We use an approach based on pseudodifferential operators on Lie groupoids to study the double layer potentials on plane polygons. Let $$\Omega $$ Ω be a simply connected polygon in $$\mathbb {R}^2$$ R2 . Denote by K the double layer potential operator on $$\Omega $$ Ω associated with the Laplace operator $$\Delta $$ Δ . We show that the operator K belongs to the groupoid $$C^*$$ C∗ -algebra that the first named author has constructed in an earlier paper (Carvalho and Qiao in Cent Eur J Math 11(1):27–54, 2013). By combining this result with general results in groupoid $$C^*$$ C∗ -algebras, we prove that the operators $$\pm I + K$$ ±I+K are Fredholm between appropriate weighted Sobolev spaces, where I is the identity operator. Furthermore, we establish that the operators $$\pm I + K$$ ±I+K are invertible between suitable weighted Sobolev spaces through techniques from Mellin transform. The invertibility of these operators implies a solvability result in weighted Sobolev spaces for the interior and exterior Dirichlet problems on $$\Omega $$ Ω .