Maximal connected

Let k k be any field and let alttext="upper G"> G encoding="application/x-tex">G a connected reductive algebraic -group. Associated to is an invariant first studied in the 1960s by Satake [Ann. of Math. (2) 71 (1960), 77–110] Tits [Théorie des Groupes Algébriques (Bruxelles, 1962), Librairie Universitaire, Louvain; Gauthier- Villars, Paris, 1962], [Algebraic Groups Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965), Amer. Soc., Providence, R.I., 1966, pp. 33–62] that called index (a Dynkin diagram along with some additional combinatorial information). showed -isogeny class uniquely determined its anisotropic kernel G Subscript a"> a encoding="application/x-tex">G_a . For cases where absolutely simple, all possibilities for have been classified 33–62]. H"> H encoding="application/x-tex">H -subgroup maximal rank We introduce left-parenthesis k right-parenthesis"> ( stretchy="false">) encoding="application/x-tex">G(k) -conjugacy embedding indices H subset-of upper ⊂ encoding="application/x-tex">H \subset G This consists map satisfies certain compatibility conditions. equivalence relation index-conjugacy on set -subgroups , observe encoding="application/x-tex">H_a show indices. simple exceptional type we classify Finally, establish existence results. In particular, consider which embeddings exist when has cohomological dimension 1 (resp. alttext="k equals double-struck R"> = R encoding="application/x-tex">k=\mathbb {R} alttext="German p"> mathvariant="fraktur">p encoding="application/x-tex">\mathfrak {p} -adic).