Irreducible representations of the generalized Jacobson-Witt algebras (Q2880063)

Let \(\mathbf{n}=(n_1,\dots,n_m)\) be an arbitrary \(m\)-tuple of positive integers and let \(L=W(m;\mathbf{n})\) denote a graded generalized Jacobson-Witt algebra over an algebraically closed field of characteristic \(p>3\). The goal of the paper under review is to use the setup of generalized restricted Lie algebras to prove that, up to finitely many exceptions, irreducible representations of \(L\) with generalized \(p\)-characters of height at most \(\min\{p^{n_i}-p^{n_i-1} \mid 1\leq i\leq m\}-1\) can be obtained as modules induced from an irreducible representation of the distinguished maximal subalgebra \(L_0\) of \(L\). (Here the \(L\)-action on the induced modules is twisted in order to ensure that the exceptions are exactly the modules induced from the irreducible \(\mathfrak{gl}_m\)-modules having as highest weight some fundamental weight.) In particular, every irreducible module of \(L\) with generalized \(p\)-character of height between \(1\) and \(\min\{ p^{n_i}-p^{n_i-1}\mid 1\leq i\leq m\}-1\) is induced from some irreducible \(L_0\)-module, and the number of isomorphism classes of irreducible \(L\)-modules with such a generalized \(p\)-character \(\chi\) is the same as the number of isomorphism classes of irreducible \(L_0\)-modules with \(p\)-character \(\chi_{\mid L_0}\). NEWLINENEWLINENEWLINENEWLINEThe main technical tool is the concept of a \(\mathfrak{C}\)-category (in the paper under review called a category of \((R,L)\)-modules). The definition of a \(\mathfrak{C}\)-category involves an \(L\)-module structure, its restriction to \(L_0\), a module structure coming from the defining divided power algebra of \(L\), and several compatibility conditions, and was introduced by \textit{S. Skryabin} [``Independent systems of derivations and Lie algebra representations'', Algebra and Analysis. Proceedings of the International Centennial Chebotarev Conference, Kazan, Russia, June 5-11, 1994. Arslanov, M.\ M.\ (ed.) et al.\ Berlin: Walter de Gruyter.\ 115--150 (1996; Zbl 0878.17004)] to study representations of restricted Jacobson-Witt algebras. The authors show that induced modules with generalized \(p\)-characters belong to such a category. This enables them to extend some of Skryabin's arguments from \(\mathbf{n}=(1,\dots, 1)\) to arbitrary \(\mathbf{n}\). In the case of a generalized \(p\)-character of height \(0\) and \(m>1\) the authors also realize the exceptional irreducible modules corresponding to a fundamental weight of \(\mathfrak{gl}_m\) in terms of a certain de Rham complex. In particular, the number of isomorphism classes of irreducible modules with a generalized \(p\)-character of height \(0\) and their dimensions are determined.NEWLINENEWLINEThe results in the paper under review generalize the results obtained by \textit{R. R. Holmes} [J.\ Algebra 237, No.\ 2, 446--469 (2001; Zbl 1005.17015)] and \textit{C. Zhang} [J.\ Algebra 290, No.\ 2, 408--432 (2005; Zbl 1137.17308)] in the restricted case \(\mathbf{n}=(1,\dots,1)\). It would be interesting to compare the authors' results for the restricted Jacobson-Witt algebra \(W(2,(1,1))\) with those obtained by \textit{N. A. Koreshkov} [Izv.\ Vyssh.\ Uchebn.\ Zaved., Mat.\ 1980, No.\ 4(215), 39--46 (1980; Zbl 0441.17008)] who gave a complete description of all isomorphism classes of the irreducible modules in this case. Finally, it should be remarked that in the meantime the authors extended their results to graded Lie algebras of types \(S\) [``Irreducible representations of the special algebras in prime characteristic, in: Representation theory, Contemp. Math. 478, 273--295 (2009; Zbl 1176.17014)] and \(H\) [J.\ Aust.\ Math.\ Soc.\ 90, No.\ 3, 403--430 (2011; Zbl 1235.17005)].