Geometric actions of classical algebraic groups

Abstract

Let k be an algebraically closed field of arbitrary characteristic p. An affine algebraic group G is an affine algebraic variety over k with a group structure such that multiplication and inversion maps are morphisms of varieties. A special class of affine algebraic groups are the so called classical groups Cl(V), groups of isometries of a finite dimensional k-vector space V with respect to a certain form on V {e.g. a zero form, a symplectic form or a non-degenerate quadratic form. These groups are: GL(V) the general linear group, Sp(V) the symplectic group and O(V) the orthogonal group. Let G = Cl(V). Various (closed) subgroups H of G can be defined naturally in terms of the geometry of V {H may be the stabiliser of a subspace of V, or a direct sumdecomposition of V, or a non-degenerate form on V, for example. Let H be such a subgroup and let = G=H be the corresponding coset space. Then is a variety with a natural algebraic action of G. We define geometric subgroups of G to be the closed subgroups arising in this manner. Consequently, for H a geometric subgroup, we say that the natural action of G on = G=H is a geometric action. We define C (x) to be the set of points in fixed by x. Then C (x) is a subvariety, and we can show that dimC (x) = dim.

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ORCID for Peter Kropholler: orcid.org/0000-0001-5460-1512

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