The [formula omitted]-invariant, Tits algebras and triality

Abstract

In the present paper we set up a connection between the indices of the Tits algebras of a semisimple linear algebraic group G and the degree one indices of its motivic J-invariant. Our main technical tools are the second Chern class map and Grothendieck’s γ-filtration.As an application we provide lower and upper bounds for the degree one indices of theJ-invariant of an algebra A with orthogonal involution σ and describe all possible values of the J-invariant in the trialitarian case, i.e., when degree of A equals 8. Moreover, we establish several relations between the J-invariant of (A,σ) and the J-invariant of the corresponding quadratic form over the function field of the Severi–Brauer variety of A.