On lifting from classical groups to GLN
On lifting from classical groups to GLN
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从经典群体提升到 GLN
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发表时间：
2001
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通讯作者：
F. Shahidi
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作者：
J. Cogdell;H. H. Kim;I. Piatetski;F. Shahidi
自守形式理论中最核心的问题之一是朗兰兹的函性或自守表示的提升[20，2]。函性的一个最基本的情况是将自守表示从分裂的经典群提升到适当的GLN。这就是我们在本文中所要解决的问题。设G是数域k上的可裂经典群。所以G是SO 2n +1、SO 2n或Sp 2n中的一个。Langlands对偶群的连通分量LG 0分别为Sp 2n（C）、SO 2n（C）或SO 2n +1（C）。在每种情况下，分别对于N = 2n、2n或2n + 1，存在LG 0到GLN（C）= GLN的自然嵌入。朗兰兹的哲学接着说，与这个对偶群映射相关联的，应该存在G（Ak）到GLN（Ak）的自守形式的提升。这个对偶群上的映射也支配G（G）到GLN（G）的不可约容许表示的局部提升，并且这些局部提升和全局提升应该是相容的。虽然目前还没有精确的整体猜想的性质提升在我们的情况下，当地的提升是很好地理解在几种情况下，全球提升可以理解的兼容性与当地电梯。若v是k的阿基米德位置，则G（n）的每个不可约容许表示πv由局部Weil群Wn到LG 0的容许同态给出[23，2].结合到GLN的映射，我们得到了GLN（λ）的不可约容许表示λ v的一个参数。Πv是πv的局部朗兰兹升力。类似地，如果v是非阿基米德空间，πv是非分歧容许表示，则πv由它的Satake参数[tv]确定，它是LG 0 [29，2]中的半单共轭类。[tv]在L-同态下的像确定了LGL 0 N中的共轭类，从而确定了GLN（N）的非分歧表示的Satake参数。同样，Πv是πv的局部朗兰兹升力。如果G（Ak）有一个不可约的自守表示π = π ′ πv，那么对于几乎所有的库所，即阿基米德库所和πv未分歧的非阿基米德库所，π的局部分支πv有一个局部提升Πv。我们可以说GLN（Ak）的一个自守表示是π的一个弱朗兰兹提升，或者简单地说是一个弱提升，如果在阿基米德位置和几乎所有非阿基米德位置，πv是非分歧的，那么实际上是πv的局部提升。
One of the most central questions in the theory of automorphic forms is that of Langlands’ functoriality or the lifting of automorphic representations [20, 2]. One of the most basic cases of functoriality would be the lifting of automorphic representations from the split classical groups to an appropriate GLN. It is this question we address in this paper. Take G to be a split classical group over a number field k. So G is one of the groups SO2n+1, SO2n, or Sp2n. The connected component LG0 of the Langlands dual group is Sp2n(C), SO2n(C), or SO2n+1(C) respectively. In each case there is a natural embedding of LG0 into GLN(C) = GLN for N = 2n, 2n, or 2n + 1 respectively. The philosophy of Langlands then says that associated to this map of dual groups there should exist a lifting of automorphic forms on G(Ak) to GLN(Ak). This map on dual groups also governs local liftings of irreducible admissible representations of G(kv) to GLN(kv) and these local and global liftings should be compatible. While there is at present no precise global conjecture on the nature of the lifting in our situation, the local lifting is well understood in several cases and the global lifting can be understood in terms of compatibility with the local lifts. If v is an archimedean place of k then every irreducible admissible representation πv of G(kv) is given by an admissible homomorphism of the local Weil group Wkv into LG0 [23, 2]. Composing with the map to GLN we get a parameter for an irreducible admissible representation Πv of GLN(kv). Πv is the local Langlands lift of πv. Similarly, if v is a non-archimedean place and πv an unramified admissible representation then πv is determined by its Satake parameter [tv] which is a semi-simple conjugacy class in LG0 [29, 2]. The image of [tv] under the L-homomorphism determines a conjugacy class in LGL 0 N and hence Satake parameters for an unramified representation Πv of GLN(kv). Again, Πv is the local Langlands lift of πv. If we have an irreducible automorphic representation π = ⊗′ πv of G(Ak) then for almost all places, namely the archimedean ones and the non-archimedean places where πv is unramified, the local component πv of π has a local lift Πv. We will say that a automorphic representation Π = ⊗Πv of GLN(Ak) is a weak Langlands lift of π, or simply a weak lift, if at the archimedean places and almost all non-archimedean places where πv is unramified Πv is in fact the local lift of πv.