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On the motive of \(G\) and the principal homomorphism \(\text{SL}_2\to\widehat G\). (English) Zbl 0942.20031

Summary: Let \(k\) be a field, and let \(G\) be a connected reductive group over \(k\). Let \(\widehat G\) be the Langlands dual group, which is a reductive group over \(\mathbb{C}\). In an earlier paper [Invent. Math. 130, No. 2, 287-313 (1997; Zbl 0904.11014)] we attached a motive \(M\) of Artin-Tate type to \(G\). In this paper, we relate \(M\) to the principal homomorphism \(SL_2\to\widehat G\), which was introduced by de Siebenthal and Dynkin, and studied extensively by Kostant. As a corollary, we relate the \(L\)-function of the dual motive \(M^\vee\), when \(k\) is a local non-Archimedean field to the Langlands \(L\)-function of the Steinberg representation of \(G(k)\), with respect to the adjoint representation of the \(L\)-group. We also construct an involution \(\theta\) of \(\widehat{\mathfrak g}=\text{Lie}(\widehat G)\) when \(k=\mathbb{R}\).

MSC:

20G25 Linear algebraic groups over local fields and their integers
11S37 Langlands-Weil conjectures, nonabelian class field theory
22E50 Representations of Lie and linear algebraic groups over local fields
20G15 Linear algebraic groups over arbitrary fields
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
14A20 Generalizations (algebraic spaces, stacks)

Citations:

Zbl 0904.11014
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