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Please forgive the provocative title, what I mean is the following:

One can find representations of Lie algebras in geometric settings, the most famous being the Bott–Borel–Weil theory. However, besides this being intersting in its own right, I am not sure what it allows to say about the representations. Is it just a curiousity or is there a deeper reason why it is a "good thing" to have geometric realisations of the irreps of $\frak{g}$?

LSpice
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Béla Fürdőház
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    Take a look at this previous MO answer by David Ben-Zvi: https://mathoverflow.net/questions/126474/new-geometric-methods-in-number-theory-and-automorphic-forms/126540#126540 – Sam Hopkins Nov 11 '23 at 14:24
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    It's G-equivariant algebraic geometry, so the 'point' of geometric representation theory is $[pt/G]$. </dad joke> – user1504 Nov 11 '23 at 20:49
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    Look up papers about orbit method of A. A. Kirillov to see how exactly geometric representation theory helps to solve algebraic questions. // Another reason: BBW theorem is only true over characteristic zero alg. closed field. There are algebraic groups over arbitrary base scheme which are often interesting to study. One can say that Margulis hyperrigidity is (in some sense) a result of geometric rep theory. – Denis T Nov 13 '23 at 00:38

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