Invariants of Centralisers in Positive Characteristic

2011/08/10 by Lewis Topley, Topley, Lewis
Mathematics · #FOS: Mathematics #Representation Theory (math.RT) #math.RT

paper · pdf · doi:10.48550/arxiv.1108.2306

arxiv created 2013/01/28 · arxiv updated 2013/01/29

Abstract

Let \q be a simple algebraic group of type A or C over a field of good positive characteristic. We show for any x ∈ \q =\Lie(Q) that the invariant algebra S(\q_x)^\q_x is generated by the p^th power subalgebra and the mod p reduction of the characteristic zero invariant algebra. The latter algebra is known to be polynomial \citePPY and we show that it remains so after reduction. Using a theory of symmetrisation in positive characteristic we prove the analogue of this result in the enveloping algebra, where the p-centre plays the role of the p^th power subalgebra. In Zassenhaus' foundational work \citeZas, the invariant theory and representation theory of modular Lie algebras were shown to be explicitly intertwined. We exploit his theory to give a precise upper bound for the dimensions of simple \q_x-modules. When \g is of type A and \g = \k ⊕ \p is a symmetric decomposition of orthogonal type we use similar methods to show that for every nilpotent e ∈ \k the invariant algebra S(\p_e)^\k_e is generated by the p^th power subalgebra and S(\p_e)^K_e which is also shown to be polynomial.

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