On the second socle level of induced modules for algebraic groups (Q752851)

Let G be a simple, simply connected algebraic group over an algebraically closed field of characteristic p. In the study of the structure of the module induced from a character \(\lambda\) on a Borel subgroup of G, calculations often are performed for \(\lambda\) only in the lowest \(p^ 2\)-alcove, with the belief that results for the lowest \(p^ 2\)-alcove determine the results for weights in the \(p^ n\)-alcove, \(n>2\) [see \textit{J. Humphreys}, Commun. Algebra 12, 2665-2677 (1984; Zbl 0546.20036), for instance]. In this paper, we illustrate that principle by showing that the second socle level of the module induced from a weight in the lowest \(p^ n\)-alcove may be computed in terms of the second socle levels of modules induced from weights in the lowest \(p^ 2\) and \(p^{n-1}\)- alcoves, under the hypothesis in Theorem 2.2.