We construct analogues of Rankin--Selberg integrals for Speh representations of the general linear group over a $p$-adic field. The integrals are in terms of the Shalika model and are expected to be the local counterparts of (suitably regularized) global integrals involving square-integrable automorphic forms and Eisenstein series on the general linear group over a global field. We relate the local integrals to the classical ones studied by Jacquet--Piatetski-Shapiro--Shalika. We also introduce a unitary structure for Speh representation on the Shalika model, as well as various other models including Zelevinsky's degenerate Whittaker model.

中文翻译：

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Speh 表示的局部 Rankin-Selberg 积分

我们为 $p$-adic 域上一般线性群的 Speh 表示构造了 Rankin--Selberg 积分的类似物。积分是在 Shalika 模型方面的，预计是（适当正则化的）全局积分的局部对应物，全局积分涉及全局域上的一般线性群的平方可积自守形式和爱森斯坦级数。我们将局部积分与 Jacquet--Piatetski-Shapiro--Shalika 研究的经典积分联系起来。我们还在 Shalika 模型以及包括 Zelevinsky 的退化 Whittaker 模型在内的各种其他模型上引入了 Speh 表示的单一结构。