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We will discuss $p$-local representation theory and conjectures of Broue and Alperin, the latter been inspired by finite groups of Lie type in characteristic $p$. The issue is the extent to which the category of representations can be reconstructed from that of proper parabolic subgroups. In different settings, a Borel suffices (Arkhipov-Bezrukavnikov-Ginzburg for rational representations, Colmez for $GL_2$ over $\mathbb Q_p$).

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