Equivariant homology theory and twisted Yangian

We study the convolution algebra $H_{\ast }^{G\times {\text{CC}}^{\ast }}(Z)$ of $G$-equivariant homology group on the Steinberg variety of type B/C and define an algebra $\tilde{Y}$ that maps to $H_{\ast }^{G\times {\text{CC}}^{\ast }}(Z)$. The Drinfeld new realization of the twisted Yangian associated to symmetric pairs is a quotient of $\tilde{Y}$. We also study the $G$-equivariant case and prove that the twisted Yangian is the deformation of the twisted current algebra.