CHARACTERISTIC CYCLES OF HOLOMORPHIC DISCRETE SERIES
CHARACTERISTIC CYCLES OF HOLOMORPHIC DISCRETE SERIES
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全纯离散级数的特征循环
DOI：
10.1090/s0002-9947-1992-1087052-3
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发表时间：
1992
影响因子：
1.3
通讯作者：
Jen
中科院分区：
文献类型：
--
作者：
Jen
研究了旗簇中闭AT -轨道产生的标准模的特征圈，并明确确定了全纯离散系列的特征圈。还计算了全纯离散系列的分布特征的渐近展开；结果在这种特殊情况下验证了D. 沃根的一个猜想。
设\(Gr\)是一个线性半单李群，\(Ar\)是\(Gr\)的一个极大紧子群。如果对称空间\(Gr/Ar\)具有一个埃尔米特结构，那么\(Gr\)有一族全纯离散系列表示。设\(I\)是任意一个全纯离散系列表示的基础哈里什 - 钱德拉模，\(\theta\)是群上相应的分布特征。\(I\)的相关圈\(AC(I)\)是在\(\mathfrak{n}^*\)（\(Gr\)的李代数复化的对偶）上代数定义的一个代数圈。另一方面，根据（巴尔巴什 - 沃根），\(\theta\)在单位元附近渐近展开的首项的傅里叶变换是\(\mathfrak{g}_{\mathbb{R}}\)中共轭\(Gr\) -轨道上典范不变测度的一个线性组合。因此它确定了\(\mathfrak{g}_{\mathbb{R}}\)上的一个圈。
在本文中，我们确定了与全纯离散系列相关的两个不变量。然后在这种特殊情况下验证了D. 沃根的一个猜想，该猜想断言在（关口）的轨道对应下，两个不变量的“主要”部分是一致的。
\(GR/AR\)在单位陪集处的复化切空间自然地等同于\(\mathfrak{g}\)的一个商空间，因此，通过不变的非退化基灵形式，全纯切空间等同于\(\mathfrak{g}^*\)中的一个子空间。这里的全纯结构是为定义\(I\)所选择的那个结构。那么相关圈\(AC(I)\)是全纯切空间，其重数是\(I\)的最低\(AR\) -型的维数（2.13）。
计算是通过\(\mathfrak{g}\)的平坦簇\(X\)上相应的\(\mathfrak{2}\) -模以及从余切丛\(T^*X\)到\(\mathfrak{g}^*\)的矩映射来进行的。一般来说，产生离散系列的\(\mathfrak{2}\) -模支撑在\(X\)上的闭\(A\) -轨道上（\(K\)是\(Ar\)的复化），法向导数然后在这些\(\mathfrak{2}\) -模上诱导一个良好的滤过。这然后将相关分次模计算为\(T^*X\)上的凝聚\(\mathfrak{c}\) -模层。尽管一般来说矩映射在相关分次模的支撑上相当复杂，但在全纯的情况下……
The characteristic cycles of standard modules arising from closed AT-orbits in a flag variety are studied and those of holomorphic discrete series are determined explicitly. Also the asymptotic expansion of the distribution characters of holomorphic discrete series are computed; the result verifies a conjecture of D. Vogan in this special case. Suppose Gr is a linear semisimple Lie group and Ar a maximal compact subgroup of Gr . In case the symmetric space Gr/Ar carries a Hermitian structure, Gr has a family of holomorphic discrete series representations. Let / be the underlying Harish-Chandra module of an arbitrary holomorphic dis- crete series representation and O be the corresponding distribution character on the group. The associated cycle 2^(7) of / is an algebraic cycle defined al- gebraically on n*, the dual to the complexification of the Lie algebra or of Gr . On the other hand, according to (Barbasch-Vogan), the Fourier transform of the leading term of the asymptotics of 8 near the identity is a linear combination of canonical invariant measures on coadjoint Gr-orbits in g.R. It therefore de- termines a cycle on gM . In this paper we determine both invariants attached to the holomorphic discrete series. It then verifies in this special case a conjecture of D. Vogan which asserts that the "major" part of both invariants coincide under the orbit correspondence of (Sekiguchi). The complexified tangent space of GR/A"R at the identity coset is naturally identified to a quotient of q , therefore, via the invariant nondegenerate Killing form, the holomorphic tangent space is identified to a subspace in g*. Here the holomorphic structure is the one chosen to define /. Then the associated cycle 2^(7) is the holomorphic tangent space with multiplicity the dimension of the lowest A"R-type of I (2.13). The computation is made through the corre- sponding 2-module on the flat variety X of 9 and the moment map from the cotangent bundle T*X to g*. Generally ^-modules giving rise to the discrete series are supported on closed A"-orbits on X (K is the complexification of Ar), the normal derivatives then induce a good filtration on these -^-modules. This then computes the associated graded modules as sheaves of coherent cf- modules on T*X. Although in general the moment map is quite complicated on the support of the associated graded module, in the situation where holo-
某些p基团诱导特征的不可约性
DOI：
--
发表时间：
2004
期刊：
Transactions of Kokushikan University Faculty Engineering 37
影响因子：
--
作者：
中島 晴久;中島 晴久;石橋 宏行;関口 勝右;Nakajima Haruhisa;Ishibashi Hiroyuki;Sekiguchi Katsusuke
通讯作者：
Sekiguchi Katsusuke