The combinatorial functor \(\mathbb V\): Graded category \(\mathcal O\), principal series and primitive ideals (Q2763521)

This thesis is motivated by the problem for the description of the composition factors of primitive quotients of the universal enveloping algebra \({\mathcal U}({\mathfrak g})\) of the semisimple complex Lie algebra \(\mathfrak g\) modulo the annihilators of simple \(\mathfrak g\)-modules. The main tool in the approach of the author is the combinatorial functor \(\mathbb V\) from a regular block of the category \(\mathcal O\) to the finite dimensional modules over the cohomology ring of the flag manifold associated with \(\mathfrak g\) [see \textit{W. Soergel}, J. Am. Math. Soc. 3, 421-445 (1990; Zbl 0747.17008)]. NEWLINENEWLINENEWLINEThe functor \(\mathbb V\) gives a structure of grading on \(\mathcal O\). The author asks the question of which modules can be lifted to graded ones and proves that this is possible for all projective, simple and standard objects. She also finds relations of the Grothendieck group of the graded objects with the Hecke algebra. The author shows that the principal series can be graded, which allows her to transfer the information for the grading to the composition factors of the primitive ideals. In particular, she generalizes the Kazhdan-Lusztig algorithm. Finally, she states in the graded context a conjecture implicitly stated by Joseph. As an illustration of her technique, the author is able to give new proofs for the descriptions of the primitive quotients for the root systems of rank 2 and of type \(A_3\).