Consider a semisimple linear algebraic group G over an arbitrary field F, and a projective homogeneous G-variety X. The geometry of such varieties has been a consistently active subject of research in algebraic geometry for decades, with significant contributions made by Grothendieck, Demazure, Tits, Panin, and Merkurjev, among others. An effective tool for the classification of these varieties is the notion of a cohomological (or alternatively, a motivic) invariant. Two such invariants are the set of Tits algebras of G defined by J. Tits, and the J-invariant of G defined by Petrov, Semenov, and Zainoulline. Quéguiner-Mathieu, Semenov and Zainoulline discovered a connection between these invariants, which they developed through use of the second Chern class map. The first goal of the present thesis is to extend this connection through the use of higher Chern class maps. Our main technical tool is the Steinberg basis, which provides explicit generators for the γ-filtration on the Grothendieck group K_0(X) in terms of characteristic classes of line bundles over X. As an application, we establish a connection between the J-invariant and the Tits algebras of a group G of inner type E6. The second goal of this thesis is to relate the indices of the Tits algebras of G to nontrivial torsion elements in the γ-filtration on K_0(X). While the Steinberg basis provides an explicit set of generators of the γ-filtration, the relations are not easily computed. A tool introduced by Zainoulline called the twisted γ-filtration acts as a surjective image of the γ-filtration, with explicit sets of both generators and relations. We use this tool to construct torsion elements in the degree 2 component of the γ-filtration for groups of inner type D2n. Such a group corresponds to an algebra A endowed with an orthogonal involution having trivial discriminant. In the trialitarian case (i.e. type D4), we construct a specific element in the γ-filtration which detects splitting of the associated Tits algebras. We then relate the non-triviality of this element to other properties of the trialitarian triple such as decomposability and hyperbolicity.

Working over an arbitrary field F of characteristic not 2, we consider linear algebraic groups over F. We view these as functors, represented by finitely generated F-Hopf algebras, from the category of commutative, associative, F-algebras Alg_F, to the category of groups. Classical examples of these groups, such as the special linear group SL_n are split, however there are also linear algebraic groups arising from central simple F-algebras which are non-split. For example, associated to a non-split central simple F-algebra A of degree n is a non-split special linear group SL(A). It is well known that central simple algebras are twisted forms of matrix algebras. This means that over the separable closure of F, denoted F_sep, we have A⊗_F F_sep ∼= M_n(F_sep) and that there is a twisted Gal(F_sep/F)-action on M_n(F_sep) whose fixed points are A. We show that a similar method of twisted Galois descent can be used to obtain all non-split semisimple linear algebraic groups associated to central simple algebras as fixed points within their split counterparts. In particular, these techniques can be used to construct the spin and half-spin groups Spin(A, τ ) and HSpin(A, τ ) associated to a central simple F-algebra of degree 4n with orthogonal involution. Furthermore, we develop a theory of twisted Galois descent for Hopf algebras and show how the fixed points obtained this way are the representing Hopf algebras of our non-split groups. Returning to the view of group schemes as functors, we discuss how the group schemes we consider are sheaves on the étale site of Alg_F whose stalks are Chevalley groups over local, strictly Henselian F-algebras. This allows us to use the generators and relations presentation of Chevalley groups to explicitly describe group scheme morphisms. After showing how the Kronecker tensor product of matrices induces maps between simply connected groups, we give an explicit description of these maps in terms of Chevalley generators. This allows us to compute the kernel of these new maps composed with standard isogenies and thereby construct new tensor product maps between non-simply connected split groups. These new maps are Gal(F_sep/F)-morphisms and so we apply our techniques of twisted Galois descent to also obtain new tensor product morphisms between non-split groups schemes. Finally, we use one of our new split tensor product maps to compute the degree three cohomological invariants of HSpin_4n for all n.