论文标题
$ \ mathrm {gl} _n $的Archimedean Newform理论
Archimedean Newform Theory for $\mathrm{GL}_n$
论文作者
论文摘要
我们介绍了一个新的不变的导体指数，该指示符是$ \ mathrm {gl} _n $的通用Casselman-Wallach表示，该代表量量化了该表示形式的影响程度。我们还确定了一个杰出的向量，即新形式，在此表示中以多种多样发生，并以导体指数自然的方式测量该矢量的复杂性。最后，我们证明了NewForm是$ \ Mathrm {Gl} _n \ Times \ Mathrm {Gl} _n $和$ \ Mathrm {Gl} _N \ Times \ Times \ Mathrm {Gl} _ {n -1} $ rankin -Selberg的二次代表时。由于Jacquet，Piatetski-Shapiro和Shalika，该理论与类似的非建筑理论类似。结合在一起，这完成了$ \ mathrm {gl} _n $在数字字段上的$ \ mathrm {gl} _n $的全局新形式理论。证明的副产品包括Stade公式的新证明以及对Archimedean Godement-Jacquet-jacquet Zeta积分的测试向量问题的新解决方案。
We introduce a new invariant, the conductor exponent, of a generic irreducible Casselman-Wallach representation of $\mathrm{GL}_n$ that quantifies the extent to which this representation may be ramified. We also determine a distinguished vector, the newform, occurring with multiplicity one in this representation, with the complexity of this vector measured in a natural way by the conductor exponent. Finally, we show that the newform is a test vector for $\mathrm{GL}_n \times \mathrm{GL}_n$ and $\mathrm{GL}_n \times \mathrm{GL}_{n - 1}$ Rankin-Selberg integrals when the second representation is unramified. This theory parallels an analogous nonarchimedean theory due to Jacquet, Piatetski-Shapiro, and Shalika; combined, this completes a global theory of newforms for automorphic representations of $\mathrm{GL}_n$ over number fields. By-products of the proofs include new proofs of Stade's formulae and a new resolution of the test vector problem for archimedean Godement-Jacquet zeta integrals.
