A specialization theorem for certain Weyl group representations and an application to the green polynomials of unitary groups
A specialization theorem for certain Weyl group representations and an application to the green polynomials of unitary groups
复制标题
某些Weyl群表示的专门化定理及其在酉群绿色多项式中的应用
DOI：
10.1007/bf01418371
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发表时间：
1977
期刊：
影响因子：
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通讯作者：
T. A. Springer
中科院分区：
文献类型：
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作者：
R. Hotta;T. A. Springer
若G是IFq上的连通约化线性代数群，且A是其李代数的幂零元，则第二作者在[12]中定义了G的Weyl群在其李代数包含A的Borel子群簇~ a的/-adic上同调中的表示（在对特征p的某些限制下）.一个简短的r6 sum ~的建设是在w 1。w的主要结果是定理1.1，这是这些表示的特化定理，它允许在某些情况下还原为A= 0的情况。作为这样一个减少的例子，我们在命题1.4中给出了一个识别的一些不可约件的外尔群表示，在这种情况下，A是一个幂零”的抛物型”，即其轨道相交密集的李代数的幂单根的一些抛物子群。这样的一个结果的存在是G. Lusztig（在2月17日的信中，1976年）。从1.1导出1.4的方法也归功于他。作为1.1的第二个应用，我们确定酉群U（0 Fq）（假设q足够大）的绿色多项式。这些绿色函数是[12]中的函数。Kazhdan [6]证明了它们与Deligne和Lusztig在[2]中定义的绿色函数是一致的。GL，（IFq）的绿色函数由q中的多项式给出。关于U_i（IF_4）的主要结果是，在GL_i（IF，O）的绿色函数中，”将q变为-q”而得到它们。精确公式在定理3.1中给出。结合最近Lusztig和Srinivasan [8]对U_i（IF_i）的不可约特征标的分类工作（也见Kawanaka的论文[5]），解决了Ennola [3]的一个猜想（p和q足够大）。3.1的证明利用了文[12]中建立的绿色函数的性质和关于旗流形上簇~ a及其补簇的上同调的一些结果。这些结果也在[12; 7.7]中提到过（但过于简略），它们是由Spaleinstein在[11]中建立的结果得出的。我们所需要的精确公式在w中给出。这一节还包含关于GL，（IFq）的绿色多项式的系数的一些注释，这些注释自然适合于讨论。
If G is a connected reductive linear algebraic group over IFq, the finite field with q elements of characteristic p, and A is a nilpotent element of its Lie algebra, the second named author defined in [12] representations of the Weyl group of G in the/-adic cohomology of the variety~ a of Borel subgroups of G whose Lie algebras contain A (under some restrictions on the characteristic p). A brief r6sum~ of the construction is given in w 1. The main result of w is Theorem 1.1, a specialization theorem for these representations, which allows one to make, in certain cases, a reduction to the case A= 0. As an example of such a reduction we give in Proposition 1.4 an identification of some of the irreducible pieces of the Weyl group representations, in the case that A is a nilpotent" of parabolic type", ie such that its orbit intersects densely the Lie algebra of the unipotent radical of some parabolic subgroup. That a result like this exists was pointed out to one of us by G. Lusztig (in a letter dated 17 Feb., 1976). The method of deriving 1.4 from 1.1 is also due to him.As a second application of 1.1 we determine the Green polynomials of the unitary groups U, 0Fq)(assuming q sufficiently large). These Green functions are the ones of [12]. It was proved by Kazhdan [6] that they coincide with the Green functions defined by Deligne and Lusztig in [2]. The Green functions of GL,,(IFq) are given by polynomials in q. The main result about those of U,(IF4) is that one obtains them" by changing q into-q" in the Green functions of GL,,(IF, O. The precise formulation is given in Theorem 3.1. Combined with the recent work on the classification of the irreducible characters of U,(IF,) by Lusztig and Srinivasan [8](see also Kawanaka's paper [5]), a conjecture of Ennola [3] has thus been settled (for p and q large enough). The proof of 3.1 uses the properties of Green functions established in [12] and some results about the cohomology of the varieties~ a and their complements in the flag manifold. These results, also mentioned (but too briefly) in [12; 7.7], follow from what is established by Spaltenstein in [11]. A precise formulation of what we need is given in w This section also contains some remarks on the coefficients of the Green polynomials of GL,(IFq), which fit naturally in the discussion.