**Shrawan Kumar**

University of North Carolina at Chapel Hill, USA

May 3, 2012

**Equivariant $K$-theory of flag varieties**:
We will discuss the $T$-equivariant $K$-theory of flag varieties
$G/B$, where $G$ is a semisimple complex algebraic group, $B$ is a Borel subgroup and $T$ is a maximal torus in $B$. The equivariant $K$-theory of $G/B$ comes equipped with two natural bases: one coming from the structure sheaves of the Schubert varieties and the other its `dual' basis. We will prove some positivity phenomenon in the $T$-equivariant $K$-theory of $G/B$ for the product structure constants in either of the above two bases. We will also discuss a generalization of these results to the flag varieties of Kac-Moody groups.