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I think I’ve once been told that under the Beilinson-Bernstein correspondence, finite-dimensional representations of a semisimple Lie algebra $\mathfrak{g}$ correspond to (twisted) D-modules on $G/B$ whose associated variety in $T^*G/B$ is the zero-section. Similarly, category O modules correspond to (twisted) D-modules microsupported on the union of conormal bundles to Schubert cells.

Can anyone point me to a reference for this?

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  • $\begingroup$ The standard reference book is: R. Hotta, K. Takeuchi, T. Tanisaki, D-modules, perverse sheaves, and representation theory. Translated from the 1995 Japanese edition by Takeuchi. Progress in Mathematics, 236. Birkhäuser Boston, Inc., Boston, MA, 2008. My AMS graduate text published also in 2008 gives mainly the algebraic preliminaries for category $\mathcal{O}$, followed by a detailed outline of the Beilinson-Bernstein arguments (without the details). $\endgroup$
    – Jim Humphreys
    Commented Sep 12, 2014 at 12:46
  • $\begingroup$ Thanks - I’m somewhat familiar with both books and didn’t have the impression these results were explicitly mentioned in either of them, but I’ll have another look. $\endgroup$
    – Frano Aleksi
    Commented Sep 12, 2014 at 13:10
  • $\begingroup$ It's true that the finite dimensional representations are not explicitly covered in the geometric setting, since they are well-studied by classical methods and probably not better understood by B-B localization. So you need to read between the lines. Even the literature on associated varieties may not be of direct help here. $\endgroup$
    – Jim Humphreys
    Commented Sep 12, 2014 at 14:58
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    $\begingroup$ Having associated variety the zero section is the same thing as being a vector bundle with flat connection. Now $\pi_1(G/B)$ is trivial, and so there are only trivial bundles whose global sections give direct sums of the trivial rep. To get all reps (and recover Borel-Weil) one instead should consider twisted $D$-modules. One possible ref for all of this is Gaitsgory's notes on geometric rep theory (on his web-page I think). $\endgroup$
    – Geordie Williamson
    Commented Sep 12, 2014 at 15:57
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    $\begingroup$ Yes, as Geordie points out the initial arguments only yield information about the principal block of $\mathcal{O}$, so twisted D-modules are needed as well. You can find Gaitsgory's 2005/2010 notes and many links including unfinished Beilinson-Drinfeld book, at math.harvard.edu/~gaitsgde/grad_2009 (but probably without getting much new insight into finite dimensional representations) $\endgroup$
    – Jim Humphreys
    Commented Sep 12, 2014 at 16:40

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