Algebraic and analytic groups

Abstract

Algebraic groups Here, algebraic group will mean a Zariski-closed subgroup of SL n ( K ), for some n ∈ ℕ and some algebraically closed field K . For background and terminology, see [B2], [H2], [PR] and [W1]. In this section, topological language refers to the Zariski topology. The following theorem, due to Merzljakov, in a sense provides the philosophical background to all the ellipticity results concerning groups of Lie type; a sharper result specific to simple groups is stated below. Theorem 5.1.1 [M2] Every algebraic group is verbally elliptic . This depends on Chevalley's concept of constructible sets . Let V = K d be the affine space, with its Zariski topology. A subset of V is constructible if it is a finite union of sets of the form C ∩ U where C is closed and U is open. A morphism from V to V 1 = K l is a mapping defined by l polynomials. The key result is Proposition 5.1.2 Let Y be a constructible subset of V, with closure Ȳ . (i) (See [W1], 14.9) The set Y contains a subset U that is open and dense in Ȳ . (ii) (Chevalley, see [H2], §4.4) If f : V → V 1 is a morphism then f(Y) is a constructible subset of V 1 .