Let F be a local field of characteristic zero. Let μ be a good character of ${\mathrm{GL}}_p(F)\times {\mathrm{GL}}_{p+1}(F)$. We study the generalized linear period problem for the pair $(G,H_{p,p+1})=({\mathrm{GL}}_{2p+1}(F),{\mathrm{GL}}_p(F)\times {\mathrm{GL}}_{p+1}(F))$ and we prove that any bi-$(H_{p,p+1},\mu )$-equivariant tempered generalized function on G is invariant under the matrix transpose. We also show that any $P\cap H_{p,p+1}$-invariant linear functional on an $H_{p,p+1}$-distinguished irreducible smooth representation of G is also $H_{p,p+1}$-invariant if F is nonarchimedean, where P is the standard mirabolic subgroup of G consisting of matrices with last row vector $(0,\cdots ,0,1)$.

中文翻译：

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广义线性周期

令F为特征为零的局部域。让μ成为${\mathrm{GL}}_p(F)\times {\mathrm{GL}}_{p+1}(F)$. 我们研究了对的广义线性周期问题$(G,H_{p,p+1})=({\mathrm{GL}}_{2p+1}(F),{\mathrm{GL}}_p(F)\times {\mathrm{GL}}_{p+1}(F))$ 我们证明任何双$(H_{p,p+1},\mu )$G上的等变缓和广义函数在矩阵转置下是不变的。我们还表明，任何$磷\cap H_{p,p+1}$- 上的不变线性泛函 $H_{p,p+1}$的不可约-distinguished光滑表示ģ也$H_{p,p+1}$-不变，如果F是非阿基米德，其中P是G的标准 mirabolic 子群，由具有最后一行向量的矩阵组成$(0,\cdots ,0,1)$.