Spectral and eigenfunction asymptotics in Toeplitz quantization

Toeplitz operators on quantized compact symplectic manifolds were introduced by Guillemin and Boutet de Monvel, who studied their spectral asymptotics in analogy with the theory developed by Duistermaat, Guillemin, and H\"{o}rmander for pseudodifferential operators. In this survey, we review some recent results concerning eigenfunction asymptotics in this context, largely based on the microlocal description of Szeg\"{o} kernels by Boutet de Monvel and Sj\"{o}strand, and its revisitation and generalization to the almost complex symplectic category by Shiffman and Zelditch. For simplicity, the exposition is restricted to the complex projective setting.