#### Alladi Ramakrishnan Hall

#### The Weyl Character Formula

#### G Arunkumar

##### IMSc

*Let $L$ be a semisimple Lie algebra over an algebraically closed field $F$ and let $H$ be a fixed Cartan subalgebra of $L$. For each $\lambda \in H^*$ there exists an irreducible standard cyclic module $V(\lambda)$ of weight $\lambda$ and if $\lambda$ is dominant then $V(\lambda)$ is finite dimensional. By using the Weyl Character Formula, in such a finite dimensional $L$-module $V(\lambda)$ we can determine which weights $\mu$ occur and with what multiplicities. In this talk, I will define the character of an $L$-module $V$ and prove the Weyl Character Formula using a theorem of Harish-Chandra. I will explain how our formula is extremely helpful by applying it to particular $L$-modules.*

Done