Representations of abelian algebraic groups
Representations of abelian algebraic groups
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阿贝尔代数群的表示
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发表时间：
1997
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通讯作者：
R. Langlands
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作者：
R. Langlands
有理由相信局部域上的约化代数群上的有理点群在调和分析意义下的不可约表示与局部域上局部域的Weil群在某个伴随复群中的表示之间有着密切的关系。在相关复群中的整体Weil群的表示和出现在自同构形空间中的Adele群的表示之间也应该有一种关系，尽管它不会那么密切。这些关系的性质将在其他地方解释。现在，我想要做的就是解释和证明当群是阿贝尔群时的关系。我应该指出，这种情况并不典型。例如，通常会有与Weil群的表示无关的代数群的表示。这些证明本身仅仅是课程场理论的练习。我之所以把它们写下来，是因为需要立即确认在几个简单情况下非常引人注目的一般原则。此外，如果不首先为阿贝尔群体解决这个问题，很可能不可能从总体上解决这个问题。如果证明看起来笨拙，过于坚持简单的事情，请记住，作者，借用一个比喻，以前没有骑过自行车，对他的车辆只有最低限度的控制。众所周知，定义在域F上且分裂在域F的Galois扩张K上的代数环面的同构类与G(K/F)作用于其上的格的等价类之间存在一一对应关系。如果T对应于L，则T上的K-有理点群Tk可以且应当被标识为具有Hom(L，K∗)的G(K/F)-模。如果K是一个整体域，A(K)是K的Adele环，如果CK是K的Idele类群，则群TA(K)/TK可以被标识为Hom(L，CK)。如果K是局部域，则CK是K的乘群。设L是格Hom(L，Z)。设C∗为非零复数乘法群，Cu为绝对值为1的复数群，则设T=Hom(L，C∗)，Tu=Hom(L，Cu)。有G(K/F)对L、T、Tu的自然作用。半直积T og(K/F)是以T u og(K/F)为实子群的复李群。如果F是局部域或全局域，则Weil群WK/F是扩展
There is reason to believe that there is a close relation between the irreducible representations, in the sense of harmonic analysis, of the group of rational points on a reductive algebraic group over a local field and the representations of the Weil group of the local field in a certain associated complex group. There should also be a relation, although it will not be so close, between the representations of the global Weil group in the associated complex group and the representations of the adele group that occur in the space of automorphic forms. The nature of these relations will be explained elsewhere. For now all I want to do is explain and prove the relations when the group is abelian. I should point out that this case is not typical. For example, in general there will be representations of the algebraic group not associated to representations of the Weil group. The proofs themselves are merely exercises in class field theory. I am writing them down because it is desirable to confirm immediately the general principle, which is very striking, in a few simple cases. Moreover, it is probably impossible to attack the problem in general without having first solved it for abelian groups. If the proofs seem clumsy and too insistent on simple things remember that the author, to borrow a metaphor, has not cocycled before and has only minimum control of his vehicle. It is well known that there is a one-to-one correspondence between isomorphism classes of algebraic tori defined over a field F and split over the Galois extension K of F and equivalence classes of lattices on which G(K/F ) acts. If T corresponds to L then TK , the group of K-rational points on T , may, and shall, be identified as a G(K/F )-module with Hom(L,K∗). If K is a global field and A(K) is the adele ring of K the group TA(K)/TK may be identified with Hom(L,CK) if CK is the idele class group of K. If K is a local field CK will be the multiplicative group of K. Suppose L is the lattice Hom(L,Z). If C∗ is the multiplicative group of nonzero complex numbers and Cu the group of complex numbers of absolute value 1 we set T = Hom(L,C∗) and Tu = Hom(L,Cu). There are natural actions of G(K/F ) on L, T , and Tu. The semidirect product T oG(K/F ) is a complex Lie group with Tu oG(K/F ) as a real subgroup. If F is a local or global field the Weil group WK/F is an extension