Date:	Thu, February 12, 2015
Time:	11:15
Place:	Seminar Room (120), Research I

Abstract: Let \(\mathfrak{g}\) be a finite-dimensional semisimple Lie algebra over the field of complex numbers with a Borel subalgebra \(\mathfrak{b}\) and a Cartan subalgebra \(\mathfrak{h}\) such that \(\mathfrak{b}\supseteq\mathfrak{h}\). Denoted by \(\mathcal{O}\) is the BGG category \(\mathcal{O}\) associated to \(\mathfrak{g}\) with respect to \((\mathfrak{b},\mathfrak{h})\). We know that a central character \(\chi\) of \(\mathfrak{U}(\mathfrak{g})\) is given by a weight \(\lambda \in \mathfrak{h}^*\) and is written as \(\chi=\chi_\lambda\). Moreover, the blocks of \(\mathcal{O}\) are of the form \(\mathcal{O}_{\lambda}:=\mathcal{O}_{\chi_\lambda}\), where \(\lambda\in \mathfrak{h}^*\).
Kazhdan-Lusztig Theory allows us to understand the structure of each block \(\mathcal{O}_\lambda\) of \(\mathcal{O}\), where \(\lambda \in \mathfrak{h}^*\) is a dominant integral weight. Information such as the composition factor multiplicities of a Verma module and the dimensions of the \(\mathrm{Ext}\)-groups can be computed from Kazhdan-Lusztig polynomials. We shall also discuss the importance of the Kazhdan-Lusztig Conjecture.