1. CJM Online first
||Order $3$ elements in $G_2$ and idempotents in symmetric composition algebras|
Order three elements in the exceptional groups of type $G_2$
are classified up to conjugation over arbitrary fields. Their
centralizers are computed, and the associated classification
of idempotents in symmetric composition algebras is obtained.
Idempotents have played a key role in the study and classification
of these algebras.
Over an algebraically closed field, there are two conjugacy classes
of order three elements in $G_2$ in characteristic not $3$ and
four of them in characteristic $3$. The centralizers in characteristic
$3$ fail to be smooth for one of these classes.
Keywords:symmetric composition algebra, Okubo algebra, automorphism group, centralizer, idempotent
Categories:17A75, 14L15, 17B25, 20G15
2. CJM 2006 (vol 58 pp. 93)
||Motivic Haar Measure on Reductive Groups |
We define a motivic analogue of the Haar measure for groups of the form
$G(k\llp t\rrp)$, where~$k$ is an algebraically closed field
of characteristic zero, and $G$ is a reductive algebraic group defined over
A classical Haar measure on such groups does not
exist since they are not locally compact.
We use the theory of motivic integration introduced by M.~Kontsevich to
define an additive function on a certain natural Boolean algebra of subsets of
$G(k\llp t\rrp)$. This function takes values in the so-called dimensional
the Grothendieck ring of the category of varieties over the base
field. It is invariant under translations by all elements of $G(k\llp t\rrp)$,
and therefore we call it a motivic analogue of Haar measure.
We give an explicit construction of the motivic Haar measure, and then prove
that the result is independent of all the choices that are made in the process.
Keywords:motivic integration, reductive group