Some properties of generalized reduced Verma modules over $\mathbb{Z}$-graded modular Lie superalgebras

This paper is primarily concerned with generalized reduced Verma modules over $\mathbb{Z}$-graded modular Lie superalgebras. Some properties of the generalized reduced Verma modules and the coinduced modules are obtained. Moreover, the invariant forms on the generalized reduced Verma modules are considered. In particular, we prove that the generalized reduced Verma module is isomorphic to the mixed product for modules of $\mathbb{Z}$-graded modular Lie superalgebras of Cartan type.