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In arithmetic, Galois representations are one of the fundamental objects
of interest and they arise quite naturally in several places. The Galois
representations coming from cuspidal automorphic forms on GL_n(A_Q) are
expected to be irreducible as representations of the absolute Galois group
of Q. However, the local representations, obtained by restricting to a
decomposition subgroup, can be reducible.
In this talk, we will show how a generalized notion of ordinariness for
automorphic representations implies the reducibility of such local
representations. We also show that non-ordinariness implies irreducibility
in certain cases. If time permits, we will also discuss the semisimplicity
of local Galois representations attached to ordinary cuspidal eigenforms,
following the approach of Ghate-Vatsal for odd primes, for n=2 and p=2.
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