Commuting varieties of -tuples over Lie algebras
arXiv:1209.1659 · doi:10.1016/j.jpaa.2013.11.024
Abstract
Let be a simple algebraic group defined over an algebraically closed field of characteristic and let $\g$ be the Lie algebra of . It is well known that for large enough the spectrum of the cohomology ring for the -th Frobenius kernel of is homeomorphic to the commuting variety of -tuples of elements in the nilpotent cone of $\g$ [Suslin-Friedlander-Bendel, J. Amer. Math. Soc, \textbf{10} (1997), 693--728]. In this paper, we study both geometric and algebraic properties including irreducibility, singularity, normality and Cohen-Macaulayness of the commuting varieties $C_r(\mathfrak{gl}_2), C_r(\fraksl_2)$ and where is the nilpotent cone of $\fraksl_2$. Our calculations lead us to state a conjecture on Cohen-Macaulayness for commuting varieties of -tuples. Furthermore, in the case when $\g=\fraksl_2$, we obtain interesting results about commuting varieties when adding more restrictions into each tuple. In the case of $\fraksl_3$, we are able to verify the aforementioned properties for $C_r(\fraku)$. Finally, applying our calculations on the commuting variety $C_r(\overline{\calO_{\sub}})$ where $\overline{\calO_{\sub}}$ is the closure of the subregular orbit in $\fraksl_3$, we prove that the nilpotent commuting variety has singularities of codimension .
To appear in Journal of Pure and Applied Algebra