AI 中文总结
本文为p-adic GL(n)的尖点表示建立Paškūnas-Stevens Whittaker函数与Whittaker新形式间的双向积分转换公式，并据此给出GL(n)×GL(n)对在Rankin-Selberg L-因子非恒为1时的显式测试向量。
AI 中文摘要
最近，Girsch-Kurinczuk获得了$p$-adic $\mathrm{GL}(n)$的深度为零和极小极大整数深度尖点表示的局部新形式的新公式。他们的主要结果之一是一个平均公式，该公式以Paškūnas-Stevens Whittaker函数给出了Whittaker新形式的积分表示。在本文中，我们首先通过建立Paškūnas-Stevens Whittaker函数以Whittaker新形式表示的积分表示，证明了一个反向平均公式。这些公式共同给出了这两个特殊向量之间的显式双向转换。最后，我们利用我们的平均公式和Kurinczuk-Matringe的早期工作，为这类尖点$\mathrm{GL}(n)\times\mathrm{GL}(n)$对给出了以Whittaker新形式表示的新的显式测试向量，只要它们的Rankin-Selberg $L$-因子不恒等于1。
英文摘要
Recently, Girsch-Kurinczuk have obtained new formulas for local newforms in depth-zero and minimax integral-depth cuspidal representations of $p$-adic $\mathrm{GL}(n)$. One of their main results is an averaging formula which provides an integral representation of the Whittaker newform in terms of the Paškūnas-Stevens Whittaker function. In the present paper, we first prove a reverse averaging formula by establishing an integral representation of the Paškūnas-Stevens Whittaker function in terms of the Whittaker newform. Together, these formulas give an explicit two-way passage between these two distinguished vectors. Finally, we use our averaging formula and earlier work of Kurinczuk-Matringe to give new explicit test vectors in terms of Whittaker newforms for such cuspidal $\mathrm{GL}(n)\times\mathrm{GL}(n)$ pairs, whenever their Rankin-Selberg $L$-factor is not identically equal to $1$.
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