On the motive of G and the principal homomorphism SL₂ → Ĝ

Let A: be a field, and let G be a connected reductive group over k.Let G be the Langlands dual group, which is a reductive group over C. In [Gr] we attached a motive M of Artin-Tate type to G.In this paper, we relate M to the principal homomorphism SL2 -t G, which was introduced by de Siebenthal [deS] and Dynkin [D], and studied extensively by Kostant [K].As a corollary, we relate the L-function of the dual motive M v , when A; is a local non-Archimedean field, to the Langlands L-function of the Steinberg representation of G(/J), with respect to the adjoint representation of the L-group.We also construct an involution 9 of § = Lie(G) when k = R.