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A note on the universal supersingular quotients of $U(2,1)$

Peng Xu

Abstract

Let $G$ be the unramified unitary group $U(2, 1)(E/F)$ defined over a non-archimedean local field $F$ of residue characteristic $p\neq 2$. In this note, we prove the universal supersingular quotients of $G$ are not irreducible in general.

A note on the universal supersingular quotients of $U(2,1)$

Abstract

Let be the unramified unitary group defined over a non-archimedean local field of residue characteristic . In this note, we prove the universal supersingular quotients of are not irreducible in general.
Paper Structure (13 sections, 22 theorems, 5 equations)

This paper contains 13 sections, 22 theorems, 5 equations.

Table of Contents

  1. Introduction
  2. Notations and Preliminary
  3. Notations
  4. The spherical Hecke algebra $\mathcal{H}(K, \sigma)$
  5. The image of $(\textnormal{ind}_K ^G \sigma)^{I_{1, K}}$ under the Hecke operator $T$
  6. The supersingular universal quotient of $G$
  7. The space $\text{Hom}_G (\text{ind}^G _K \sigma, \text{ind}^G _K \sigma')$
  8. The universal quotient $\text{ind}^G _K \sigma/ (T_\sigma)$ is not irreducible
  9. The degenerate case
  10. The regular case that $K=K_1$ and $q=p$
  11. Appendix: the $I_{1, K}$-invariant maps $S_K$ and $S_-$
  12. Definition of $S_K$ and $S_-$
  13. The images of $(\textnormal{ind}_K ^G \sigma)^{I_{1, K}}$ under $S_K$ and $S_-$

Key Result

Theorem 1.1

(Corollary reducibility of supersingular quotient, Corollary reducibility of supersingular quotient: the case K=K_1 and q=p) Suppose $K$ is special but non-hyperspecial, and the size of the residue field of $F$ is $p$. For any weight $\sigma$ of $K$, the universal supersingular quotient $\textnormal

Theorems & Definitions (50)

  • Theorem 1.1
  • Lemma 2.1
  • Proposition 2.2
  • proof
  • Theorem 2.3
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Remark 3.3
  • ...and 40 more