Question

Is it known that the Galois representation constructed by Harris and Taylor in their book is semisimple? I can't see this proven in the book, but on the other hand, everywhere else the representation is taken to be semisimple... Are they considering its semisimplification?

Sorry for the simple question.

Thanks

Answer 1

Do you mean the local Galois representations or the local Galois representations ?

The global Galois representations they are constructing correspond to cuspidal automorphic representations of GL(n). They are expected to be always irreducible, though I'm not sure when this is known exactly. But it is known in the case that Harris and Taylor consider (when the automorphic representation is square integrable at a finite place), cf corollary 1.3 of the article "Compatibility of local and global Langlands correspondences" by Taylor and Yoshida.

The local Galois representations are not expected to be semi-simple in general. They are expected to be Frobenius semi-simple (ie, the Frobenius elements are supposed to act semi-simply), but this is not known for $n\geq 3$. So, if you mean the local representations, then yes, very often people are just taking the Frobenius semi-simplifications of the representations that appear in the cohomology of Shimura varieties.