THE INVARIANT HOLONOMIC SYSTEM ON A SEMISIMPLE LIE-ALGEBRA
THE INVARIANT HOLONOMIC SYSTEM ON A SEMISIMPLE LIE-ALGEBRA
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DOI：
10.1007/bf01388568
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发表时间：
1984-01-01
影响因子：
3.1
通讯作者：
KASHIWARA, M
中科院分区：
文献类型：
--
作者：
HOTTA, R;KASHIWARA, M
我们的动机是从李代数上完整系统的观点来研究Weyl群表示的几何理论，即所谓的Springer表示(见[21]及其参考文献)。在这一尝试中，Harish-Chandra广泛研究的定义不变特征分布的微分方程组通过Riemann-Hilbert对应相当于与定义Springer表示的交上同调复相对应的正则完整系统。由此，我们得到了这个完整系统根据Weyl群的作用进行的分解，这也与根据单程分解有关。通过傅立叶变换，这种分解给出了Borho和MacPherson([4]，[5])最近的一个定理的“解析”证明，该定理已首次用Bernstein-Beilinson-Deligne-Gabber的深层定理证明。其次，应用这一结果，我们可以研究这一微分方程组的解的结构。特别地，我们可以统一地推广Barbasch和Vogan([7]，[8])最近关于幂零轨道积分的傅里叶变换的结果。我们将深入了解更多细节。设g是具有连通群G的复半单李代数，固定g的一个Cartan子代数b，记为[3*[3]的对偶空间，通常通过Killing型与[3]相一致。让S(G)(分别Ii；[g]G)是G-不变对称张量环。G-不变多项式)。对于2e[3“，我们考虑以下微分方程组：和
Our motivation is to study the geometric theory of Weyl group representations, so called Springer's representations (see [21] and its references), from the view point of holonomic systems on Lie algebras. In this attempt, the system of differential equations defining invariant eigendistributions, which was extensively investigated by Harish-Chandra, occurs quite naturally as the regular holonomic system corresponding to the intersection cohomology complex defining Springer's representations, through the Riemann-Hilbert correspondence. Thus we obtain a decomposition of this holonomic system according to the action of the Weyl group, which is also related to the decomposition according to the monodromies. Through the Fourier transform, this decomposition gives an" analytic" proof of a recent theorem of Borho and MacPherson ([4],[5]), which has been first proved by using a deep theorem of Bernstein-Beilinson-Deligne-Gabber. Secondly, applying this result, we can investigate structures of solutions to this system of differential equations9 In particular, we can extend, in a unified way, a recent result of Barbasch and Vogan ([7],[8]) on the Fourier transforms of the nilpotent orbital integrals. We are going into more details. Let g be a complex semisimple Lie algebra with connected group G. Fix a Cartan subalgebra b of g and denote by [3* the dual space of [3 which is often identified with [3 through the Killing form. Let S (g)(resp. II;[g] G) be the ring of G-invariant symmetric tensors (resp. G-invariant polynomials) on g. For 2e [3", we consider the following systems of differential equations: and