## Experimental Mathematics

- Experiment. Math.
- Volume 13, Number 1 (2004), 13-30.

### Series of Nilpotent Orbits

J. M. Landsberg, Laurent Manivel, and Bruce W. Westbury

#### Abstract

We organize the nilpotent orbits in the exceptional complex Lie algebras into
series and show that within each series the dimension of the orbit is a linear
function of the natural parameter {\small $a=1, 2, 4, 8$}, respectively for
{\small $\ff_4,\fe_6,\fe_7,\fe_8$}. We observe similar regularities for the
centralizers of nilpotent elements in a series and grade components in the
associated grading of the ambient Lie algebra. More strikingly, we observe that
for {\small $a\geq 2$} the numbers of {\small $\FF_q$}-rational points on the
nilpotent orbits of a given series are given by polynomials that have uniform
expressions in terms of *a*. This even remains true for the degrees of
the unipotent characters associated to these series through the Springer
correspondence. We make similar observations for the series arising from the
other rows of Freudenthal's magic chart and make some observations about the
general organization of nilpotent orbits, including the description of and
dimension formulas for several universal nilpotent orbits (universal in the
sense that they occur in almost every simple Lie algebra).

#### Article information

**Source**

Experiment. Math., Volume 13, Number 1 (2004), 13-30.

**Dates**

First available in Project Euclid: 10 June 2004

**Permanent link to this document**

https://projecteuclid.org/euclid.em/1086894087

**Mathematical Reviews number (MathSciNet)**

MR2065565

**Zentralblatt MATH identifier**

1108.14037

**Subjects**

Primary: 20C33: Representations of finite groups of Lie type 17B45: Lie algebras of linear algebraic groups [See also 14Lxx and 20Gxx]

Secondary: 14L40: Other algebraic groups (geometric aspects) 22E46: Semisimple Lie groups and their representations

**Keywords**

Nilpotent orbits exceptional Lie algebra unipotent character

#### Citation

Landsberg, J. M.; Manivel, Laurent; Westbury, Bruce W. Series of Nilpotent Orbits. Experiment. Math. 13 (2004), no. 1, 13--30. https://projecteuclid.org/euclid.em/1086894087