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Branching rules for principal series representations of unramified U(1,1)

Ekta Tiwari

TL;DR

The paper resolves branching rules for principal series representations of the unramified quasi-split unitary group $G=\mathbb{U}(1,1)(F)$ when restricted to the maximal compact subgroup $\mathcal{K}$. It develops a canonical, multiplicity-free decomposition by constructing an explicit family of irreducible $\mathcal{K}$-representations $\mathcal{S}_d(X,\zeta)$ and identifying the constituents $\mathcal{W}_{d,\chi}$ with these modules, as well as providing a detailed description in terms of depth via Moy–Prasad filtrations. The work introduces positive-depth $\mathcal{K}$-representations, extends Clifford theory to this setting, and connects the higher-depth components to depth-zero data and nilpotent-orbit representations, yielding explicit formulas for restrictions such as $\mathrm{Res}_{\mathcal{K}_{2r+1}}\pi_{\chi}$. Applications of these decompositions address conjectures about near-identity representations and the relation between depth-zero and higher-depth components, laying groundwork for a follow-up treatment of all irreducible smooth representations. The results advance understanding of $p$-adic branching beyond rank-one groups and have potential implications for local Langlands correspondence in this context.

Abstract

Let $G$ denote the unramified quasi-split unitary group $\mathbb{U}(1,1)(F)$ over a $p$-adic field $F$ with residual characteristic $p \neq 2$. In this paper, we first construct a large family of irreducible representations of the maximal compact subgroup $\mathcal{K} = \mathbb{U}(1,1)(\mathcal{O}_F)$ of $G$. We then describe the branching rules for all principal series representations of $G$ upon restriction to $\mathcal{K}$ in terms of these representations. The resulting decomposition is multiplicity-free and is characterized by distinct degrees. Finally, we present two important applications of this decomposition that address certain recent open conjectures in the literature. This is the first in a series of two articles in which we provide branching rules for all irreducible smooth representations of the $G$ upon restriction to $\mathcal{K}$.

Branching rules for principal series representations of unramified U(1,1)

TL;DR

The paper resolves branching rules for principal series representations of the unramified quasi-split unitary group when restricted to the maximal compact subgroup . It develops a canonical, multiplicity-free decomposition by constructing an explicit family of irreducible -representations and identifying the constituents with these modules, as well as providing a detailed description in terms of depth via Moy–Prasad filtrations. The work introduces positive-depth -representations, extends Clifford theory to this setting, and connects the higher-depth components to depth-zero data and nilpotent-orbit representations, yielding explicit formulas for restrictions such as . Applications of these decompositions address conjectures about near-identity representations and the relation between depth-zero and higher-depth components, laying groundwork for a follow-up treatment of all irreducible smooth representations. The results advance understanding of -adic branching beyond rank-one groups and have potential implications for local Langlands correspondence in this context.

Abstract

Let denote the unramified quasi-split unitary group over a -adic field with residual characteristic . In this paper, we first construct a large family of irreducible representations of the maximal compact subgroup of . We then describe the branching rules for all principal series representations of upon restriction to in terms of these representations. The resulting decomposition is multiplicity-free and is characterized by distinct degrees. Finally, we present two important applications of this decomposition that address certain recent open conjectures in the literature. This is the first in a series of two articles in which we provide branching rules for all irreducible smooth representations of the upon restriction to .

Paper Structure

This paper contains 19 sections, 37 theorems, 155 equations.

Table of Contents

  1. Introduction
  2. Notations and background
  3. The field
  4. Some structure theory
  5. The group $\mathrm{U}(1,1)$
  6. Filtrations on $T$ and on $\mathcal{K}$
  7. Representation theory
  8. Construction of representations
  9. A canonical decomposition upon restriction to $\mathcal{K}$
  10. Positive depth representations of $\mathcal{K}$
  11. Key concepts involved in the construction
  12. Positive-depth representations of $\mathcal{K}$
  13. Representations associated to nilpotent orbits
  14. An explicit description of $\mathcal{W}_{d,\chi}$
  15. Applications
  16. ...and 4 more sections

Key Result

Theorem 1

Let $\chi$ be a character of $T$ of minimal depth $r$, and let $(\pi_{\chi}, V_{\chi})$ denote the principal series representation associated to $\chi$. Then for $d\geq r+1$, there exists irreducible representations $\mathcal{W}_{d, \chi}$ of degree $(q^{2}-1)q^{d-1}$ such that where $V_{\chi}^{\mathcal{K}_{1}}=\mathbbm{1}_{q}+\mathrm{St}_{q}$ if $\chi=\mathbbm{1}$, and $V_{\chi}^{\mathcal{K}_{r+

Theorems & Definitions (65)

  • Theorem 1: Theorem \ref{['thm2']}
  • Theorem 2: Theorem \ref{['Rep of K']}
  • Theorem 3: Corollary \ref{['prop2']}
  • Theorem 4: Theorem \ref{['keyidentificationp']}
  • Theorem 5: Theorem \ref{['principalseriesrestrictedtoK2r']}
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • ...and 55 more