We describe two conjectures, one strictly stronger than the other, that give descriptions of the integral Bernstein center for GL_n(F) (that is, the center of the category of smooth W(k)[GL_n(F)]-modules, for F a p-adic field and k an algebraically closed field of characteristic l different from p) in terms of Galois theory. Moreover, we show that the weak version of the conjecture (for m at most n) implies the strong version of the conjecture. In a companion paper [HM] we show that the strong conjecture for n-1 implies the weak conjecture for n; thus the two papers together give an inductive proof of both conjectures. The upshot is a description of the integral Bernstein center for GL_n in purely Galois- theoretic terms; previous work of the author shows that such a description implies the conjectural "local Langlands correspondence in families" of Emerton and the author.

中文翻译：

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Curtis 同态和 GLn 的积分 Bernstein 中心

我们描述了两个猜想，一个比另一个强，它们描述了 GL_n(F) 的积分 Bernstein 中心（即平滑 W(k)[GL_n(F)]-模的范畴的中心，对于根据伽罗瓦理论，F 是 p 进场，k 是特征 l 与 p) 不同的代数闭场。此外，我们证明了猜想的弱版本（对于 m 至多 n）意味着猜想的强版本。在一篇配套论文 [HM] 中，我们证明了 n-1 的强猜想意味着 n 的弱猜想；因此，这两篇论文共同对这两个猜想进行了归纳证明。结果是用纯粹的伽罗瓦理论术语描述了 GL_n 的积分伯恩斯坦中心；作者之前的工作表明，这样的描述意味着推测“