**S. Eswara Rao**

TIFR, Mumbai

March 9, 2017

**Generalized Casimir Operators**:
Let gg be symmetrizable Kac-Moody Lie algebra and let Omega be Casimir operator. Let A be commutative associative finitely generated algebra with unit. In this talk we consider highest weight modules for gg otimes A. We define operators Omega (a,b), a, b in A (Omega (1,1)=Omega) which operate on gg otimes A module and commutes with gg action. We will specialize on tensor product modules (for gg) which can be given a gg otimes CC[t,t^{-1}] module structure.
We will use similar ideas for gl_{N} otimes A and construct more
operators which commute with gl_N. It is well known that the center of
U(gl_N) is finitely generated as an algebra. Around 1950 Gelfand defined central elements (called Gelfand invariants) T_k for each positive integer k. It is known that first N generates the center. The decomposition of tensor product module for reductive Lie algebra is a classical problem. It is known that each Gelfand invariant acts as scalar on an irreducible submodule of a tensor product modul