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General Bruhat decomposition (with parabolic not necessarily Borel)

As @JimHumphreys has pointed out, [BT2] Borel and Tits - Compléments à l'article: «Groupes réductifs», specifically Proposition 3.16(i, iv), gives the decomposition $G/P_I = \bigsqcup_{w \in [W_I\...
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conjugacy in adjoint representation

Here is a partial answer. Let first $G=PGL(n)$. Let $g$ have eigenvalues $\lambda_1,\ldots,\lambda_n$ (up to a scalar). Then $\rho(g)$ has eigenvalues $\lambda_i\lambda_j^{-1}$ with $i,j=1,\ldots,n$. ...
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Does the "building of parabolics" of a semisimple group have a simplex corresponding to the entire group?

@UriyaFirst points out that every abstract simplicial complex must have an empty face of dimension $-1$, and it would make sense for this to correspond to the entire group. (This is consistent with ...
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Tits building of a linear algebraic group

I asked a related question Does the "building of parabolics" of a semisimple group have a simplex corresponding to the entire group? recently, at which point MO reminded me of yours. @...
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