Let $\mathfrak{g}$ be a semisimple lie algebra over a field of characteristic $0$, $G$ its adjoint group, with Borel group $B$. I am trying to understand the theory of algebraic $(g,B)$ modules as defined by Borho and Brylinski in https://link.springer.com/content/pdf/10.1007%2FBF01388547.pdf. More precisely, given an object in BGG category $O$ with trivial central character I would like to define an action of the borel group $B$ that respect the axioms in 2.1.

Borho and Brylinski state that the Verma modules $M(w(-\rho)-\rho)$ and their unique simple quotients $L(w(-\rho)-\rho)$ for any $w \in W$ are in this category and refer to Dixmier's Book, Enveloping Algebra. Another reference I tried to look at is Bernstein-Gelfand: Tensor Products of finite and infinite dimensional representations of lie algebras, but in neither of this place the action of the Borel group is defined.

Another idea was to think of these $U(\mathfrak{g})$ modules as modules over the distribution algebra Dist$(G)$ and use the correspondence between Dist($G$)-modules and $G$-modules, but I do not think that if restricted to B the axioms will be satisfied.

Thanks for any ideas or references.

  • $\begingroup$ Crossposted at MSE. $\endgroup$ – Dietrich Burde Jul 3 '17 at 13:09

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