On conjugacy of unipotent elements in finite groups of Lie type (original) (raw)

Rational points on generalized flag varieties and unipotent conjugacy in finite groups of Lie type

Transactions of the American Mathematical Society, 2008

Let G be a connected reductive algebraic group defined over the finite field F q , where q is a power of a good prime for G. We write F for the Frobenius morphism of G corresponding to the F q -structure, so that G F is a finite group of Lie type. Let P be an F -stable parabolic subgroup of G and U the unipotent radical of P . In this paper, we prove that the number of U F -conjugacy classes in G F is given by a polynomial in q, under the assumption that the centre of G is connected. This answers a question of J. Alperin in .

CONJUGACY CLASSES IN PARABOLIC SUBGROUPS OF SEMISIMPLE ALGEBRAIC GROUPS

Let G be a connected sernisimple algebraic group defined over an algebraically closed field K, let P be a parabolic subgroup of G and let V denote the unipotent radical of P. Assume that the set of unipotent conjugacy classes of G is finite. In this note we shall show that there exists veV such that C P (v), the conjugacy class of v in P, is an open subset of V.

Counting conjugacy classes in the unipotent radical of parabolic subgroups of GL n ( q )

Pacific Journal of Mathematics, 2010

Let q be a power of a prime p. Let P be a parabolic subgroup of the general linear group GL n (q) that is the stabilizer of a flag in F n q of length at most 5, and let U = O p (P ). In this note we prove that, as a function of q, the number k(U ) of conjugacy classes of U is a polynomial in q with integer coefficients.

On the coadjoint orbits of maximal unipotent subgroups of reductive groups

2014

Let G be a simple algebraic group defined over an algebraically closed field of characteristic 0 or a good prime for G. Let U be a maximal unipotent subgroup of G and u its Lie algebra. We prove the separability of orbit maps and the connectedness of centralizers for the coadjoint action of U on (certain quotients of) the dual u * of u. This leads to a method to give a parametrization of the coadjoint orbits in terms of so-called minimal representatives which form a disjoint union of quasi-affine varieties. Moreover, we obtain an algorithm to explicitly calculate this parametrization which has been used for G of rank at most 8, except E 8 .

A Correspondence for Conjugacy Classes in Certain Extensions of Order 2 of Finite Groups of Lie Type

2003

Let G be a connected linear algebraic group over an algebraically closed field of prime characteristic. Let F : G ! G denote a standard Frobe- nius mapping of G. Let be an involutory automorphism of G which commutes with F and let F denote the corresponding twisted Frobenius mapping. Let H denote either GF or GF and let Hhti denote the extension of H by a cyclic group of order 2, generated by t, that induces the automorphism on H. We show that there is a one-to-one correspondence between the conjugacy classes of GFhti \ GF and those of GF hti \ GF . If xt and yt are elements in corresponding conjugacy classes of the two groups, then xt and yt have the same order and the centralizer of xt in GF is isomorphic to the centralizer of yt in GF . We also discuss numerical evidence for the existence of a related correspondence of characters of the two extension groups.

On Unipotent Radicals of Pseudo-Reductive Groups

The Michigan Mathematical Journal

We establish some results on the structure of the geometric unipotent radicals of pseudo-reductive k-groups. In particular, our main theorem gives bounds on the nilpotency class of geometric unipotent radicals of standard pseudo-reductive groups, which are sharp in many cases. A major part of the proof rests upon consideration of the following situation: let k be a purely inseparable field extension of k of degree p e and let G denote the Weil restriction of scalars R k /k (G) of a reductive k-group G. When G = R k /k (G) we also provide some results on the orders of elements of the unipotent radical Ru(Gk) of the extension of scalars of G to the algebraic closurē k of k. 2010 Mathematics Subject Classification. 20G15. 1 Note that is not surjective when Z G has non-étale fibre at a factor field k i of k that is not separable over k.