Abstract
For a cyclic group A and a connected Lie group G with an A-module structure (with the additional assumptions that G is compact and the A-module structure on G is 1-semisimple if \(A\cong{\mathbb{Z}}\)), we define the twisted Weyl group W = W(G,A,T), which acts on T and H 1(A,T), where T is a maximal compact torus of \(G_0^A\) , the identity component of the group of invariants G A. We then prove that the natural map \(W\backslash H^1(A,T)\rightarrow H^1(A,G)\) is a bijection, reducing the calculation of H 1(A,G) to the calculation of the action of W on T. We also prove some properties of the twisted Weyl group W, one of which is that W is a finite group. A new proof of a known result concerning the ranks of groups of invariants with respect to automorphisms of a compact Lie group is also given.
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An, J. Twisted Weyl groups of Lie groups and nonabelian cohomology. Geom Dedicata 128, 167–176 (2007). https://doi.org/10.1007/s10711-007-9188-y
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DOI: https://doi.org/10.1007/s10711-007-9188-y