On blocks in restricted representations of Lie superalgebras of Cartan type
Let \(\mathbf{k}\) be an algebraically closed field of finite characteristic \(p>3\). And let \(\mathfrak{g}\) be a either one of the simple Lie superalgebras \(W(n)\), \(S(n)\), \(H(n)\) of Cartan type or toral extensions \(\bar S(n)\), \(\bar H(n)\) of the last two cases. In all these cases the Lie superalgebra \(\mathfrak{g}\) is naturally equipped with the structure of a restricted Lie superalgebra. The authors consider the category \(U_0(\mathfrak{g})\mathbf{-mod}\) of restricted representations of \(\mathfrak{g}\) and prove that is it of one block for all five cases of \(\mathfrak{g}\) above. The case of \(\mathfrak{g}=W(n)\) was already treated in [\textit{Y. Yao} and \textit{B. Shu}, Chin. Ann. Math., Ser. B 34, No. 6, 921--926 (2013; Zbl 1343.17013)]. In the given paper the authors first prove that \(U_0(\mathfrak{g})\mathbf{-mod}\) is of one block modulo parities for \(\mathfrak{g}\) being \(\bar S(n)\) (Theorem 3.4) or \(\bar H(n)\), \(n\ge 5\) (Theorem 4.5). Then they prove that the same statement is true for \(S(n)\) and \(H(n)\). Finally, they complete the proof by using the Degenerate Parity Lemma (Lemma 2.6), which states that in the category \(U_0(\mathfrak{g})\mathbf{-mod}\) the irreducible modules \(L(\lambda)\) and \(\Pi L(\lambda)\) are of the same block. Here \(\Pi\) denotes the parity change functor in the category of \(\mathbf{k}\) superspaces.
scientific article; zbMATH DE number 7343748