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In this paper, we give a simple formula for sectional curvatures on the general linear group, which is also valid for many other matrix groups. Similar formula is given for a reductive Lie group. We also discuss the relation between commuting matrices and zero sectional curvature.
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We prove that for a connected, semisimple linear Lie group $G$ the spaces of generating pairs of elements or subgroups are well-behaved in a number of ways: the set of pairs of elements generating a dense subgroup is Zariski-open in the compact case, Euclidean-open in general, and always dense. Similarly, for sufficiently generic circle subgroups $H_i$, $i=1,2$ of $G$, the space of conjugates of $H_i$ that generate a dense subgroup is always Zariski-open and dense. Similar statements hold for pairs of Lie subalgebras of the Lie algebra $Lie(G)$.
The nilpotent bicone of a finite dimensional complex reductive Lie algebra g is the subset of elements in g x g whose subspace generated by the components is contained in the nilpotent cone of g. The main result of this note is that the nilpotent bicone is a complete intersection. This affirmatively answers a conjecture of Kraft-Wallach concerning the nullcone. In addition, we introduce and study the characteristic submodule of g. The properties of the nilpotent bicone and the characteristic submodule are known to be very important for the understanding of the commuting variety and its ideal of definition. In order to study the nilpotent bicone, we introduce another subvariety, the principal bicone. The nilpotent bicone, as well as the principal bicone, are linked to jet schemes. We study their dimensions using arguments from motivic integration. Namely, we follow methods developed in http://arxiv.org/abs/math/0008002v5 .
Let $G$ be a reductive algebraic group---possibly non-connected---over a field $k$ and let $H$ be a subgroup of $G$. If $G= GL_n$ then there is a degeneration process for obtaining from $H$ a completely reducible subgroup $H$ of $G$; one takes a limit of $H$ along a cocharacter of $G$ in an appropriate sense. We generalise this idea to arbitrary reductive $G$ using the notion of $G$-complete reducibility and results from geometric invariant theory over non-algebraically closed fields due to the authors and Herpel. Our construction produces a $G$-completely reducible subgroup $H$ of $G$, unique up to $G(k)$-conjugacy, which we call a $k$-semisimplification of $H$. This gives a single unifying construction which extends various special cases in the literature (in particular, it agrees with the usual notion for $G= GL_n$ and with Serres $G$-analogue of semisimplification for subgroups of $G(k)$). We also show that under some extra hypotheses, one can pick $H$ in a more canonical way using the Tits Centre Conjecture for spherical buildings and/or the theory of optimal destabilising cocharacters introduced by Hesselink, Kempf and Rousseau.
The commuting variety of a reductive Lie algebra ${goth g}$ is the underlying variety of a well defined subscheme of $gg g{}$. In this note, it is proved that this scheme is normal. In particular, its ideal of definition is a prime ideal.
The generalized commuting and isospectral commuting varieties of a reductive Lie algebra have been introduced in a preceding article. In this note, it is proved that their normalizations are Gorenstein with rational singularities. Moreover, their canonical modules are free of rank 1. In particular, the usual commuting variety is Gorenstein with rational singularities and its canonical module is free of rank 1.
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