  The symmetric coinvariant algebra $C[x_1, dots, x_n]_{S_n}$ is the quotient
algebra of the polynomial ring by the ideal generated by symmetric polynomials
vanishing at the origin. It is known that the algebra is isomorphic to the
regular representation of $S_n$. Replacing $C[x]$ with $A = C[x,y]/(xy)$, we
introduce another symmetric coinvariant algebra $A^{otimes n}_{S_n}$ and
determine its $S_n$-module structure. As an application, we determine the
$sl_{r+1}$-module structure of the local Weyl module at a double point for
$sl_{r+1} otimes A$.
