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Exceptional dual pair correspondences; case of real groups of split rank one

Petar Bakic, Hung Yean Loke, Gordan Savin

TL;DR

The work investigates exceptional real groups with quaternionic structures by analyzing the theta correspondence for dual pairs $(G,G')$ where $G'$ is the split group of type ${\mathrm G}_2$ and $G$ is the split form of a compact-type automorphism group $\mathrm{Aut}(J)$. Central to the approach is the minimal representation $V_{min}$ of the ambient quaternionic group, together with a generalized theta construction and a see-saw framework that connects ${\mathrm Spin}$ and ${\mathrm U}$-type subgroups to ${\mathrm G}_2$ and ${\mathrm PU}(2,1)$ representations. The authors establish finite-length results, identify minimal $K$-types $F_{\\tau}$ that generate the theta lifts, and provide explicit correspondences across several rich dual pairs in $E_6$, $E_7$, $E_8$, and $F_4$, including detailed lifts between ${\mathrm PU}(2,1)$ and ${\mathrm G}_2$. These results yield a concrete realization of functorial theta correspondences in the exceptional setting, with potential applications to automorphic forms and unitary representations, and they supply new branching rules and computational tools for analyzing representations in these highly structured groups.

Abstract

Exceptional real groups have quaternionic forms of split rank 4 that contain dual pairs $G\times G'$, where $G'$ is the split Lie group of the type $G_2$, and $G$ a Lie group of split rank one. In this paper we restrict the minimal representation of the quaternionic group to the dual pair and prove some significant results for the resulting correspondence of representations.

Exceptional dual pair correspondences; case of real groups of split rank one

TL;DR

The work investigates exceptional real groups with quaternionic structures by analyzing the theta correspondence for dual pairs where is the split group of type and is the split form of a compact-type automorphism group . Central to the approach is the minimal representation of the ambient quaternionic group, together with a generalized theta construction and a see-saw framework that connects and -type subgroups to and representations. The authors establish finite-length results, identify minimal -types that generate the theta lifts, and provide explicit correspondences across several rich dual pairs in , , , and , including detailed lifts between and . These results yield a concrete realization of functorial theta correspondences in the exceptional setting, with potential applications to automorphic forms and unitary representations, and they supply new branching rules and computational tools for analyzing representations in these highly structured groups.

Abstract

Exceptional real groups have quaternionic forms of split rank 4 that contain dual pairs , where is the split Lie group of the type , and a Lie group of split rank one. In this paper we restrict the minimal representation of the quaternionic group to the dual pair and prove some significant results for the resulting correspondence of representations.

Paper Structure

This paper contains 45 sections, 30 theorems, 185 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Main result on dual pair correspondences
  3. Two definitions of $\Theta(\pi)$
  4. Main result
  5. Our dual pair
  6. Quaternionic groups
  7. General case
  8. Example of ${\mathrm{E}}_6$
  9. Quaternionic representations
  10. Construction
  11. Figuring out $K$-types
  12. Lie algebra action
  13. Restriction to a subalgebra
  14. Restriction of quaternionic representations from ${\mathrm{Spin}}(4,3)$ to $G_2$
  15. $D_4$ dual pairs and correspondence in $E_6$
  16. ...and 30 more sections

Key Result

Theorem 1.1

Assume that the following two hold: Let $\pi$ and $\pi'$ be irreducible $(\mathfrak g,K)$ and $(\mathfrak g',K')$-modules, respectively. Then:

Figures (5)

  • Figure 1: The cones of $K$-types for regular $\lambda$. Pictured: $(a,b,c)=(2,1,-3)$ and $A_\mathfrak{q}(\lambda)$ modules A (red), B (green), and C (blue).
  • Figure 2: The cones of $K$-types for cases Ia and IIa on the wall $a=b$: the $A_\mathfrak{q}(\lambda)$ modules A (red), AC (purple), C (blue) with $\lambda = (1,1,-2)$, and B (green) with $\lambda =(2,-1,-1)$
  • Figure 3: The cones of $K$-types for cases Ib and IIb on the wall $b=0$. Here we see $A_\mathfrak{q}(\lambda)$ modules A (red), AB (orange), B (green) with $\lambda = (1,0,-1)$, and C (blue) with $\lambda =(0,1,-1)$.
  • Figure 4: The $K$-types for $(a,b,c)=(1,0,-1)$. The one thickened dot in each representation is the type $\tau$ that we lift in Section \ref{['S:correspondence']}.
  • Figure 5: A 'visual proof' of Theorem \ref{['G2toPU3regular']}: Most $F_\tau$'s miss the cones entirely, whereas $F_{\tau_A}$ (resp. $F_{\tau_C}$) intersects the cone of $\pi_A'$ (resp. $\pi_C'$) precisely in the minimal type.

Theorems & Definitions (46)

  • Theorem 1.1
  • Proposition 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Theorem 2.4
  • proof
  • Corollary 2.5
  • Lemma 4.1
  • ...and 36 more