Extensions of Restricted Simple Modules for Restricted Lie Algebras of Cartan Type

We determine the first extension groups between simple restricted modules for the restricted simple Lie algebras of Cartan type over an algebraically closed field of characteristic $p\ge 5$, which answers the restricted case of a problem of Benkart and Feldvoss. Extensions from parabolically induced modules are computed by finite restricted-cohomology complexes of the grading parabolic, and for the exceptional modules we obtain closed formulas in all four families. Neither the trivial module nor any nontrivial exceptional simple has self-extensions. In characteristic seven we determine the quiver of $S(3;1{)}^{(1)}$, which has 49 vertices and 566 arrows.