Let F be a non-archimedean local field of residue characteristic p, let Ĝ be a split connected reductive group over Z[ 1 p ] with an action of W F , and let G L denote the semidirect product Ĝ ⋊ W F . We construct a moduli space of Langlands parameters W F → G L , and show that it is locally of finite type and flat over Z[ 1 p ], and that it is a reduced local complete intersection. We give parameterizations of the connected components of this space over algebraically closed fields of characteristic zero and characteristic ℓ = p, as well as of the components of the space over Z ℓ and (conjecturally) over Z[ 1 p ]. Finally, we study the functions on this space that are invariant under conjugation by Ĝ (or, equivalently, the GIT quotient of this space by Ĝ) and give a complete description of this ring of functions after inverting an explicit finite set of primes depending only on G L .
Let G be a reductive group over a non-archimedean local field F of residue characteristic p. We prove that the Hecke algebras of G(F ) with coefficients in any noetherian Z ℓ -algebra R with ℓ = p, are finitely generated modules over their centers, and that these centers are finitely generated R-algebras. Following Bernstein's original strategy, we then deduce that "second adjointness" holds for smooth representations of G(F ) with coefficients in any Z[ 1 p ]-algebra. These results had been conjectured for a long time. The crucial new tool that unlocks the problem is the Fargues-Scholze morphism between a certain "excursion algebra" defined on the Langlands parameters side and the Bernstein center of G(F ). Using this bridge, our main results are representation theoretic counterparts of the finiteness of certain morphisms between coarse moduli spaces of local Langlands parameters that we also prove here, which may be of independent interest. Contents 1. Main results 1 2. Finiteness on the parameters side 4 3. Finiteness on the group side 8 3.1. Proof of Theorem 1.2 10 4. Second Adjointness 11 Acknowledgements 15 References 15
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