Multiplier Modules of extended Rees algebras

Given a local ring $(R, \mathfrak{m})$ and an ideal $\mathfrak{a}$ of positive height, we give a way of computing multiplier module ${J}(\omega_{{T}}, t^{-\lambda})$ for the extended Rees algebra ${T} =R[\mathfrak{a} t, t^{-1}]$ for an ideal $\mathfrak{a}$ by proving a decomposition theorem for ${J}(\omega_{{T}}, t^{-\lambda})$, (also see the works of Budur, Musta\c{t}\u{a} and Saito). We compute the multiplier module ${J}(\omega_{{S}}, (\mathfrak{a} \cdot {S})^{\lambda})$ for the Rees algebra ${S} =R[\mathfrak{a} t]$ as well (also see the works of Hyry and Kotal-Kummini). We use these decompositions to understand relationships between associated graded rings, Rees and extended Rees algebras having rational singularities (also see the works of Hara, Watanabe, and Yoshida).