复Kac-Moody群的离散逼近
Discrete approximations for complex Kac-Moody groups
摘要：
构造了p元域的代数闭包上的离散Kac-Moody群的分类空间到复拓扑Kac-Moody群的分类空间的映射，并证明了它在素数q处是同调等价的，推广了Quillen-Friedlander-Mislin关于李群的经典结果.作为应用，我们构造了一般Kac-Moody群的不稳定Adams算子，该算子与Frobenius同态相容.与Lie情形相反，这些不稳定Adams运算的同伦不动点不能在q处实现为有限域上Kac-Moody群的分类空间.我们的结果依赖于新的积分同调分解的某些无限维幂幺子群的离散Kac-Moody群。
英文摘要：
We construct a map from the classifying space of a discrete Kac-Moody group over the algebraic closure of the field with p elements to the classifying space of a complex topological Kac-Moody group and prove that it is a homology equivalence at primes q different from p. This generalises a classical result of Quillen-Friedlander-Mislin for Lie groups. As an application, we construct unstable Adams operations for general Kac-Moody groups compatible with the Frobenius homomorphism. In contrast to the Lie case, the homotopy fixed points of these unstable Adams operations cannot be realized at q as the classifying spaces of Kac-Moody groups over finite fields. Our results rely on new integral homology decompositions for certain infinite dimensional unipotent subgroups of discrete Kac-Moody groups.
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