Steinberg modules and Donkin pairs (Q5940020)

For a semisimple algebraic group \(G\) and a semisimple subgroup \(H\) over an algebraically closed field \(k\), the pair \((G,H)\) is called a Donkin pair if the restriction to \(H\) of any rational \(G\)-module \(M\) with a good filtration has a good filtration. It is equivalent that the restriction of every induced \(G\)-module to \(H\) has a good filtration. Examples of the subgroups \(H\) are the Levi types (the commutator subgroups of Levi subgroups) and the fixed point subgroups of an involution. The main result of the paper is to prove a conjecture of \textit{J. Brundan} [in Algebraic groups and their representations, NATO ASI Ser., Ser. C, Math. Phys. Sci. 517, 259-274 (1998; Zbl 0933.20038)] that \((G,H)\) is a Donkin pair if \(H\) is the centralizer of a graph automorphism (for all characteristics) or the centralizer of an involution of \(G\) with characteristic not 2. Most of the cases were settled by \textit{S. Donkin} [Rational representations of algebraic groups, Lect. Notes Math. 1140 (1985; Zbl 0586.20017)] and Brundan. The present paper uses a more geometric approach to settle the remaining cases (for exceptional groups of type \(E\) and \(F\)). A pair \((G,H)\) is said to satisfy the pairing criterion if there is an \(H\)-module homomorphism \(St_H^*\otimes St_H\to St_G^*\otimes St_G\) with image not being contained in the kernel of the evaluation map \(St_G^*\otimes St_G\to k\). By using the concept of canonical Frobenius splitting and its relations with the existence of good filtrations established by \textit{O. Mathieu} in proving that the tensor product of modules with good filtration also has a good filtration [Ann. Sci. Éc. Norm. Supér., IV. Sér. 23, No. 4, 625-644 (1990; Zbl 0748.20026)], the author proves that the pairing criterion implies the Donkin pair condition. This result is used to prove that the pair \((E_6,F_4)\) is a Donkin pair. The remaining cases: \((E_8,D_8)\), \((E_8,E_7A_1)\), \((E_7,A_7)\), \((E_7,D_6A_1)\), \((E_6,A_5A_1)\), \((E_6,C_4)\), \((F_4,B_4)\), \((F_4,C_3A_1)\), are settled using mainly the representation theory as used by Donkin.