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M. S. Raghunathan ¹

Affiliations
1 Tata Institute of Fundamental Research, Colaba, Bombay-400 005, IN

Abstract

Let k be a field. We will say that the field k has property (A) if the following holds.

Let V be a vector space defined over k and G ⊂ GL(V) a reductive algebraic k-subgroup. Let V_k denote the k-rational points of V. Let v ∈ V_k and X = Gv.

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M. S. Raghunathan ¹

Affiliations
1 School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba, Mumbai-400005, IN

Abstract

Let G be a connected quasi-split semisimple algebraic group over a field k. Let S be a maximal k-split torus and T=Z(S) its centraliser. Then T is a maximal k-torus in G. The normaliser N(S)of S is also the normaliser N(T) of T . All these groups are defined over k and hence also N(T)/Z(T)=W, a finite group. Suppose now that T'⊂G is any maximal k-torus. Then there is an element g∈G(k_s)/ such that gTg-¹=T' where k_s is a separable closure of k. If G is the Galois group of k_s over k and σ∈G, evidently σ(g)Tσ(g)^-1=σ(gTg^-1)=T'.

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