By the unfolding method, Rankin–Selberg L-functions for \({{\,\mathrm{GL}\,}}(n)\times {{\,\mathrm{GL}\,}}(n^{\prime })\) can be expressed in terms of period integrals. These period integrals actually define invariant forms on tensor products of the relevant automorphic representations. By the multiplicity-one theorems due to Sun–Zhu and Chen–Sun such invariant forms are unique up to scalar multiples and can therefore be related to invariant forms on equivalent principal series representations. We construct meromorphic families of such invariant forms for spherical principal series representations of \({{\,\mathrm{GL}\,}}(n,{\mathbb {R}})\) and conjecture that their special values at the spherical vectors agree in absolute value with the archimedean local L-factors of the corresponding L-functions. We verify this conjecture in several cases. This work can be viewed as the first of two steps in a technique due to Bernstein–Reznikov for estimating L-functions using their period integral expressions.

通过展开方法，Rankin–Selberg L函数对\（{{\\ mathrm {GL} \，}}（n）\ times {{\\ mathrm {GL} \，}}（n ^ {\质数}）\）可以用周期积分表示。这些周期积分实际上在相关自守表示的张量积上定义了不变形式。通过归因于Sun–Zhu和Chen–Sun的多重一定理，这种不变量形式在标量倍数之前是唯一的，因此可以与等效主级表示形式的不变量形式相关。我们为\（{{\，\ mathrm {GL} \，}}（n，{\ mathbb {R}}）\）的球面主序列表示构造此类不变形式的亚纯族，并推测它们在球形向量的绝对值与阿基米德局部值一致相应的L函数的L因子。我们在几种情况下验证了这个猜想。由于Bernstein–Reznikov使用其周期积分表达式估算L函数，因此这项工作可以看作是该技术中两个步骤的第一步。