Derived Functors and Intertwining Operators for Principal Series Representations of $SL_2({\pmb {\mathbb {R}}})$

Abstract

We consider the principal series representations $I_\nu$ induced from a character $\nu$ of the upper triangular matrices B and its realization on the Frechet space of $C^\infty$-sections of a line bundle over G / B. Its continuous dual is denoted by $I_\nu ^*$. Let $N \subset B$ be the nilpotent subgroup whose diagonal entries are 1 and denote by ${\mathfrak n }$ its Lie algebra. We determine $H^0({\mathfrak n }, I_\nu ^*)$ and $H^1({\mathfrak n },I_\nu ^*)$ and conclude that space of the intertwining operators $T:I_\nu \rightarrow I_{-\nu }$ is 2 dimensional for some integral parameter, otherwise it is one dimensional. The intertwining operators are identified with distributions. We show that for certain parameters the support of this distribution is a point, i.e. that the intertwining operator is a differential intertwining operator.