Adeles and Algebraic Groups

We always denote by k a field of algebraic numbers or a field of algebraic functions of one variable over a finite constant field. We denote by k the completion of k with respect to a valuation v of k; if v is discrete, we use often the notation p and denote by \( \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}\to {o} _p \) the ring of p-adic integers in k_p. We denote by S any finite set of valuations which contains all the non discrete valuations (infinite places).

We normalize the Haar measure dx_v of the complete field k_v in the following way:

As formerly, whenever V is a variety, defined over a field k, we denote by V_k the set of points of V, rational over k; a vectorspace of dimension d over k can always be denoted by R_k, where R is an affine space of dimension d in the sense of algebraic geometry. In particular, any algebra over k can be so written; the obvious extension to R of the multiplication-law on the algebra R_k makes R into an algebra-variety, defined over k (which means that the multiplication-law on R is defined over k). The given algebra R_k over k is absolutely semisimple if and only if R is so, i.e. if and only if R is isomorphic (over the universal domain) to a direct sum of matrix algebras.

We consider only the algebraic groups, over a groundfield k, which, over the universal domain, are isogenous to products of simple groups of the three “classical” types : “special” linear, orthogonal and symplectic. Excluding the case of characteristic 2 (which has not been fully investigated) and certain “exceptional” forms of the orthogonal group in 8 variables (depending upon the principle of triality), such groups, up to isogeny, can be reduced to the following types, which will be called “classical” (the letter indicates the type over the universal domain, and K denotes any separably algebraic extension of k):