Symplectic Dirac cohomology and lifting of characters to metaplectic groups

We formulate the transfer factor of character lifting from orthogonal groups to symplectic groups by Adams in the framework of symplectic Dirac cohomology for the Lie superalgebras and the Rittenberg-Scheunert correspondence of representations of the Lie superalgebra $\text{fro}\text{frsp}(1\mathrm{\mid }2n)$ and the Lie algebra $\text{fro}(2n+1)$. This leads to formulation of a direct lifting of characters from the linear symplectic group $Sp(2n,\text{bbR})$ to its nonlinear covering metaplectic group $Mp(2n,\text{bbR})$.