[Paper Review] Principal series representations of metaplectic groups over local fields

Peter J. McNamara|arXiv (Cornell University)|Nov 11, 2009

Advanced Algebra and Geometry9 references4 citations

TL;DR

This paper develops the theory of principal series representations for metaplectic groups—central extensions of split reductive groups over non-archimedean local fields by a cyclic group of order $ n $, under tameness conditions. It establishes a metaplectic Satake isomorphism and introduces a combinatorial dual group for metaplectic groups, extending Langlands functoriality to this setting via Hecke algebra structures and categorified geometric constructions.

ABSTRACT

Let G be a split reductive algebraic group over a non-archimedean local field. We study the representation theory of a central extension $\G$ of G by a cyclic group of order n, under some mild tameness assumptions on n. In particular, we focus our attention on the development of the theory of principal series representations for $\G$ and applications of this theory.

Motivation & Objective

• To extend the theory of principal series representations to metaplectic groups, which are central extensions of split reductive groups by $ \mu_n $, under tameness assumptions on $ n $.
• To generalize the Satake isomorphism to the metaplectic setting, establishing a link between spherical Hecke algebras and the dual group of the metaplectic cover.
• To define a combinatorial dual group for metaplectic groups, extending the classical Langlands dual group construction to non-reductive, non-simply connected covers.
• To provide a framework for relating representations of metaplectic groups via Hecke algebra isomorphisms, particularly in the Iwahori and spherical cases.
• To connect the algebraic Hecke algebra structure with geometric constructions, such as perverse sheaves on torsors over the affine Grassmannian.

Proposed method

• Constructs the metaplectic group $ \widetilde{G} $ as a central extension of $ G $ by $ \mu_n $, using cocharacter and coroot data from the reductive group $ \mathbf{G} $ over $ O_F $.
• Applies the Stone–von Neumann theorem to show that irreducible representations of the metaplectic torus are finite-dimensional, despite non-abelian structure.
• Induces principal series representations from the inverse image of a Borel subgroup in $ \widetilde{G} $, generalizing the classical construction to metaplectic covers.
• Develops Jacquet modules and intertwining operators for these induced representations, adapting standard representation-theoretic tools to the metaplectic setting.
• Uses the Iwahori and maximal compact subgroups $ K = \mathbf{G}(O_F) $ to define Hecke algebras of anti-genuine, compactly supported functions, and presents them via generators and relations.
• Establishes a metaplectic Satake isomorphism (Theorem 10.1) and constructs the dual group $ \widetilde{G}^\vee $ from the root datum of the cover, generalizing the Langlands dual group.

Experimental results

Research questions

• RQ1 How can the theory of principal series representations be extended to metaplectic groups, which are non-split central extensions of reductive groups?
• RQ2 What is the structure of the Iwahori and spherical Hecke algebras for metaplectic groups, and how do they relate to the dual group?
• RQ3 Can a combinatorial dual group be defined for metaplectic groups that extends the classical Langlands dual group, even though metaplectic groups are not reductive?
• RQ4 How do the Satake isomorphism and Hecke algebra isomorphisms facilitate a correspondence between representations of different metaplectic groups?
• RQ5 What is the geometric realization of the metaplectic dual group, and how does it relate to perverse sheaves on torsors over the affine Grassmannian?

Key findings

• The metaplectic Satake isomorphism (Theorem 10.1) provides a canonical isomorphism between the spherical Hecke algebra $ \mathcal{H}(\widetilde{G}, K) $ and the representation ring of the dual group $ \widetilde{G}^\vee $.
• The Iwahori-Hecke algebra $ \mathcal{H}(\widetilde{G}, I) $ admits a presentation via generators and relations, with explicit relations involving $ T_\lambda $ and $ T_s $, as shown in Theorem 12.4.
• The dual group $ \widetilde{G}^\vee $ is a complex reductive group whose root system is derived from the metaplectic cover's quadratic form $ Q $, extending the classical dual group construction.
• Isomorphic dual groups $ \widetilde{G}^\vee \cong \widetilde{H}^\vee $ imply isomorphic Iwahori-Hecke algebras $ \mathcal{H}_\epsilon(\widetilde{G}, I) \cong \mathcal{H}_\epsilon(\widetilde{H}, I) $, as shown in Corollary 13.1.
• The categorified metaplectic Satake isomorphism, via perverse sheaves on a $ \mathbb{G}_m $-torsor over the affine Grassmannian, realizes $ \widetilde{G}^\vee $ as the group of symmetries of the tensor category of equivariant perverse sheaves.
• The construction of the dual group is valid under the assumption that $ F^\times $ contains $ 2n $-th roots of unity and $ 2n $ is coprime to the residue characteristic, ensuring tameness.