Genus theta series, Hecke operators and the basis problem for Eisenstein   series
  We derive explicit formulas for the action of the Hecke operator $T(p)$ on the genus theta series of a positive definite integral quadratic form and prove a theorem on the generation of spaces of Eisenstein series by genus theta series. We also discuss connections of our results with Kudla's matching principle for theta integrals. 
Hidenori Katsurada, Rainer Schulze-Pillot