Dear Thomas,

This "landscape" is, I think, a sketch of the proof of the following theorem of Taylor and Yoshida:  if $\Pi$ is a self-dual cuspdidal automorphic form on $GL_n$ over $E$ (a CM field)
(satisfying some further technical conditions) and $\rho$ is the associated $n$-dimensional
Galois representation (constructed by Harris and Taylor in their book "The geometry and cohomology of some simple Shimura varieties"), then the local factors of $\Pi$ at any prime
$p$ of $E$ matches, via local Langlands, with the restriction of $\rho$ to a decomposition group at $p$.

What was proved in Harris--Taylor was that this matching is correct, up to the question
of matching the $N$ on each side.  What Taylor and Yoshida verified is that the $N$ on each side
also matches.  

The top part of the landscape represents the reduction to the unipotent situation (i.e. the context in which $\Pi$, locally at $p$, has an Iwahori fixed vector --- the analogue
of classical level $\Gamma_0(p)$), by
base-changing to an extension of $E$.  On the automorphic side, this is the process
labelled by the names Arthur--Clozel--Selberg; on the Galois side, this is just restricting
from $Gal(\overline{E}/E)$ to $Gal(\overline{E}/E')$ for some appropriately chosen
extension $E'$ of $E$.  If one thinks of $\rho$ as appearing in the cohomology of a Shimura
variety (which is, after all, how it is constructed), then this just corresponds to
base-changing the variety from $E$ to $E'$, which is why this process is labelled as
geometric base change in the "landscape".


On the lower right of the diagram, one has the generalized Ramanujan conjecture, stating that the local factor of $\Pi$ at $p$ should be tempered,
which in turn implies that the local (at $p$) Galois representation (more precisely,
Weil--Deligne representation) attached to this local factor satisfies the monodromy
weight conjecture.   This temperedness was proved by Harris--Taylor.

Now we already know that $\rho$ locally at $p$ matches with the local factor of $\Pi$ at
$p$ up to $N$, and it is not hard to show that there is a unique way to add $N$ so as
to obtain a local Weil--Deligne representation satisfying the monodromy weight conjecture.
So to prove their theorem, Taylor and Yoshida are reduced to proving that $\rho$, locally
at $p$, satisfies the monodromy weight conjecture.  
They do this via an application of the Rappoport--Zink spectral sequence and a careful analysis of the bad reduction of the Shimura variety in whose cohomology $\rho$ lives.  This is represented in the lower left part of the diagram. 

The very bottom part of the diagram on the left represents the fact that a priori one knows
that $\rho$ is mixed, i.e. its semi-simplification has Frobenius eigenvalues that are
Weil numbers of various weights.    But the main part of the landscape here is the monodromy weight conjecture, which describes the precise relationship between the Frobenius eigenvalues and the $N$ operator.


To prove the MWC in their context, Taylor and Yoshida
use the interplay between the geometry of the
special fibre of the Shimura variety and the representation theory of $\Pi$ that is the main subject of Harris and Taylor's book, as well as Section 2 of Taylor and Yoshida's article.  In particular, the fact that $\Pi$, locally at $p$, is tempered unitary, and is assumed to have an Iwahori fixed vector, puts strong restrictions on its structure (using the known classification of unitary reps. of $GL_n$ over local fields; this is why Tadic's name appears on the right hand side of the landscape), which when fed over to
the geometric side implies that the RZ spectral sequence degenerates at $E_1$, giving the desired monodromy weight conjecture.

My suggestion is that if you want to understand this in more detail, you should read [Taylor and Yoshida's article](http://www.math.harvard.edu/~rtaylor).  Start at the very end of the paper, where the main theorem is proved (i.e. the paragraph beginning "We can now conclude ... ").  The lemmas referred to in section 1 are more or less elementary.  Theorem 3.2 is the heart of the argument; it is where degeneration of the RZ spectral sequence is proved.  One can try to read it more or less formally, taking various assertions as a black box,
at least to see what role the temperedness of $\Pi$ locally at $p$ plays.  The earlier parts of Section 3 are just establishing notation, and making contact with the book of Harris--Taylor.   Section 2 is devoted to establishing the basic properties of the semi-stable models of the Shimura varieties in whose cohomology $\rho$ lives; I recommend treating it as a black box on first reading.