On a theta lift related to the Shintani lift

We study the Millson theta lift which maps weight −2k to weight 1/2−k harmonic weak Maass forms for k∈Z,k≥0, and which is closely related to the classical Shintani lift from weight 2k+2 to weight k+3/2 cusp forms. We compute the Fourier expansion of the theta lift and show that it involves twisted traces of CM values and geodesic cycle integrals of the input function. As an application, we recover Zagier's generating series of twisted traces of singular moduli of weight 1/2 as Millson lifts.