Good filtrations of representations of Lie algebras of Cartan type
The authors continue the investigation of \(D(G)\mathfrak u(\mathfrak g)\)-modules, where \(\mathfrak g\) is a Lie \(p\)-algebra of Cartan type, \(G=\Aut\mathfrak g\), \(D(G)\) is the distribution algebra of \(G\), \(\mathfrak u(\mathfrak g)\) is the restricted enveloping algebra of \(\mathfrak g\), \(D(G)\mathfrak u(\mathfrak g)\) is a natural Hopf algebra containing \(\mathfrak u(\mathfrak g)\) as a normal Hopf subalgebra [see \textit{Z. Lin} and \textit{D. K. Nakano}, J. Algebra 179, 852-888 (1996; Zbl 0862.17015)]. Denote \(D(G)\mathfrak u(\mathfrak g)\) by \(A\). Let \(\mathfrak g=\bigoplus _{i=-s}^r \mathfrak g_i\) be the standard grading of \(\mathfrak g, \mathfrak n^-=\bigoplus _{i<0}\mathfrak g_i, \mathfrak n^+=\bigoplus _{i>0}\mathfrak g_i, \mathfrak b^{\pm }=\mathfrak g_0\oplus \mathfrak n^{\pm }.\) It is well known that \(G\cong G_0\ltimes U^+\) where \(G_0\) is reductive (\(G_0\) is one of the groups \(GL_m, GSP_{2r}\)), \(U^+\) is unipotent. Let \(T\) be a maximal torus in \(G_0\), \(\lambda \in X(T)_+\), \[ Z^-(\lambda)=H^0(\lambda)\cong \text{Ind}_{D(G_0)\mathfrak u(\mathfrak b^-)}^A H^0(\lambda), \] where \(H^0(\lambda)\) is the classical induced \(G_0\)-module. Note that \(Z^-(\lambda)\) is infinite-dimensional and \(Z^-(\lambda)\cong H^0(\lambda)\otimes k[U^+]\) as \(D(G_0)\)-module. A \(G_0\)-module \(M\) may be considered as the trivial \(U^+\)-module and one can build the coinduced module \(\text{Coind}_{D(G)}^AM=A\bigotimes _{D(G_0)}M.\) Let \(w_0\) be the longest element in the Weyl group of \(G_0, V(\lambda)=H^0(-w_0 (\lambda))^*\) the Weyl module of highest weight \(\lambda \) for \(G_0\) and \(Z^+(\lambda)=\text{Coind}_{D(G)}^AV(\lambda).\) The latter is finite-dimensional and isomorphic to \(\mathfrak u(\mathfrak n^-)\bigotimes_kV(\lambda)\) as \(D(G_0)\)-module. \(Z^+(\lambda)\) is an analog of the Weyl module for \(A\)-modules. Let \(\mathcal P\) be a collection of \(A\)-modules. A module \(M\) is said to have a good \(\mathcal P\)-filtration if there is a filtration \(0=M_0\subseteq M_1 \subseteq M_2\subseteq \ldots\) such that \(M=\cup M_i\) and \(M_i/M_{i-1}\) is isomorphic to a member of \(\mathcal P.\) In the paper under review for an \(A\)-module \(M\) the connection between a good \(Z^-\)-filtration of \(M\) and a good \(H^0\)-filtration of \(M^{U^+}\) is established (Th. 3.3). It is proved that if an \(A\)-module \(V\) admits a filtration \(0=V_0\subseteq V_1\subseteq V_2\subseteq \ldots, V=\cup V_i\) such that \(V_i/V_{i-1}\) have a good \(Z^-\)-filtration then \(V\) also admits a good \(Z^-\)-filtration (Th. 3.4). The authors prove that under certain natural restrictions an \(A\)-module \(M\) admits a good \(Z^-\)-filtration iff \(\text{Ext}_A^1(Z^+(\lambda), M)=0\) for all \(\lambda \in X(T)_+.\) For \(M\) having a good \(Z^-\)-filtration the multiplicities \([M: Z^-(\lambda)]\) are found (Th. 3.5). It is shown that the injective hulls of simple \(A\)-modules admit a good \(Z^-\)-filtration. For finite-dimensional \(A\)-modules the analogous criterion of the existence of a good \(Z^+\)-filtration (Weyl filtration) is obtained (Prop. 4.1). Sufficient conditions for the case when \(Z^+(\lambda)\otimes Z^+(\gamma)\) admits a \(Z^+\)-filtration are given (Lemma 4.2). It is shown that these conditions are fulfilled for \(\mathfrak g=W(1:1)\), \(W(2:1)\), \(H(2:1).\) However, for \(\mathfrak g=W(3:1)\) the \(A\)-module \(Z^+(0)\otimes Z^+(0)\) does not admit a \(Z^+\)-filtration. For a finite-dimensional \(A\)-module a character formula in terms of Weyl characters of the reductive group \(G_0\) and ``Kazhdan-Lusztig type'' coefficients is given.
scientific article; zbMATH DE number 1308165
scientific article