BGG sequences from infinite-rank bundles

I will present a generalization of Calderbank-Diemer construction that works for bundles whose fiber is unitarizable highest weight module. These modules exists only when $(G, K)$ is Hermitian symmetric pair. The resulting BGG sequences of Verma modules are (after a twist) generalizations of minimal free resolutions of determinantal ideals. This suggests that these sequences of differential operators are in fact resolutions for interesting differential operators such as Yamabe or Dirac on $G^\mathb{C}/P$. Moreover these differential operators still obey $A_\infty$ relations as in the classical case of finite-dimensional bundles.