Real homogeneous spaces, Galois cohomology, and Reeder puzzles

Abstract

Let G be a simply connected absolutely simple algebraic group defined over the field of real numbers R. Let H be a simply connected semisimple R-subgroup of G. We consider the homogeneous space X = G/H. We ask: how many connected components has X(R)?We give a method of answering this question. Our method is based on our solutions of generalized Reeder puzzles.

Galois cohomology of real semisimple groups via Kac labelings

“…Since all automorphisms of G are inner, the real forms of E 7 correspond to the elements of H 1 (R, G ad ), and by Corollary 13.7 they correspond to the orbits of C = P ∨ /Q ∨ ≃ Z/2Z in the set K( D) of Kac labelings of D. These orbits are: {q (1) , q (2) }, {q (3) }, {q (4) , q (5) }, {q (6) }, hence #H 1 (R, G ad ) = 4. We write G q for the real form of G defined by the Kac labeling q.…”

“…For G q = EV = E 7(7) (the split form) we take q = q (6) , and for G q = EVII = E (7(−25) (the Hermitian form) we take q = q (3) ; see [19,Table 7]. Both labelings q (6) and q (3) are odd.…”

“…For G q = EV = E 7(7) (the split form) we take q = q (6) , and for G q = EVII = E (7(−25) (the Hermitian form) we take q = q (3) ; see [19,Table 7]. Both labelings q (6) and q (3) are odd. In both cases the set K (q) is the set K odd = {q (3) , q (6) } of all odd labelings of D. The set H 1 (R, G q ) is in a bijection with the set K odd .…”

“…A real algebraic group G is called R-simple if it has no positive-dimensional normal algebraic subgroups defined over R. A real algebraic group G is called absolutely simple if it remains simple after extension of scalars from R to C. The Galois cohomology sets H 1 (R, G) of the classical groups are well known. Recently the sets H 1 (R, G) were computed for "most" of the absolutely simple R-groups by Adams and Taïbi [1], in particular, for all simply connected absolutely simple R-groups by Adams and Taïbi [1] and by Borovoi and Evenor [6]. A simply connected R-simple group G that is not absolutely simple is isomorphic to a group obtained by the Weil restriction of scalars from a simply connected simple C-group.…”

Galois cohomology and component group of a real reductive group

“…The Galois cohomology sets H 1 G of the classical groups G are well known. The sets H 1 G were computed for "most" of the absolutely simple R-groups by Adams and Taïbi [AT18], in particular, for all simply connected absolutely simple R-groups by Adams and Taïbi [AT18] and by Borovoi and Evenor [BE16]. Thus H 1 G is known for all simply connected semisimple R-groups G; see [BT21,Introduction] for details.…”