REPRESENTATION GROWTH OF COMPACT LINEAR GROUPS
REPRESENTATION GROWTH OF COMPACT LINEAR GROUPS
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DOI：
10.1090/tran/7618
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发表时间：
2019-07-15
影响因子：
1.3
通讯作者：
Stasinski, Alexander
中科院分区：
文献类型：
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作者：
Hasa, Jokke;Stasinski, Alexander
我们研究了简单紧李群和SLn（O）的表示增长，其中O是紧离散赋值环，以及GL（n）（O）的扭曲表示增长。这相当于一个研究的横坐标的收敛相应的（扭曲）表示zeta函数。我们确定了包括维滕zeta函数在内的一类Mellin zeta函数的横坐标。作为一种特殊情况，我们得到了Larsen和Lubotzky定理的一个新的证明，即维滕zeta函数的横坐标是r/kappa，其中r是秩和.正根的个数。然后我们证明了GL（n）（O）的扭曲zeta函数的存在性，并且当n不整除charO时，它与SLn（O）的zeta函数具有相同的收敛横坐标.我们计算了GL（2）（O）在剩余特征p为奇数时的扭曲zeta函数，并在p = 2时对zeta函数进行了近似，得出横坐标为1.最后，我们构造了SL ~ 2（F-q[[t]]），q偶的大部分表示，并推导出其横坐标位于区间[1，5/2]。
We study the representation growth of simple compact Lie groups and of SLn(O), where O is a compact discrete valuation ring, as well as the twist representation growth of GL(n)(O). This amounts to a study of the abscissae of convergence of the corresponding (twist) representation zeta functions. We determine the abscissae for a class of Mellin zeta functions which include the Witten zeta functions. As a special case, we obtain a new proof of the theorem of Larsen and Lubotzky that the abscissa of Witten zeta functions is r/kappa, where r is the rank and. the number of positive roots. We then show that the twist zeta function of GL(n)(O) exists and has the same abscissa of convergence as the zeta function of SLn(O), provided n does not divide charO. We compute the twist zeta function of GL(2)(O) when the residue characteristic p of O is odd and approximate the zeta function when p = 2 to deduce that the abscissa is 1. Finally, we construct a large part of the representations of SL2(F-q[[t]]), q even, and deduce that its abscissa lies in the interval [1, 5/2].