#### Alladi Ramakrishnan Hall

#### Kostant module and KPRV Refinement

#### Mrigendra Singh Kushwaha

##### IMSc

*Peter Littelmann has given the notion of Lakkshmibai - Seshadri paths $B_\lamda$ for given a dominat weight $\lamda$ of a symmetrizable Kac-Moody Algebra, which carries more information than (like Young Tableaux) just the character of the representation. I will define Kostant module which is submodule of the tensor product of two irreducible modules. We give filtration of concatenation of sets $B_\lamda$ and $B_\mu$ by the double cosets $W_\lamda \W/W_\mu$, and we also give filtration of Kostant modules by the double cosets $W_\lambda\W/W_\mu$, where $W$ is Weyl group and $\lambda, \mu$ are dominant weights. Also we will give decomposition rule for kostant modules by paths for finite dimensional semi-simple lie algebra. Next we see that set of paths corresponding to fix double coset is path model for Kostant module corresponding to the same double coset. And lastly we give lower bound of existance of certain irreducible modules in Kostant modules, which is generalisation of PRV refinement, and also we see existance of generalised PRV component in Kostant modules. This is joint work with Prof K N Raghavan and Prof S Viswanath.*

Done