We prove that for every Bushnell–Kutzko type that satisfies a certain rigidity assumption, the equivalence of categories between the corresponding Bernstein component and the category of modules for the Hecke algebra of the type induces a bijection between irreducible unitary representations in the two categories. Moreover, we show that every irreducible smooth G-representation contains a rigid type. This is a generalization of the unitarity criterion of Barbasch and Moy for representations with Iwahori fixed vectors.

中文翻译：

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还原 p-adic 群的类型和幺正表示

我们证明，对于每个满足一定刚性假设的 Bushnell-Kutzko 类型，对应的 Bernstein 分量和该类型的 Hecke 代数的模范畴之间的范畴的等价性在两个范畴中的不可约幺正表示之间引入了双射。此外，我们证明了每个不可约的平滑 G 表示都包含一个刚性类型。这是 Barbasch 和 Moy 对 Iwahori 固定向量表示的幺正性标准的概括。