Symmetric Invariants of Centralisers in Classical Lie Algebras and the KW1 Conjecture

Details Symmetric Invariants of Centralisers in Classical Lie Algebras and the KW1 Conjecture Symmetric Invariants of Centralisers in Classical Lie Algebras and the KW1 Conjecture

2011-08-10

arxiv.org/abs/1108.2306v2

Mathematics

Representation Theory

Scientific paper

Let $G$ denote a simple algebraic group of rank $l$ over an algebraically closed field $\K$ of positive characteristic $p$. Let $\g = \text{Lie}(G)$ and choose $x \in \mathfrak{g}$. In this paper we give explicit presentations for the invariant subalgebras $S(\mathfrak{g}_x)^{G_x}$ and $S(\mathfrak{g}_x)^{\mathfrak{g}_x}$ when $G$ is of type $A$ or $C$. In particular, $S(\mathfrak{g}_x)^{G_x}$ is shown to be graded polynomial in $l$ variables and $S(\mathfrak{g}_x)^{\mathfrak{g}_x}$ to be the tensor product of $S(\mathfrak{g}_x)^{G_x}$ with $S(\mathfrak{g}_x)^p$ over their intersection. When $G$ is of type $B$ or $D$ we have $\mathfrak{gl}_m(\K) = \mathfrak{g} \oplus W$ where $m$ is the dimension of the natural representation for $\mathfrak{g}$ and $W$ is a $\mathfrak{g}$-module. In case $x$ is nilpotent with associated partition $\lambda$ we show that $S(W_x)^{G_x}$ is polynomial on $(m + |\{i : \lambda_i \text{odd} \}|)/2$ variables, and that $S(W_x)^{\mathfrak{g}_x}$ is the tensor product of $S(W_x)^p$ with $S(W_x)^{G_x}$ over their intersection. For types $A$ and $C$ we go on to give an analogous description of the centre of the enveloping algebra $Z(\mathfrak{g}_x)$, and use this to confirm the first Kac-Weisfieler conjecture for $\mathfrak{g}_x$. As an immediate consequence we are able to give an algebraic characterisation of the singular points on the Zassenhaus variety $\text{Specm} Z(\g_x)$.

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