Hecke Operators and the Stable Homology of GL(n)
Hecke Operators and the Stable Homology of GL(n)
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Hecke 算子和 GL(n) 的稳定同调
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发表时间：
2012
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通讯作者：
A. Ash
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作者：
A. Ash
设R是一个具有任意特征的域，a是一个主理想定义域。我们推测断言任何Hecke算子T徒准时Hecke eigenclass稳定的同源性与琐碎的系数R的主同余组ΓGL (n),也就是说,乘法的单一叠合组包含的数量在T =ℤ,这种猜想意味着之和组成的伽罗瓦表示第0、1日⋅n−1分圆的字符是附加到任何的权力稳定Hecke eigenclass。当A = 0和Γ =GL（n, 0）时，这个猜想已经由Calegari和Venkatesh提出。我们得到了证明这个猜想的部分结果。这些结果暗示Hecke代数的某些部分是准时作用的。如果R的特征为0，他们表明整个Hecke代数是准时性的，这已经用a . Borel的结果以完全不同的方式知道了。
Let R be a field of any characteristic and A a principal ideal domain. We make a conjecture that asserts that any Hecke operator T acts punctually on any Hecke eigenclass in the stable homology with trivial coefficients R of a principal congruence subgroup Γ in GL(n, A), i.e., as multiplication by the number of single cosets contained in T. In the case where A = ℤ, this conjecture implies that the Galois representation consisting of the sum of the 0th, 1st, ⋅, n − 1st powers of the cyclotomic character is attached to any stable Hecke eigenclass. When A = ℤ and Γ =GL(n, ℤ), this conjecture was already made by Calegari and Venkatesh. We obtain partial results giving evidence for the conjecture. These results imply that some part of the Hecke algebra acts punctually. If the characteristic of R is 0, they show that the entire Hecke algebra acts punctually, which was already known in a completely different way using a result of A. Borel.