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THE LOEWY STRUCTURE OF $G_{1}T$ -VERMA MODULES OF SINGULAR HIGHEST WEIGHTS

  • Noriyuki Abe (a1) and Masaharu Kaneda (a2)

Abstract

Let $G$ be a reductive algebraic group over an algebraically closed field of positive characteristic, $G_{1}$ the Frobenius kernel of $G$ , and $T$ a maximal torus of $G$ . We show that the parabolically induced $G_{1}T$ -Verma modules of singular highest weights are all rigid, determine their Loewy length, and describe their Loewy structure using the periodic Kazhdan–Lusztig $P$ - and $Q$ -polynomials. We assume that the characteristic of the field is sufficiently large that, in particular, Lusztig’s conjecture for the irreducible $G_{1}T$ -characters holds.

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