10
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ Does Havlíčková's dissertation contain the sort of result you want? dspace.mit.edu/handle/1721.1/43796 $\endgroup$ – S. Carnahan Sep 25 '14 at 14:23
  • $\begingroup$ She references the Blattner formula, but doesn't derive it, as far as I could tell. $\endgroup$ – Allen Knutson Sep 29 '14 at 20:26
3
$\begingroup$

"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.

$\endgroup$
  • $\begingroup$ 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 Mar 21 '15 at 15:45

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.