Search for: All records

The goal of this paper is extend Kottwitz’s theory of B(G) for global fields. In particular, we show how to extend the definition of “B(G) with adelic coefficients” from tori to all connected reductive groups. As an application, we give an explicit construction of certain transfer factors for non-regular semisimple elements of non-quasisplit groups. This generalizes some results of Kaletha and Taibi. These formulas are used in the stabilization of the cohomology of Shimura and Igusa varieties. 

Abstract We construct a moduli space $$\textsf {LP}_{G}$$ of $$\operatorname {SL}_{2}$$-parameters over $${\mathbb {Q}}$$, and show that it has good geometric properties (e.g., explicitly parametrized geometric connected components and smoothness). We construct a Jacobson–Morozov morphism$$\textsf {JM}\colon \textsf {LP}_{G}\to \textsf {WDP}_{G}$$ (where $$\textsf {WDP}_{G}$$ is the moduli space of Weil–Deligne parameters considered by several other authors). We show that $$\textsf {JM}$$ is an isomorphism over a dense open of $$\textsf {WDP}_{G}$$, that it induces an isomorphism between the discrete loci $$\textsf {LP}^{\textrm {disc}}_{G}\to \textsf {WDP}_{G}^{\textrm {disc}}$$, and that for any $${\mathbb {Q}}$$-algebra $$A$$ it induces a bijection between Frobenius semi-simple equivalence classes in $$\textsf {LP}_{G}(A)$$ and Frobenius semi-simple equivalence classes in $$\textsf {WDP}_{G}(A)$$ with constant (up to conjugacy) monodromy operator. 

Scholze and Shin [J. Amer. Math. Soc. 26 (2013), pp. 261–294] gave a conjectural formula relating the traces on the automorphic and Galois sides of a local Langlands correspondence. Their work generalized an earlier formula of Scholze, which he used to give a new proof of the local Langlands conjecture for GL_n. Unlike the case for GL_n, the existence of non-singleton L-packets for more general reductive groups constitutes a serious representation-theoretic obstruction to proving that such a formula uniquely characterizes such a correspondence. We show how to overcome this problem, and demonstrate that the Scholze–Shin equation is enough, together with other standard desiderata, to uniquely characterize the local Langlands correspondence for discrete parameters.