TY - JOUR

T1 - On the conjugacy classes in maximal unipotent subgroups of simple algebraic groups

AU - Goodwin, Simon

PY - 2006/3/1

Y1 - 2006/3/1

N2 - Let G be a simple algebraic group over the algebraically closed field k of characteristic p >= 0. Assume p is zero or good for G. Let B be a Borel subgroup of G; we write U for the unipotent radical of B and u for the Lie algebra of U. Using relative Springer isomorphisms we analyze the adjoint orbits of U in u. In particular, we show that an adjoint orbit of U in u contains a unique so-called minimal representative. In case p > 0, assume G is defined and split over the finite field of p elements F-p. Let q be a power of p and let G(q) be the finite group of F-q-rational points of G. Let F be the Frobenius morphism such that G(q) = G(F). Assume B is F-stable, so that U is also F-stable and U(q) is a Sylow p-subgroup of G(q). We show that the conjugacy classes of U(q) axe in correspondence with the F-stable adjoint orbits of U in u. This allows us to deduce results about the conjugacy classes of U(q).

AB - Let G be a simple algebraic group over the algebraically closed field k of characteristic p >= 0. Assume p is zero or good for G. Let B be a Borel subgroup of G; we write U for the unipotent radical of B and u for the Lie algebra of U. Using relative Springer isomorphisms we analyze the adjoint orbits of U in u. In particular, we show that an adjoint orbit of U in u contains a unique so-called minimal representative. In case p > 0, assume G is defined and split over the finite field of p elements F-p. Let q be a power of p and let G(q) be the finite group of F-q-rational points of G. Let F be the Frobenius morphism such that G(q) = G(F). Assume B is F-stable, so that U is also F-stable and U(q) is a Sylow p-subgroup of G(q). We show that the conjugacy classes of U(q) axe in correspondence with the F-stable adjoint orbits of U in u. This allows us to deduce results about the conjugacy classes of U(q).

UR - http://www.scopus.com/inward/record.url?scp=33645574130&partnerID=8YFLogxK

U2 - 10.1007/s00031-005-1104-7

DO - 10.1007/s00031-005-1104-7

M3 - Article

SN - 1083-4362

VL - 11

SP - 51

EP - 76

JO - Transformation Groups

JF - Transformation Groups

IS - 1

ER -