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Transfer using Fourier transform and minimal representation of $E_7$

Nhat Hoang Le, Bryan Peng Jun Wang

Abstract

In this paper, we study the Sakellaridis-Venkatesh conjecture for the rank-1 spherical variety $X=\text{Spin}_9\backslash F_4$ using an exceptional theta correspondence. We establish the correct transfer map satisfying relative character identities in this case and show that our transfer map agrees with the formula in (Sakellaridis, 2021). We also formulate the local relative characters for the degenerate Whittaker period of (Mao-Wan-Zhang, 2026a) associated with $X$. Moreover, we show how our techniques lead to a characterization of $X$-relatively cuspidal representations.

Transfer using Fourier transform and minimal representation of $E_7$

Abstract

In this paper, we study the Sakellaridis-Venkatesh conjecture for the rank-1 spherical variety using an exceptional theta correspondence. We establish the correct transfer map satisfying relative character identities in this case and show that our transfer map agrees with the formula in (Sakellaridis, 2021). We also formulate the local relative characters for the degenerate Whittaker period of (Mao-Wan-Zhang, 2026a) associated with . Moreover, we show how our techniques lead to a characterization of -relatively cuspidal representations.

Paper Structure

This paper contains 25 sections, 17 theorems, 176 equations.

Table of Contents

  1. Introduction
  2. The relative Langlands program
  3. Relative cuspidality
  4. Degenerate Whittaker periods
  5. Organization of this paper
  6. Preliminaries
  7. The exceptional groups $F_4$ and $E_7$
  8. Dual pairs $PGL_2\times F_4$ and $(SL_2\times SL_2)/\mu_2^\Delta \times \text{Spin}_9$ in $E_7$
  9. Measures
  10. The group $PGL_2$
  11. The cone $\Omega$
  12. Exceptional Theta correspondences
  13. The minimal representation of $E_7$
  14. Smooth exceptional theta correspondence
  15. $L^2$-exceptional theta correspondence
  16. ...and 10 more sections

Key Result

Theorem 2.1

Let $\sigma$ be an irreducible smooth representation of $\text{PGL}_2(F)$. Recall that $P_1$ and $P_2$ are maximal parabolic subgroups of $F_4$ defined in Section 2.1.

Theorems & Definitions (30)

  • Theorem 2.1
  • Lemma 3.1
  • proof
  • Definition 3.2
  • Lemma 3.3
  • proof
  • Proposition 3.4
  • proof
  • Theorem 3.5
  • proof
  • ...and 20 more