The paper under review investigates finite dimensional representations of Lie algebras over algebraically closed fields of characteristic \(p>0\). The Lie algebras are assumed to be restricted. The most essential results concern Lie algebras of the form \({\mathfrak g}=Lie(G)\) where G is a semisimple algebraic group. However, the representations in question are not assumed to be restricted. An important role plays the central subalgebra \({\mathcal O}={\mathcal O}({\mathfrak g})\) of the universal enveloping algebra \({\mathcal U}({\mathfrak g})\) generated by elements \(X^ p-X^{[p]}\) (X\(\in {\mathfrak g})\). Let \(\chi\) : \({\mathcal O}\to k\) be a character and \(A_{\chi}={\mathcal U}({\mathfrak g})\otimes_{{\mathcal O}}k_{\chi}\) where \(k_{\chi}\) is the \({\mathcal O}\)-algebra of dimension 1 generated by \(\chi\). The authors prove in section 1 that \(A_{\chi}\) is a Frobenius algebra of finite dimension. Moreover \(A_{\chi}\) is symmetric provided tr(ad x)\(=0\) for every \(x\in {\mathfrak g}.\)
If \({\mathfrak g}=Lie(G)\) and p is a good prime for \({\mathfrak g}\) then a unique element \(a_{\chi}\in {\mathfrak g}\) corresponds to \(\chi\). A character is called semisimple (resp. nilpotent, resp. regular) if \(a_{\chi}\) is a semisimple (resp. nilpotent, resp. regular) element.
The section 2 is devoted to a detailed consideration of the case \({\mathfrak g}={\mathfrak sl}_ 2\). The authors prove that \(A_{\chi}\) is semisimple provided \(\chi\) is a non-zero semisimple character. If \(\chi\neq 0\) is nilpotent then \(A_{\chi}\) has \((p+1)/2\) classes of isomorphic irreducible modules. One of them contains projective modules only. Projective covers of others have two isomorphic composition factors.
In Section 3 it is proved for Lie algebras of the form Lie(G) that \(A_{\chi}\) is semisimple if and only if \(a_{\chi}\) is a regular semisimple element. In section 4 the case of arbitrary regular character is considered under some restrictions on the type of G. Section 5 studies cohomologies of \(A_{\chi}\). In sections 6 and 7 so-called support varieties of \(A_{\chi}\)-modules of finite dimension are studied. The authors’ results extend their previous ones [Invent. Math. 86, 553-562 (1986; Zbl 0626.17010)]. Section 8 contains a new proof of a refined version of the Kac-Weisfeiler theorem [see V. Kac and B. Weisfeiler, Funkts. Anal. Prilozh. 5, No.2, 28-36 (1971; Zbl 0237.17003)].