Smooth transfer of Kloosterman integrals (the Archimedean case)
Smooth transfer of Kloosterman integrals (the Archimedean case)
复制标题
Kloosterman 积分的平滑传递（阿基米德情况）
DOI：
10.1353/ajm.2013.0000
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发表时间：
2010
影响因子：
1.7
通讯作者：
D. Gourevitch
中科院分区：
文献类型：
--
作者：
Avraham Aizenbud;D. Gourevitch
We establish the existence of a transfer, which is compatible with Kloosterman
integrals, between Schwartz functions on ${\rm GL}_n(\Bbb{R})$ and Schwartz functions on
the variety of non-degenerate Hermitian forms. Namely, we consider an integral
of a Schwartz function on ${\rm GL}_n(\Bbb{R})$ along the orbits of the two sided action
of the groups of upper and lower unipotent matrices twisted by a non-degenerate
character. This gives a smooth function on the torus. We prove that the space of
all functions obtained in such a way coincides with the space that is constructed
analogously when ${\rm GL}_n(\Bbb{R})$ is replaced with the variety of non-degenerate hermitian
forms. We also obtain similar results for $\frak{gl}_n(\Bbb{R})$. The non-Archimedean case was
done by H. Jacquet ({\it Duke Math. J.}, 2003) and our proof is based on the ideas
of this work. However we have to face additional difficulties that appear only in the
Archimedean case. Those results are crucial for the comparison of the Kuznetsov trace
formula and the relative trace formula of ${\rm GL}_n$ with respect to the maximal unipotent
subgroup and the unitary group, as done by H. Jacquet, and by B. Feigon, E. Lapid, and O. Offen.
We establish the existence of a transfer, which is compatible with Kloosterman
integrals, between Schwartz functions on ${\rm GL}_n(\Bbb{R})$ and Schwartz functions on
the variety of non-degenerate Hermitian forms. Namely, we consider an integral
of a Schwartz function on ${\rm GL}_n(\Bbb{R})$ along the orbits of the two sided action
of the groups of upper and lower unipotent matrices twisted by a non-degenerate
character. This gives a smooth function on the torus. We prove that the space of
all functions obtained in such a way coincides with the space that is constructed
analogously when ${\rm GL}_n(\Bbb{R})$ is replaced with the variety of non-degenerate hermitian
forms. We also obtain similar results for $\frak{gl}_n(\Bbb{R})$. The non-Archimedean case was
done by H. Jacquet ({\it Duke Math. J.}, 2003) and our proof is based on the ideas
of this work. However we have to face additional difficulties that appear only in the
Archimedean case. Those results are crucial for the comparison of the Kuznetsov trace
formula and the relative trace formula of ${\rm GL}_n$ with respect to the maximal unipotent
subgroup and the unitary group, as done by H. Jacquet, and by B. Feigon, E. Lapid, and O. Offen.