Orbit structure of a distinguished Stein invariant domain in the complexification of a Hermitian symmetric space

Recently, Bruinier and Ono proved that the coefficients of certain weight \(-1/2\) harmonic weak Maaß forms are given as “traces” of singular moduli for harmonic weak Maaß forms. Here, we prove that similar results hold for the coefficients of harmonic weak Maaß forms of weight \(3/2+k\) , \(k\) even, and weight \(1/2-k\) , \(k\) odd, by extending the theta lift of Bruinier–Funke and Bruinier–Ono. Moreover, we generalize these results to include twisted traces of singular moduli using earlier work of the author and Ehlen on the twisted Bruinier–Funke-lift. Employing a general duality result between weight \(k\) and \(2-k\) , we obtain formulas for all half-integral weights. We also show that the non-holomorphic part of the theta lift in weight \(1/2-k\) , \(k\) odd, is connected to the vanishing of the special value of the \(L\) -function of a certain derivative of the lifted function.