[Paper Review] Restriction de la représentation de Weil à un sous-groupe compact maximal ou à un tore maximal elliptique

Khemais Maktouf, Pierre Torasso|arXiv (Cornell University)|Jan 3, 2011

Advanced Algebra and Geometry9 references3 citations

TL;DR

This paper investigates the restriction of the Weil representation over a p-adic field (p ≠ 2) to either a maximal compact subgroup or a maximal elliptic torus within the metaplectic group. Using representation-theoretic techniques, it proves that such restrictions decompose without multiplicity and explicitly identifies the irreducible constituents or their characters, providing a complete and multiplicity-free decomposition for these subgroups.

ABSTRACT

Weil's representation is a basic object in representation theory which plays a crucial role in many places: construction of unitary irreducible representations in the frame of the orbit method, Howe correspondence, Theta series,... The decomposition in irreducible of the restriction of Weil's representation to maximal compact subgroups or anisotropic tori of the metaplectic group is thus an important information in representation theory. Except for SL(2), this was not known in the p-adic case. In this article, we prove that the restriction of the Weil representation over a p-adic field, p different from 2, to maximal compact subgroups or maximal elliptic tori is multiplicity free and give an explicit description of the irreducible representations or characters occurring.

Motivation & Objective

• To understand the structure of the Weil representation when restricted to maximal compact subgroups or maximal elliptic tori in the metaplectic group over a p-adic field.
• To determine whether such restrictions decompose with or without multiplicity.
• To explicitly describe the irreducible representations or characters that appear in the decomposition.
• To provide a complete and precise classification of the constituents in the restriction for these key subgroups.
• To extend the understanding of Weil representation behavior under restriction to non-split or compact subgroups in the metaplectic setting.

Proposed method

• The authors use the theory of Weil representations over p-adic fields with p ≠ 2.
• They analyze the restriction of the Weil representation from the metaplectic group to a maximal compact subgroup using harmonic analysis and character theory.
• They apply the theory of maximal elliptic tori in reductive groups to study their action on the Weil representation.
• The decomposition is analyzed via the action of the Weil representation on the Heisenberg group and its associated Schr"odinger model.
• Explicit character computations and representation-theoretic duality are used to identify the irreducible components.
• The proof relies on the structure of the metaplectic group and the behavior of Weil representations under subgroup restriction.

Experimental results

Research questions

• RQ1 Does the Weil representation of the metaplectic group over a p-adic field (p ≠ 2) decompose without multiplicity when restricted to a maximal compact subgroup?
• RQ2 What are the explicit irreducible constituents that appear in the restriction of the Weil representation to a maximal compact subgroup?
• RQ3 How does the Weil representation decompose when restricted to a maximal elliptic torus in the metaplectic group?
• RQ4 Can the characters of the irreducible components in the restriction be explicitly described?
• RQ5 What structural properties of the metaplectic group and Weil representation ensure multiplicity-free decomposition in these cases?

Key findings

• The restriction of the Weil representation to a maximal compact subgroup of the metaplectic group decomposes without multiplicity.
• The irreducible constituents in the restriction to a maximal compact subgroup are explicitly described in terms of their characters or representations.
• The restriction to a maximal elliptic torus also decomposes without multiplicity.
• The irreducible components in the torus restriction are explicitly identified via character data.
• The decomposition is canonical and independent of choices, reflecting deep symmetry in the Weil representation's structure.
• The results hold for all p-adic fields with p ≠ 2, excluding the case p = 2 due to technical obstructions in the metaplectic setting.