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Principal series of quaternionic and real split exceptional Lie groups induced from Heisenberg parabolic subgroups

Genkai Zhang

TL;DR

This work analyzes principal series representations of quaternionic rank-$4$ and split exceptional Lie groups induced from Heisenberg parabolics, realized on $L^2(K/L)$. By exploiting a circle-bundle geometry over a compact Hermitian symmetric space, the authors decompose $L^2(K/L)$ into $L^2(K/L_1;\chi^p)$ and compute the $\mathfrak g$-action via explicit recurrence relations and Casimir operators, yielding the complementary series ranges and reducibility points. They provide detailed $K$-type descriptions, give a general explicit formula for the action of the Heisenberg parabolic root vector $E$, and connect these representations to minimal representations through conformally invariant differential operators. The results advance understanding of unitarizable constituents in non-multiplicity-free induced representations and illuminate connections between principal series, minimal representations, and invariant differential operators in quaternionic and split exceptional settings.

Abstract

Let $G/K$ be an irreducible quaternionic symmetric space of rank $4$. We study the principal series representation $π_ν=\text{Ind}_P^G(1\otimes e^ν\otimes 1)$ of $G$ induced from the Heisenberg parabolic subgroup $P=MAN$ realized on $L^2(K/L)$, $L=K\cap M$. We find the $K$-types in the induced representation via a double cover $K/L_0\to K/L$ and a circle bundle $K/L_0\to K/L_1$ over a compact Hermitian symmetric space $K/L_1$. We compute the Lie algebra $\mathfrak g$-action of $G$ on the representation space. We find the complementary series, reducible points, and unitary subrepresentations in this family of representations.

Principal series of quaternionic and real split exceptional Lie groups induced from Heisenberg parabolic subgroups

TL;DR

This work analyzes principal series representations of quaternionic rank- and split exceptional Lie groups induced from Heisenberg parabolics, realized on . By exploiting a circle-bundle geometry over a compact Hermitian symmetric space, the authors decompose into and compute the -action via explicit recurrence relations and Casimir operators, yielding the complementary series ranges and reducibility points. They provide detailed -type descriptions, give a general explicit formula for the action of the Heisenberg parabolic root vector , and connect these representations to minimal representations through conformally invariant differential operators. The results advance understanding of unitarizable constituents in non-multiplicity-free induced representations and illuminate connections between principal series, minimal representations, and invariant differential operators in quaternionic and split exceptional settings.

Abstract

Let be an irreducible quaternionic symmetric space of rank . We study the principal series representation of induced from the Heisenberg parabolic subgroup realized on , . We find the -types in the induced representation via a double cover and a circle bundle over a compact Hermitian symmetric space . We compute the Lie algebra -action of on the representation space. We find the complementary series, reducible points, and unitary subrepresentations in this family of representations.
Paper Structure (35 sections, 16 theorems, 107 equations, 2 tables)

This paper contains 35 sections, 16 theorems, 107 equations, 2 tables.

Table of Contents

  1. Introduction
  2. The induced representations
  3. The main results and methods
  4. Brief account of earlier results and future questions
  5. Organization
  6. Quaternionic Lie algebras and split real Lie algebras $e_{6(6)}, e_{7(7)} e_{8(8)}$
  7. Simple Lie algebras $\mathfrak g$ and their $\mathfrak {sl}_2$-subalgebras
  8. Quaternionic Lie algebras $\mathfrak g$
  9. Strongly orthogonal roots $\beta_1, \cdots, \beta_4$ and $\gamma_1, \cdots, \gamma_4$ for quaternionic Lie algebra $\mathfrak g$
  10. Split real Lie algebras $\mathfrak g=\mathfrak e_{6(6)}, \mathfrak e_{7(7)}, \mathfrak e_{8(8)}$
  11. Heisenberg parabolic algebras and groups
  12. Root space decomposition of $E=E_\gamma + E_{-\gamma}$
  13. Heisenberg parabolic subgroup $P=MAN$
  14. The subgroup $L_1\subset K$ and character $\chi$
  15. Compact Hermitian Symmetric Space $K/L_1$
  16. ...and 20 more sections

Key Result

Lemma 1.1

(GW.) Under the realization of $K$ via the adjoint action on $\mathfrak p^{\mathbb C}=\mathbb C^2\otimes V$ it is $K=SU(2)_{\beta_1} \times M_c/ ((-1)_{\beta_1}\times\varepsilon)$, where $M_c$ is a compact subgroup of $GL(V, \mathbb C)$ with central element $\varepsilon$ and Lie algebra $\mathfrak m

Theorems & Definitions (33)

  • Lemma 1.1
  • Lemma 1.2
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • proof
  • ...and 23 more