## 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.

### MSC:

 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

### Citations:

Zbl 1031.11028; Zbl 1321.11055
Full Text:

### References:

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