel upper half plane of degree r, but extended to the Siegel upper half plane of degree g by projecting this to the smaller Siegel upper half plane.

The cusp form f is the image of the Klingen Eisenstein series under the operator Φg−r, where Φ is the Siegel operator.