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Multiplicity free Weil representations arising from exceptional groups

Marcela Hanzer, Gordan Savin

Abstract

Using exceptional theta correspondences, we prove that certain Weil representations of $p$-adic groups are multiplicity free and determine irreducible quotients.

Multiplicity free Weil representations arising from exceptional groups

Abstract

Using exceptional theta correspondences, we prove that certain Weil representations of -adic groups are multiplicity free and determine irreducible quotients.

Paper Structure

This paper contains 15 sections, 25 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. Theta correspondence for principal series
  3. A degenerate principal series for $G$
  4. Minimal representation
  5. Computing the theta correspondence for principal series representations of $G$.
  6. Computing the theta correspondence for principal series representations of $(G,V)=({\mathrm{SL}}_2^3,V_2^{\otimes 3})$
  7. On cuspidal support of the lift
  8. Classical theta for ${\mathrm{SL}}_2 \times {\mathrm{Spin}}_{2n}$
  9. Exceptional correspondence
  10. Lift from $G$ to ${\mathrm{SL}}_2$
  11. Mini theta
  12. Two ${\mathrm{SL}}_2$
  13. Three ${\mathrm{SL}}_2$
  14. Four ${\mathrm{SL}}_2$
  15. Computing parameters of lifts of tempered representations

Key Result

Proposition 2.1

Let $r=1,2,4$, so that $\dim N=3+6r$. Then ${\mathrm{Ind}}_P^G(\chi)$ is irreducible unless The trivial representation $1_G$ is the unique quotient for $s=1+2r$. The minimal representation $\Pi_G$ is the unique quotient for $s=1+r$.

Theorems & Definitions (45)

  • Proposition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • ...and 35 more