$(\mathfrak{g},K)$-modules and parabolic category $\mathcal{O}$ - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T13:15:05Z http://mathoverflow.net/feeds/question/84980 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/84980/mathfrakg-k-modules-and-parabolic-category-mathcalo $(\mathfrak{g},K)$-modules and parabolic category $\mathcal{O}$ robot 2012-01-05T18:02:54Z 2012-01-05T18:02:54Z <p>I am trying to get acquainted with various infinite dimensional representations of Lie groups. So a general reference would be appreciated. Right now I am trying to figure out the following question.</p> <p>Consider a real Lie group $G$ and its maximal compact subgroup $K$. Let $P$ be a parabolic subgroup of the complexification of $G$ such that its Levi factor is a complexification of lie algebra of $K$. What is the relation between category of $(\mathfrak{g},K)$-modules and the parabolic category $\mathcal{O}_P$?</p> <p>Is there a canonical way to extend the representation of $K$ on the $(\mathfrak{g},K)$-module to a representation of $P$ at least in some special cases (for example in hermitian symmetric setting)?</p> <p>Of course one can drop the condition on $K$ being the maximal compact subgroup and ask basically the same thing.</p>