On the degenerate principal series of $G_{2(2)}$ induced from a Heisenberg parabolic subgroup

TL;DR

This paper analyzes the structure of degenerate principal series representations of the split real group G_{2(2)}, identifying reducibility points, complementary series, and special subrepresentations like minimal and discrete series.

Contribution

It provides a detailed Lie algebra analysis of these representations, including reducibility, complementary series, and the identification of minimal and discrete series as kernels of intertwining operators.

Findings

Identified points of reducibility and complementary series.

Characterized the minimal representation and limit of discrete series.

Showed occurrence of quaternionic discrete series as subrepresentations.

Abstract

We study degenerate principal series representations of the split real group $G_{2(2)}$ induced from a character of a maximal parabolic subgroup whose unipotent radical is a Heisenberg group. Using the Lie algebra action on the space of $K$-finite vectors, we find the points of reducibility and the complementary series. The minimal representation and a limit of discrete series are identified as kernel of the corresponding Knapp-Stein intertwining operator. Moreover, we show that some quaternionic discrete series representations occur as the subrepresentation on which the family of intertwining operators vanishes of order two.