Representations of The Weil Group
Representations of the Weil group do not quite correspond to irreducible smooth representations of general linear groups. To get a bijection, one has to slightly modify the notion of a representation of the Weil group, to something called a Weil–Deligne representation. This consists of a representation of the Weil group on a vector space V together with a nilpotent endomorphism N of V such that wNw−1=||w||N, or equivalently a representation of the Weil–Deligne group. In addition the representation of the Weil group should have an open kernel, and should be (Frobenius) semisimple.
For every Frobenius semisimple complex n-dimensional Weil–Deligne representations ρ of the Weil group of F there is an L-function L(s,ρ) and a local ε-factor ε(s,ρ,ψ) (depending on a character ψ of F).
Read more about this topic: Local Langlands Conjectures
Famous quotes containing the words representations of the, representations of, weil and/or group:
“These marbles, the works of the dreamers and idealists of old, live on, leading and pointing to good. They are the works of visionaries and dreamers, but they are realizations of soul, the representations of the ideal. They are grand, beautiful, and true, and they speak with a voice that echoes through the ages. Governments have changed; empires have fallen; nations have passed away; but these mute marbles remain—the oracles of time, the perfection of art.”
—Herman Melville (1819–1891)
“These marbles, the works of the dreamers and idealists of old, live on, leading and pointing to good. They are the works of visionaries and dreamers, but they are realizations of soul, the representations of the ideal. They are grand, beautiful, and true, and they speak with a voice that echoes through the ages. Governments have changed; empires have fallen; nations have passed away; but these mute marbles remain—the oracles of time, the perfection of art.”
—Herman Melville (1819–1891)
“The most important part of teaching = to teach what it is to know.”
—Simone Weil (1909–1943)
“Remember that the peer group is important to young adolescents, and there’s nothing wrong with that. Parents are often just as important, however. Don’t give up on the idea that you can make a difference.”
—The Lions Clubs International and the Quest Nation. The Surprising Years, I, ch.5 (1985)