Filtration patterns for representations of algebraic groups and their Frobenius kernels
type: Item
instance of: scholarly article
title: Filtration patterns for representations of algebraic groups and their Frobenius kernels
zbMATH Open document ID: 0601.20041
DOI: 10.1007/BF01161764
author: S. R. Doty
author: John Brendan Sullivan
published in: Mathematische Zeitschrift
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/183700
review text: Let T be a maximal torus contained in a Borel subgroup B of a simply connected semisimple algebraic group G over an algebraically closed field of characteristic \(p\neq 0\). For a character \(\lambda\) on T, let \({\mathcal L}(\lambda)\) be the corresponding induced line bundle on the homogeneous space G/B, (or on \(TG_ n/TB_ n\), where \(G_ n\) is the kernel of the nth power of the Frobenius map on G). In a recent paper, J.-C. Ye determined the set of composition factors of the \(TG_ 1\) module \(H^ 0(TG_ 1/TB_ 1,{\mathcal L}(\lambda))\) associated to any p-regular weight \(\lambda\). We extend this result to arbitrary weight \(\lambda\). We then determine the set of composition factors for \(H^ 0(TG_ n/TB_ n,{\mathcal L}(\lambda))\), for \(n>1\), and show that their multiplicities in a composition series for \(H^ 0(TG_ n/TB_ n,{\mathcal L}(\lambda))\) would be determined if those multiplicities for \(n=1\) were known. We also relate the multiplicities of the composition factors of \(H^ 0(G/B,{\mathcal L}(\lambda))\) to those of \(H^ 0(TG_ n/TB_ n,{\mathcal L}(\lambda))\), extending results of Jantzen and Chastkofsky.
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20G10
Mathematics Subject Classification ID: 14L30
zbMATH DE Number: 3968900
zbMATH Keywords: maximal torus
zbMATH Keywords: Borel subgroup
zbMATH Keywords: simply connected semisimple algebraic group
zbMATH Keywords: induced line bundle
zbMATH Keywords: homogeneous space
zbMATH Keywords: Frobenius map
zbMATH Keywords: composition factors
zbMATH Keywords: p-regular weight
zbMATH Keywords: composition series
zbMATH Keywords: character
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3968900