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On the last page of Schmid's article "Discrete Series", he says

"In the Beilinson-Bernstein picture, discrete series modules are attached to closed $K$-orbits in $X$... the $K_{\mathbb R}$-structure of discrete series modules [i.e. Blattner's conjecture, which Schmid proved] is almost obvious from this point of view."

I do indeed see how to prove Blattner's conjecture from this point of view, though there is enough battle with $\rho$-shifts that I didn't find it "almost obvious". (Added: I posted this proof on the arXiv.)

Is there a place where Blattner's conjecture is derived in this algebraic setting of $(\mathfrak g,K)$-modules, rather than Schmid's original analytic setting?

(I did write Schmid, though I suspect if he knew of such a reference he would have referenced it. Of course one may have come out in the intervening time, and if he has an answer I will include it here.)

ADDED: Schmid's derivation sounded (on the phone) pretty isomorphic to mine, except where I write out a page (being an outsider to the field, and not clear on which parts are considered easy and which hard) he often puts just a sentence. Nothing wrong with that, of course, if you know your audience and know what will suffice for them. Anyway that doesn't answer my reference-request.

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  • $\begingroup$ Does Havlíčková's dissertation contain the sort of result you want? dspace.mit.edu/handle/1721.1/43796 $\endgroup$
    – S. Carnahan ♦
    Commented Sep 25, 2014 at 14:23
  • $\begingroup$ She references the Blattner formula, but doesn't derive it, as far as I could tell. $\endgroup$
    – Allen Knutson
    Commented Sep 29, 2014 at 20:26

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"Geometric Methods in Representation Theory", by Gregg Zuckerman p.283, in 'Representation Theory of Reductive Groups' ed. Peter C Trombi, Progress in Math. 290 Birkhauser(1983) Procedings of University of Utah conference may have what you are looking for.

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  • $\begingroup$ It's definitely got some of the ingredients, and in particular, draws attention to the importance of K-orbits with smooth closure. It doesn't reference Blattner directly though. Thanks! $\endgroup$
    – Allen Knutson
    Commented Mar 21, 2015 at 15:45

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