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Also, let me add something about the role of the classification of unitary representations of $GL_n$, which will help illuminate the structure of the lower part of the landscape: First, one uses the general result that geometrically obtained Galois representations are mixed (i.e. have Frobenius eigenvalues that are Weil numbers), coupled with the local-global compatibility without the $N$, to deduce that the Frobenius eigenvalues of the Weil--Deligne representation attached to the local factor of $\Pi$ at $p$ are Weil numbers. Second, results of Tadic on the classification of unitary representations of $GL_n$ of $p$-adic fields greatly restrict the structure of this local factor (since it is both unitary and generic, being the local factor of a cuspidal automorphic representation for $GL_n$). Finally, when this is combined with the fact that the Weil--Deligne rep'n associated to it via local Langlands is mixed, ones sees that this local factor is forced to be tempered. This is how Harris--Taylor deduce temperedness; it is an analogue at $p$ of an argument at the archimedean prime made by Clozel in his Ann Arbor paper.

Also, let me add something about the role of the classification of unitary representations of $GL_n$, which will help illuminate the structure of the lower part of the landscape: First, one uses the general result that geometrically obtained Galois representations are mixed (i.e. have Frobenius eigenvalues that are Weil numbers), coupled with the local-global compatibility without the $N$, to deduce that the Frobenius eigenvalues of the Weil--Deligne representation attached to the local factor of $\Pi$ at $p$ are Weil numbers. Second, results of Tadic on the classification of unitary representations of $GL_n$ of $p$-adic fields greatly restrict the structure of this local factor (since it is both unitary and generic, being the local factor of a cuspidal automorphic representation for $GL_n$). Finally, when this is combined with the fact that the Weil--Deligne rep'n associated to it via local Langlands is mixed, ones sees that this local factor is forced to be tempered. This is how Harris--Taylor deduce temperedness; it is an analogue at $p$ of an argument at the archimedean prime made by Clozel in his Ann Arbor paper. (See Lemma 4.9 on p.144 of volume I of the Ann Arbor conference. A scan is available on Jim Milne's web-page here, but note that it is quite a big file.)

Added: First, I should not that the references I make are to the version of Taylor--Yoshida that is currently posted on Taylor's web-page, which I am told may differ quite substantially in its organization from the published version of the paper.

Added: First, I should note that the references I make are to the version of Taylor--Yoshida that is currently posted on Taylor's web-page, which I am told may differ quite substantially in its organization from the published version of the paper.

Added: First, I should not that the references I make are to the version of Taylor--Yoshida that is currently posted on Taylor's web-page, which I am told may differ quite substantially in its organization from the published version of the paper.

Also, let me add something about the role of the classification of unitary representations of $GL_n$, which will help illuminate the structure of the lower part of the landscape: First, one uses the general result that geometrically obtained Galois representations are mixed (i.e. have Frobenius eigenvalues that are Weil numbers), coupled with the local-global compatibility without the $N$, to deduce that the Frobenius eigenvalues of the Weil--Deligne representation attached to the local factor of $\Pi$ at $p$ are Weil numbers. Second, results of Tadic on the classification of unitary representations of $GL_n$ of $p$-adic fields greatly restrict the structure of this local factor (since it is both unitary and generic, being the local factor of a cuspidal automorphic representation for $GL_n$). Finally, when this is combined with the fact that the Weil--Deligne rep'n associated to it via local Langlands is mixed, ones sees that this local factor is forced to be tempered. This is how Harris--Taylor deduce temperedness; it is an analogue at $p$ of an argument at the archimedean prime made by Clozel in his Ann Arbor paper.

Finally, let me now explain how to correctly read the landscape: one starts in the lower left, following the arrow. When you come to a bridge, you cross the river from the motivic/Galois side to the automorphic side, or back again, continuing to follow the arrows. The areas marked with x's are impenetrable (or, at least, you don't try to cross them directly --- e.g. you don't prove GRC directly, you don't prove WMC directly, you don't study the general semi-stable reduction problem directly); the only way to proceed is by crossing the river. This then gives the structure of the Taylor--Yoshida argument.