Whittaker models and the integral Bernstein center for \(\mathrm{GL}_n\). (English) Zbl 1343.11053

Let \(F\) denote a \(p\)-adic field, and let \(k\) stand for an algebraically closed field of characteristic \(l\), where \(p\) and \(l\) are distinct primes. Let \(W(k)\) denote a ring of Witt vectors.
The first purpose of the paper under review is to provide an integral version of the results of C. J. Bushnell and G. Henniart [Am. J. Math. 125, No. 3, 513–547 (2003; Zbl 1031.11028)] on the Whittaker models in the complex representation theory of \(p\)-adic groups in the case of \(\mathrm{GL}(n, F)\), using known results on the structure theory of the category of smooth \(W(k)[\mathrm{GL}(n,F)]\)-modules.
In the second part of the paper, the author uses the previous techniques to discuss the existence of certain representations which appear in a conjectural local Langlands correspondence in families, introduced by M. Emerton and the author [Ann. Sci. Éc. Norm. Supér. (4) 47, No. 4, 655–722 (2014; Zbl 1321.11055)]. The existence of such representations is reduced to existence of a certain map defined on the integral Bernstein center.


11F70 Representation-theoretic methods; automorphic representations over local and global fields
11F33 Congruences for modular and \(p\)-adic modular forms
22E50 Representations of Lie and linear algebraic groups over local fields
11S37 Langlands-Weil conjectures, nonabelian class field theory
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