Complexity of Lie algebra representations and nilpotent elements of the stabilizers of linear forms
Let \(G\) be a reductive group over an algebraically closed field of characteristic \(p>0,\) \(\mathfrak g= \text{Lie}(G).\) Let \(u(\mathfrak{g}, \chi)\) be the quotient of the universal enveloping algebra \(U(\mathfrak{g})\) over the two-sided ideal generated by all elements \(x^p-x^{[p]}-\chi (x)^p\cdot 1, x\in \mathfrak{g},\) where \(\chi \) is a linear function on \(\mathfrak{g}.\) In the paper the support varieties \(\mathcal N _{\mathfrak{g}}(N)\) of the modules \(N\) over \(u(\mathfrak{g}, \chi)\) are investigated under the assumptions that \(G^{(1)}\) is simply connected and \(p\) is a good prime number for the root system \(R=R(G),\) i.e. \(p\) is bigger than any coefficient of any positive root of \(R\) with respect to a basis of \(R.\) Let \(N_p(\mathfrak{g})\) denote the closed, conical subset of \(\mathfrak{g}\) consisting of all \(x\in \mathfrak{g}\) such that \(x^{[p]}=0,\) \(\mathfrak{z}_{\mathfrak{g}}(\chi)\) denote the stabilizer of \(\chi \) in \(\mathfrak{g}.\) Earlier the author proved that for \(\mathfrak{g}\) semisimple, \(\mathcal N _{\mathfrak{g}}(N)\subset N_p(\mathfrak{g})\cap \mathfrak{z}_{\mathfrak{g}}(\chi)\) for any \(u(\mathfrak{g}, \chi)\)-module \(N\) and conjectured that there exists a finite dimensional \(u(\mathfrak{g}, \chi)\)-module \(E\) such that \(\mathcal N _{\mathfrak{g}}(E)=N_p(\mathfrak{g})\cap \mathfrak{z}_{\mathfrak{g}}(\chi)\) [see \textit{A. Premet}, J. Lond. Math. Soc. (2) 55, 236-250 (1997; Zbl 0889.17019)]. In the present paper this conjecture is proved under the above assumptions on \(p\) and \(G.\) Since any closed, conical subset of \(\mathcal N _{\mathfrak{g}}(E)\) is equal to \(\mathcal N _{\mathfrak{g}}(N)\) for some \(u(\mathfrak{g}, \chi)\)-module \(N\), this result gives the description of all support varieties for \(u(\mathfrak{g}, \chi)\)-modules. Further it is applied to investigate the \(\chi \in \mathfrak{g}^*\) such that \(u(\mathfrak{g}, \chi)\) is of finite or tame type. In particular, all forms \(\chi \) for which \(u(\mathfrak{g}, \chi)\) is of finite type are described. Let \(\chi =\chi _s+\Phi (e, -)\) be the Jordan decomposition of \(\chi \in \mathfrak{g}^*.\) Here \(\Phi \) is the trace form of some special representation of \(\mathfrak{g}\), \(e\) is a nilpotent element of \(\mathfrak{g}.\) The author shows how the type of \(u(\mathfrak{g}, \chi)\) depends on the properties of nilpotent \(e\) and the stabilizer \(Z_G(\chi _s)\) of \(\chi _s\) in \(G.\)
scientific article; zbMATH DE number 1186180