Symmetrisation of Saito-Kurokawa and Borcherds lifts

There is a unique symmetry property characterizing Saito-Kurokawa lifts (also called Maass lifts) and Borcherds lifts.
The additive version determines SK lifts and gives a new approach to study the Maass Spezialschar and Fourier coefficients.
The multiplicative version determines Borcherds lifts, and gives new results on Heegner Divisors (joint work with Kaiser and Murase).
In this talks I will give some basic applications of the symmetry principle.
Recently the symmetry principle was successfully applied to the Maass Spezialschar of level N with Character, based on
work of Ibukiyama and joint work with Murase. On this case and known results will be reported in more detail.