Semisimplicity and Jacobson Radicals of Reduced Enveloping Algebras of Cartan Type

Let k be an algebraically closed field of characteristic p > 3, let L be a restricted simple Lie algebra of Cartan type, and let u_chi(L) be the reduced enveloping algebra attached to chi in L*. We determine the semisimple locus and give a pointwise formula for the Jacobson radical. The radical formula is uniform: for every finite-dimensional symmetric algebra A in characteristic p, with T_n(A) = {a : a^(p^n) in [A,A]} and d = dim A/[A,A], one has T_d(A)^perp = Z(A) and Soc(A), and Jac(A) = (A T_d(A)^perp)^perp. For the Cartan families the positive loci occur in W(1) and in a single maximal-height chart of W(2). Outside these Witt cases, semisimple characters can occur only on the Reeb-open part of the Contact family, where semisimplicity is equivalent to nonvanishing of the determinant of the first Frobenius on a finite projective Morita corner. All higher Witt, Special and Hamiltonian reduced enveloping algebras are nonsemisimple for every character, and every Reeb-zero Contact algebra is likewise nonsemisimple. This answers Problem 5 of Benkart and Feldvoss.