# Derivation of Blattner's conjecture in the Beilinson-Bernstein picture

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.

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