The Extension groups of one $\rm K$-type representations of graded Hecke algebra

Abstract

We study the extension groups between one $\rm K$-type representations of the graded Hecke algebra $\mathbb H$. A one $\rm K$-type representation is a Hermitian representation of $\mbb H$ whose restriction to the Weyl group is irreducible. Using a Koszul-type resolution of $\mathbb H$-modules, we reduce the computation of $\operatorname{Ext}_{\mathbb H}^j(X,Y)$ to a combinatorial count, yielding explicit formulas for the classical types and a computer computation for the exceptional types.