Whittaker modules for the derivation Lie algebra of torus with two variables

Let $mathcal{L}$ be the derivation Lie algebra of ${mathbb C}[t_1^{pm 1},t_2^{pm 1}]$. Given a triangle decomposition $mathcal{L} =mathcal{L}^{+}oplusmathfrak{h}oplusmathcal{L}^{-}$, we define a nonsingular Lie algebra homomorphism $psi:mathcal{L}^{+}rightarrowmathbb{C}$ and the universal Whittaker $mathcal{L}$-module $W_{psi}$ of type $psi$. We obtain all Whittaker vectors and submodules of $W_{psi}$, and all simple Whittaker $mathcal{L}$-modules of type $psi$.