Abstract
This article is a survey on the cohomology of a reductive algebraic group with coefficients in twisted representations. A large part of the paper is devoted to the advances obtained by the theory of strict polynomial functors initiated by Friedlander and Suslin in the late 90s. The last section explains that the existence of certain ‘universal classes’ used to prove cohomological finite generation is equivalent to some recent ‘untwisting theorems’ in the theory of strict polynomial functors. We actually provide thereby a new proof of these theorems.
This work was supported in part by the Labex CEMPI (ANR-11-LABX-0007-01).
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Notes
- 1.
Indeed, that the \(GL_d\)-module \(G(\Bbbk ^d)\) has a costandard filtration, which is equivalent to the fact that the functor G has a Schur filtration, i.e. a filtration whose subquotients are direct sums of Schur functors as defined in [1]. It then follows from [1, Theorem II.2.16] that the parametrized functor \(G_W\) also has a Schur filtration. The Ext condition follows by a highest weight category argument.
- 2.
Moreover conditions (2) and (3) for bifunctors can also be deduced from computations already published in the literature. Indeed, for \(E=\Bbbk \) (concentrated in degree zero) the statement follows from [12, Theorem 1.8], [32, Theorem Proposition 5.4] or the computations of [4, p. 781]. For an arbitrary E, the computations can be deduced from the case \(E=\Bbbk \) by using the isomorphism \(\mathrm {H}^*_{E,\mathrm {gl}}(B)\simeq \mathrm {H}^*_{\mathrm {gl}}(B_E)\) explained at the end of Sect. 5.6.4.
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The author thanks the anonymous referee for very carefully reading a first version of the article and detecting several mistakes.
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Touzé, A. (2018). Cohomology of Algebraic Groups with Coefficients in Twisted Representations. In: Carlson, J., Iyengar, S., Pevtsova, J. (eds) Geometric and Topological Aspects of the Representation Theory of Finite Groups. PSSW 2016. Springer Proceedings in Mathematics & Statistics, vol 242. Springer, Cham. https://doi.org/10.1007/978-3-319-94033-5_18
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