On certain representations of semi-simple algebraic groups and the arithmetic of the corresponding invariants (1)
On certain representations of semi-simple algebraic groups and the arithmetic of the corresponding invariants (1)
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关于半单代数群的某些表示及其相应不变量的算术（1）
DOI：
10.1007/bf01389828
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发表时间：
1971
影响因子：
3.1
通讯作者：
J. Igusa
中科院分区：
文献类型：
--
作者：
J. Igusa
得到齐次空间GA/Gk上的连续函数。我们说，如果每一个这样的函数在GA/ gk上对Haa r测度d g是可积的，那么表示p在k上是可积的，为了我们后面的目的，在总测度为1的条件下。如果p在k的每一个有限代数扩展上都是可容许的，我们说p在k上是绝对可容许的。设G O表示G中单位元的连通分量。然后，如果p在k上是绝对可容许的，则X上的G~环是由X上有限个代数无关的齐次多项式f~ . . . . .生成的设I(X)表示仿射n空间，f: X ~ I(X)表示由f(X) =(fl(X) . . . . .定义的态射则i (x)中每一点i上的纤维f t (i)包含一个主G~如U(i)，而U(i)对所有i的并集x′是定义在k上的x的Zariski开子集。我们称x′为x的主子集。设Z表示一个“基本字符”，即ka/k的非平凡字符，令(i, i *) = Z (il i′~ +…+i N i*) =(il ... .， i:r i*=(i* . . . . .i~v) in (kA) ~r设dx表示XA上的Haar测度，以XA /Xk的总测度为1为归一化条件。然后，如果X上的每一个G~都是一个变元，一个推测的西格尔公式表明
gives rise to a continuous function on the homogeneous space GA/Gk. We say that the representation p is admissible over k if every such function is Ll-integrable with respect to the Haa r measure d g on GA/G k normalized, for our later purpose, by the condition that the total measure is 1. We say that p is absolutely admissible over k if it is admissible over every finite algebraic extension of k. Let G O denote the connected component of the identity in G. Then, if p is absolutely admissible over k, the ring of G~ on X is generated by a finite number of algebraically independent homogeneous polynomials on X, say f~ . . . . . f~r with coefficients in k. Let I(X) denote the affine N-space and f : X ~ I(X) the morphism defined by f(x)=(fl(x) . . . . . fN(x)) for every x in X. Then the fiber f t ( i ) over every point i of I(X) contains a principal G~ say U(i), and the union X' of U(i) for all i is a Zariski open subset of X defined over k. We call X ' the principal subset of X. Let Z denote a "basic character", i.e., a non-trivial character of ka/k, and put (i, i * ) = z(il i'~ + ... +i N i*) for every i=(il ... . , i:r i*=(i* . . . . . i~v) in (kA) ~r Let dx denote the Haar measure on XA normalized by the condition that the total measure of Xa/Xk is 1. Then, if every G~ on X is a Ginvariant, a conjectural Siegel formula states that