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Nov 18, 2013 at 11:01 comment added Alexander Braverman Let me make one comment about it. Although, the construction is indeed as described above, there is one subtle thing about it. Namely, on the representation theory side, the restriction functor depends only on $L$ but not a choice of a parabolic $P$. However, geometrically the functor depends on the parabolic very heavily (note that you can have non-conjugate parabolics with the same Levi). I don't know any geometric way (without using Satake equivalence) to prove independence of $P$ - this is actually a very good problem (but probably it has no reasonable solution).
Sep 24, 2013 at 13:13 vote accept Oliver Straser
Sep 23, 2013 at 15:08 history answered user40391 CC BY-SA 3.0