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Series of nilpotent orbits. (English) Zbl 1108.14037

Given two real normed algebras \(\mathbb{A}\) and \(\mathbb{B}\) the Tits-Freudenthal construction gives a \(\mathbb{Z}_{2}\)-graded Lie algebra \( \mathfrak{g}( \mathbb{A},\mathbb{B}) .\) Also, for any real normed algebra \(\mathbb{A}\) one can construct the triality algebra \(t( \mathbb{ A}) \) and hence one can also construct a \(( \mathbb{Z}_{2}\times \mathbb{Z}_{2}) \)-graded semi-simple Lie algebra \(\mathfrak{\tilde{g}} ( \mathbb{A},\mathbb{B}) .\) Either of these constructions gives rise to the so-called “Freudenthal’s magic square”, from which one can easily see \(\mathfrak{g}( \mathbb{R},\mathbb{O}) =\mathfrak{f} _{4},\;\mathfrak{g}( \mathbb{C},\mathbb{O}) =\mathfrak{e}_{6},\; \mathfrak{g}( \mathbb{H},\mathbb{O}) =\mathfrak{e}_{7},\) and \( \mathfrak{g}( \mathbb{O},\mathbb{O}) =\mathfrak{e}_{8}\) from the bottom row of the square. This paper focuses on these four exceptional Lie algebras - specifically the series of nilpotent orbits one obtains when starting with a nilpotent orbit in \(\mathfrak{f}_{4}.\)
Using the fact that \(\mathfrak{so}_{8}\subset \mathfrak{f}_{4}\subset \mathfrak{e}_{6}\subset \mathfrak{e}_{7}\subset \mathfrak{e}_{8},\) the triality model defines a weight for each of these exceptional Lie algebras. Given a nilpotent orbit \(\mathcal{O=O}_{1}\) in \(\mathfrak{f}_{4}\) one obtains a series of orbits \(\mathcal{O}_{2},\mathcal{O}_{4},\) and \(\mathcal{O }_{8}\) in \(\mathfrak{e}_{6},\mathfrak{e}_{7},\;\)and \(\mathfrak{e}_{8}\) respectively. The first result of the paper is that the dimension of \( \mathcal{O}_{a}\) is linear in \(a\). This is obtained by showing the codimension of the centralizer \(c( X) _{a}\) of \(X\in \mathfrak{f} _{4}\) in \(\mathfrak{g}( \mathbb{A},\mathbb{O}) \) is linear in \(a\).
Next, the authors show that the dimension of the nilpotent radical \( \mathfrak{r}( a) \) of the stabilizer of an element of \(\mathcal{O} _{a}\) is also linear in \(a\), as is the dimension of the \(i^{\text{th}}\) part of the induced gradation of \(\mathfrak{g}( \mathbb{A},\mathbb{B}).\) Next, it is shown that \(\mathcal{\bar{O}}_{a}\) can be desingularized by a homogeneous vector bundle whose dimensions of both the base and fiber are linear in \(a\). Following this the authors prove that the number of \(\mathbb{F }_{q}\)-rational points (for sufficiently large \(\mathbb{F}_{q})\) on \( \mathcal{O}_{a}\) is given by a polynomial – the largest and smallest are given explicitly as rational functions with each denominator a polynomial consisting of products of the form \(q^{n}-1.\) This polynomial is not linear in \(a\) but it has, as the paper states, “a uniform expression in \(a\)”.
Finally, the authors show that the unipotent characters of the finite groups of exceptional Lie type, which are associated to the orbits \(\mathcal{O}_{a}\) through the Springer correspondence, have degrees given by polynomials that can be expressed as rational functions with uniform expressions in \(a\). These are listed in great detail in the paper and depend on the chosen series.
After the study of these exceptional Lie algebras, other rows of the square are considered, and this leads to results for other Lie algebras. It is shown that a few orbits appear in every (or almost every) simple Lie algebra.

MSC:

17B25 Exceptional (super)algebras
17B45 Lie algebras of linear algebraic groups
17B08 Coadjoint orbits; nilpotent varieties
14L40 Other algebraic groups (geometric aspects)
20C33 Representations of finite groups of Lie type
22E46 Semisimple Lie groups and their representations
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