In this talk, we are concerned with sharp estimate for the spectral projection $P_\mu$ associated with the twisted Laplacian in the Lebesgue spaces. We provide a complete characterization of the sharp $L^p-L^q$ bound for $P_\mu$, which is similar to that for the spectral projection associated with the Laplacian. As an application, we discuss the resolvent estimate for the twisted Laplacian. This talk is based on a joint work with Sanghyuk Lee and Jaehyeon Ryu.