Let G be a connected reductive linear algebraic group over an algebraically closed field of characteristic not 2. Let 0 be an automorphism of order 2 of the algebraic group G. Denote by K the fixed point group of 0 and by B a Borel group of G.It is known that the number of double cosets BgK is finite. This paper gives a combinatorial description of the inclusion relations between the Zariski-closures of such double cosets. The description can be viewed as a generalization of Chevalley's description of the inclusion relations between the closures of double cosets BgB, which uses the Bruhat order of the corresponding Weyl group.

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A.M.S. {1980) subject classification: 20G 15.We summarize some standard results on G-invariants for morphic actions of G on an affine variety X. For more details on some of these results see [5,10,11,13]. These references generally deal with the case in which G is reductive (and hence connected). However, the extension to the case of non-connected G follows easily from the results of [17, pp. 57-60]. The references [10,11] deal with the characteristic-zero case, but many of the foundational results in these references carry over to characteristic p because of the affirmative solution of the Mumford Conjecture by Haboush. See the appendix to [11] for a discussion of this point.2.1. The quotient variety X/G. Let G act morphically on an affine variety X. Then the algebra of invariants k[X~\ G is a finitely generated /c-algebra. We denote by X/G the affine variety Spm^A'] 0 ) and by n x : X -> X/G the morphism of varieties corresponding to the inclusion homomorphism k[X] G -> /c [X]; when reference to G is necessary we write n x G instead of n x .The morphism n x has the following properties.2.1.1. The morphism n x is a surjective morphism.2.1.2. If V x and V 2 are disjoint, G-stable closed subsets of X, then there exists an invariant fe k[X] G such that / = 0 on V x and / = 1 on V 2 . (a)If c e X/G, then the fibre n x~l (£,) contains a unique closed orbit. We denote by T(£) the unique closed orbit in n x~l {Q.(b) n x determines a bijective mapping from the set of closed G-orbits on X to X/G.2.1.4. If V is a G-stable closed subset of X, then n x (V) is a closed subset of X/G. 2.1.5. The topology on X/G is the quotient topology. That is, a subset U of X/G is open if and only if n x~l (U) is open in X.