Block representation type of reduced enveloping algebras. (Q2781373)

Let \(G\) be a connected, reductive algebraic group over an algebraically closed field \(K\) of characteristic \(p\), \({\mathfrak g}=\text{Lie}(G)\) the corresponding Lie \(p\)-algebra satisfying the conditions: (A) the derived group \(G^{(1)}\) of \(G\) is simply-connected; (B) \(p\) is good for \(G\); (C) \(\mathfrak g\) has a nondegenerate \(G\)-invariant bilinear form.NEWLINENEWLINE Given \(\chi\in{\mathfrak g}^*\), a reduced enveloping algebra \(U_\chi(\mathfrak g)\) is the quotient algebra of \(U(\mathfrak g)\) over the ideal \(I_\chi\) generated by \(x^p-x^{[p]}-\chi (x)^p\) for \(x\in{\mathfrak g}\). All blocks of \(U_\chi(\mathfrak g)\) having finite and tame representation type are determined.NEWLINENEWLINE Using (C) one can obtain the Jordan decomposition of \(\chi\), \(\chi=\chi_s+\chi_n\) and, possibly after conjugation by \(g\in G\), have \(\chi_s({\mathfrak n}^\pm)=0\), \(\chi_n({\mathfrak h})=0\) where \({\mathfrak g}={\mathfrak n}^-\oplus{\mathfrak h}\oplus{\mathfrak n}^+\) is the triangular decomposition of \(\mathfrak g\). Set NEWLINE\[NEWLINE\Lambda_{\chi_s}=\{\lambda\in{\mathfrak h}^*\mid\lambda(h)^p-\lambda(h^{[p]})=\chi_s(h)^p,\;h\in{\mathfrak h}\}.NEWLINE\]NEWLINE According to \textit{K. A. Bown} and \textit{I. Gordon} [Math. Z. 238, No. 4, 733-779 (2001; Zbl 1037.17011)] blocks \(B_{\chi,\lambda}\) of \(U_\chi(\mathfrak g)\) are parametrised by the set \(W\Lambda_{\chi_s}/W\) where \(W\) is the Weyl group of \(\mathfrak g\). In particular, by \textit{J. C. Jantzen} [NATO ASI Ser., Ser. C, Math. Phys. Sci. 514, 185-235 (1998; Zbl 0974.17022)] the baby Verma module \(Z_\chi(\lambda)\) belongs to \(B_{\chi,\lambda}\).NEWLINENEWLINE According to Proposition 2.7 there is an algebra isomorphism NEWLINE\[NEWLINEB_{\chi,\lambda}\cong\text{Mat}_{p^d}(B_{\chi_n,\lambda '}({\mathfrak z}_{\mathfrak g}(\chi_s)))NEWLINE\]NEWLINE which reduces the representation type problem for \(B_{\chi,\lambda}\) to that for \(B_{\chi_n,\lambda'}({\mathfrak z}_{\mathfrak g}(\chi_s))\) corresponding to a nilpotent \(\chi_n\). Here \({\mathfrak z}_{\mathfrak g}(\chi_s)\) is the stabilizer of \(\chi_s\) in \(\mathfrak g\). For semisimple \(\chi_s\) \({\mathfrak z}_{\mathfrak g}(\chi_s)=\text{Lie}(Z_G(\chi_s))\) where \(Z_G(\chi_s)\) is the stabilizer of \(\chi_s\) in \(G\), and \(Z_G(\chi_s)\) is a connected, reductive algebraic group satisfying conditions (A)-(C) [see \textit{J. E. Humphreys}, Conjugacy classes in semisimple algebraic groups (1995; Zbl 0834.20048)].NEWLINENEWLINE The classification of blocks \(B_{\chi,\lambda}\) of finite and tame representation type is given in Theorems 5.2 and 5.3 for nilpotent \(\chi\) in terms of the simple normal subgroups of \(G^{(1)}\). In the case when \(G\) is simple, the theorems state that \(B_{\chi,\lambda}\) is of finite representation type if and only if one of the following occurs: 1. \(W=W(\lambda)\) (\(W(\lambda)\) is the stabilizer of \(\lambda\) in \(W\)); 2. \(\chi\) is regular and one of the following holds: (a) \(W\) is of type \(A_n\) and \(W(\lambda)\) is of type \(A_{n-1}\); (b) \(W\) is of type \(B_n\) (or \(C_n\)) and \(W(\lambda)\) is of type \(B_{n-1}\) (or \(C_{n-1}\)); (c) \(W\) is of type \(G_2\) and \(W(\lambda)\) is of type \(A_1\).NEWLINENEWLINE The block \(B_{\chi,\lambda}\) is of tame representation type if and only if one of the following occurs: 1. \(\chi\) is regular and one of the following holds: (a) \(W\) has rank 2; (b) \(W\) is of type \(A_3\) and \(W(\lambda)\) is of type \(A_1\times A_1\); (c) \(W\) is of type \(B_3\) (or \(C_3\)) and \(W(\lambda)\) is of type \(A_2\); (d) \(W\) is of type \(D_n\) and \(W(\lambda)\) is of type \(D_{n-1}\); 2. \(\chi\) is subregular, \(W\) is of type \(A_n\) and \(W(\lambda)\) is of type \(A_{n-1}\).NEWLINENEWLINE It is shown (Theorem 4.2) that the rank variety of \(B_{\chi,\lambda}\) coincides with the intersection of the rank variety \(B_{\chi_s,\lambda}\) with the \(p\)-nilpotent cone of \({\mathfrak z}_{\mathfrak g}(\chi)\). It follows that the rank variety of \(U_\chi(\mathfrak g)\) is equal to the \(p\)-nilpotent cone of \({\mathfrak z}_{\mathfrak g}(\chi)\). Theorem 4.2 allows to compute the maximum value of the rate of growth of minimal projective resolutions of finite dimensional \(B_{\chi,\lambda}\)-modules. This information as well as results on block degeneration and on representation type of partial coinvariant algebras play an essential role in the proof of Theorems 5.2 and 5.3.