A note on the adjoint action of a semisimple group on the coordinate algebra of an infinitesimal subgroup
Let \(G\) be a connected, semisimple, simply connected affine algebraic group over an algebraically closed field \(K\) of prime characteristic \(p\). Let \(T\) be a maximal torus, \(B \supset T\) be a Borel subgroup corresponding to the negative roots, and \({\mathfrak g}\) denote the Lie algebra of \(G\). Let \(X^{+}(T)\) denote the dominant weights. Each \(\lambda \in X^{+}(T)\) determines a one-dimensional \(B\)-module \(K_{\lambda}\). Inducing to \(G\), one obtains a costandard module \(\nabla(\lambda) := \text{ind}_{B}^{G}(K_{\lambda})\), which is dual to a Weyl module. A finite-dimensional rational \(G\)-module \(M\) is said to have a good filtration if there exists a filtration \(0 = M_0 \leq M_1 \leq M_2 \leq \cdots \leq M_n = M\), such that each (non-zero) quotient \(M_{i}/M_{i-1} \cong \nabla (\lambda_i)\) for some \(\lambda_i \in X^{+}(T)\). If the dual of the module \(M\) also admits a good filtration (or, equivalently, if \(M\) admits a Weyl filtration), then \(M\) is said to be a tilting module. Let \(K[G]\) denote the coordinate algebra of \(G\). Given a positive integer \(r\), let \(G_r\) denote the \(r\)th Frobenius kernel and \(K[G_r]\) denote its coordinate algebra, which may be identified with a quotient of \(K[G]\). The conjugation action of \(G\) on itself, gives \(K[G]\) and, hence, \(k[G_r]\) the structure of \(G\)-modules. Let \(\rho\) denote the half-sum of the positive roots and set \(\text{St}_r := \nabla((p^r - 1)\rho)\), the \(r\)th Steinberg module. The module \(\text{St}_r\) is simple as a \(G\)-module and projective (and injective) over \(G_r\). Given a finite-dimensional rational \(G\)-module \(M\), the author has a long-standing conjecture on a condition for the tensor product \(\text{St}_r\otimes M\) to admit a good filtration. While investigating that conjecture, \textit{D. Nakano} et al. [``On tensoring with the Steinberg module'', Transform. Groups, to appear, \url{arXiv:1804.00613}] raised the question of whether the module \(\text{St}_r\otimes K[G_r]\) admits a good filtration. The main result of the paper under review is an affirmative answer to this question under the mild assumption that \(p\) is good for the root system (i.e., \(p\) is not 2 in types B, C, D; \(p\) is not 2 or 3 in types \(E_6\), \(E_7\), \(F_4\), or \(G_2\); and \(p\) is not 2, 3, or 5 in type \(E_8\)). To prove the result, the author works in a slightly more general setting. Let \(X\) be an irreducible, nonsingular affine \(G\)-variety and \(x \in X\) be a fixed point. Then one may consider the associated \(r\)th infinitesimal subscheme \(X_{x,r}\) of \(X\) and its coordinate algebra \(k[X_{x,r}]\). Let \(M_x\) denote the ideal of \(x\) in \(k[X]\). The key result of the paper is that if \(\text{St}_r\otimes M_x^d/M_x^{d+1}\) has a good filtration for all \(d \geq 0\), then \(\text{St}_r\otimes K[X_{x,r}]\) is a tilting module for \(G\). Note that \(M_x^d/M_x^{d+1}\) may be identified with the \(d\)th symmetric power of the module \(M_x/M_x^2\) (the cotangent space). Under the assumption that \(X\) is defined over the finite field \({\mathbb F}_p\) and \(x\) is a rational point, then the hypothesis can be weakened to knowing that \(\text{St}_1\otimes M_{x,1}^d/M_{x,1}^{d+1}\) has a good filtration for all \(d \geq 0\), where \(M_{x,1}\) denotes the maximal ideal in \(K[X_{x,1}]\). The claim that \(\text{St}_r\otimes K[G_r]\) has a good filtration follows as a special case by taking \(X = G\) and using the fact that \(\text{St}_1\otimes S^d({\mathfrak g}^*)\) admits a good filtration when the prime \(p\) is good. The author concludes the paper by noting that the ideas may also be applied to the general linear group and exploring connections with a conjecture of Doty about the matrix of decomposition numbers for simple modules in certain tensor products of truncated symmetric powers of the standard module.
scientific article; zbMATH DE number 6988690
scientific article