A pr 2 00 8 REALITY PROPERTIES OF CONJUGACY CLASSES IN G 2
نویسنده
چکیده
Let G be an algebraic group over a field k. We call g ∈ G(k) real if g is conjugate to g−1 in G(k). In this paper we study reality for groups of type G2 over fields of characteristic different from 2. Let G be such a group over k. We discuss reality for both semisimple and unipotent elements. We show that a semisimple element in G(k) is real if and only if it is a product of two involutions in G(k). Every unipotent element in G(k) is a product of two involutions in G(k). We discuss reality for G2 over special fields and construct examples to show that reality fails for semisimple elements in G2 over Q and Qp. We show that semisimple elements are real for G2 over k with cd(k) ≤ 1. We conclude with examples of nonreal elements in G2 over k finite, with characteristic k not 2 or 3, which are not semisimple or unipotent.
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