Determinants of Intertwining Operators between Genuine Principal Series Representations of Nonlinear Real Split Groups
Determinants of Intertwining Operators between Genuine Principal Series Representations of Nonlinear Real Split Groups
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非线性实分裂群真主级数表示间交织算子的行列式
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发表时间：
2012
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通讯作者：
Seung Won Lee
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作者：
Seung Won Lee
作者：Lee，Seung Won|摘要：利用Clifford代数完成了单Lie型的连通单连通分裂真实的形式的dim g 1的所有小K型的列表，得到了C_n型以外的单Lie型的连通单连通分裂真实的形式的“小K型”的分类.对含有小K型的K型主级数V_[xi]，定义了Kostant P[gamma]矩阵的一个类似P[xi]，并证明了P[xi]在对应于约化限制根的秩1子群上的行列式的乘积公式.乘积公式和P[xi]与交织算子在真主级数表示之间的关系给出了计算Vogan和Wallach推广的Leslie Cohn行列式公式的移位因子的方法，该公式用于将交织算子限制到以经典Gamma函数的比给出的K-同构分量。得到了宽波浪号[SL（n，R）]（ng 3）的真主级数表示之间的交织算子的行列式作为经典Gamma函数的比
Author(s): Lee, Seung Won | Abstract: Classification of "small K-types" for the connected, simply connected split real form of simple Lie type other than type C_n is obtained via Clifford algebras which completes the list of all small K-types of dim g 1 for the connected, simply connected split real form of simple Lie types. An analog, P[xi], of Kostant's P[gamma] matrix is defined for a K-type V_[xi] of principal series admitting a small K-type, and a product formula of the determinant of P[xi] over the rank one subgroups corresponding to the reduced restricted roots is proved. The product formula and the relationship between P[xi] and intertwining operator between the genuine principal series representations give a method to compute the shift factors of Vogan and Wallach's generalization of Leslie Cohn's determinant formula for the restriction of the intertwining operator to a K-isotypic component given in terms of ratios of classical gamma functions. The determinant of the intertwining operator between the genuine principal series representations of widetilde [SL(n,R)] (n g 3) is obtained as a ratio of classical gamma functions