Abstract
We prove a descent criterion for certain families of smooth representations of {text {GL}}_n(F) (F a p-adic field) in terms of the gamma -factors of pairs constructed in Moss (Int Math Res Not 2016(16):4903–4936, 2016). We then use this descent criterion, together with a theory of gamma -factors for families of representations of the Weil group W_F (Helm and Moss in Deligne–Langlands gamma factors in families, arXiv:1510.08743v3, 2015), to prove a series of conjectures, due to the first author, that give a complete description of the center of the category of smooth W(k)[{text {GL}}_n(F)]-modules (the so-called “integral Bernstein center”) in terms of Galois theory and the local Langlands correspondence. An immediate consequence is the conjectural “local Langlands correspondence in families” of Emerton and Helm (Ann Sci Éc Norm Supér (4) 47(4):655–722, 2014).
Highlights
The center of an abelian category A is the ring of natural transformations from the identity functor of A to itself
Paskunas [12] gives a complete description of the center of the category of finite length representations of GL2(Qp) over p-adic integer rings, and uses this description to describe the image of the Colmez functor
We show that if the γ -factors attached to the pairs V × Wν have coefficients in A ⊗ Zν, V arises via base change from a co-Whittaker module over A
Summary
The center of an abelian category A is the ring of natural transformations from the identity functor of A to itself. In [11], the second author developed a theory of zeta integrals and γ -factors for co-Whittaker modules that is compatible with the classical theory over algebraically closed fields of characteristic zero and that satisfies a suitable local functional equation The first is that it is easy to show that the weak conjecture holds after inverting , using the Bernstein–Deligne description of the Bernstein center for fields of characteristic zero (this is Theorem 10.4 of [8]) This gives us a map Zν → Rν[ 1 ], compatible with local Langlands; the problem is to show that its image lies in Rν.
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