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

TL;DR

The paper analyzes the degenerate principal series for the split real group $G_{2(2)}$ induced from a Heisenberg parabolic, identifying reducibility points and unitarity via an explicit Lie algebra action on $K$-finite vectors. A key methodological advancement is the explicit use of Rankin–Cohen brackets to decompose tensor products and to construct a basis of multiplicity spaces in which the Knapp–Stein intertwiners act in an upper-triangular form, yielding closed-form eigenvalues. The work reveals a detailed landscape: the complementary series $(1,2)$, the minimal ladder and double ladder subrepresentations as kernels of intertwiners, and realizations of quaternionic discrete series as subrepresentations or Jantzen-filtration levels of these induced modules. These results tie into broader themes in automorphic and theta correspondence contexts and provide tools for studying exceptional groups via explicit intertwiner analysis.

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.

Figures (2)

• Figure 1: The complementary series (grey) for $\operatorname{Ind}_{M_{\mathfrak{p}} A_{\mathfrak{p}} N_{\mathfrak{p}}}^G(1 \otimes \operatorname{exp}(\mu) \otimes 1)$ (left) and $\operatorname{Ind}_{M_{\mathfrak{p}} A_{\mathfrak{p}} N_{\mathfrak{p}}}^G(\xi \otimes \operatorname{exp}(\mu) \otimes 1)$ (right) and the parameters for which the complementary series of $I(\varepsilon,s)$ occurs as Langlands quotient (red)
• Figure 2: The complementary series (grey) for $\operatorname{Ind}_{M_{\mathfrak{p}} A_{\mathfrak{p}} N_{\mathfrak{p}}}^G(1 \otimes \operatorname{exp}(\mu) \otimes 1)$ (left) and $\operatorname{Ind}_{M_{\mathfrak{p}} A_{\mathfrak{p}} N_{\mathfrak{p}}}^G(\xi \otimes \operatorname{exp}(\mu) \otimes 1)$ (right) and the parameters for which the complementary series of $I(\varepsilon,s)$ occurs as irreducible subquotient (red)

Theorems & Definitions (53)

• Theorem A: see Propositions \ref{['prop:irredtriv']} and \ref{['prop:sgnirred']}
• Corollary B: see Corollary \ref{['cor:ComplementarySeries']}
• Theorem C: see Propositions \ref{['prop:ladderker']} and \ref{['prop:doubleladderker']}
• Theorem D: see Theorems \ref{['thm:LDS']} and \ref{['thm:QDS']}
• Remark 1.1
• Proposition 2.1
• proof
• Lemma 2.2
• proof
• Proposition 2.3