$\ell$ -模Zelevinski对合
The $\ell$-modular Zelevinski involution
摘要：
设F是一个具有剩余特征p的非阿基米德局部紧域，G是GL（n，F）的一个内形式，n是正整数，R是一个特征不同于p的代数闭域.当R具有特征$\ell>0$时，G的不可约光滑R-表示$π$的Aubert对合象不一定是不可约的.我们证明了这个图像（在G的Grothendieck群）包含一个唯一的不可约项$π$ * 具有相同的尖点支持$π$。这定义了G的不可约R-表示的同构类集合上的对合，当R是复数域时，这与Zelevinski对合相一致。我们所用的方法也适用于特征为p的有限域F，在这种情况下，我们得到了类似的结果。
英文摘要：
Let F be a non-Archimedean locally compact field with residual characteristic p, let G be an inner form of GL(n,F) for a positive integer n and let R be an algebraically closed field of characteristic different from p. When R has characteristic $\ell>0$, the image of an irreducible smooth R-representation $π$ of G by the Aubert involution need not be irreducible. We prove that this image (in the Grothendieck group of G) contains a unique irreducible term $π$* with the same cuspidal support as $π$. This defines an involution on the set of isomorphism classes of irreducible R-representations of G, that coincides with the Zelevinski involution when R is the field of complex numbers. The method we use also works for F a finite field of characteristic p, in which case we get a similar result.
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