[Paper Review] The Metaplectic Casselman-Shalika Formula

Peter J. McNamara|arXiv (Cornell University)|Mar 24, 2011

Advanced Algebra and Geometry12 references4 citations

TL;DR

This paper establishes a metaplectic Casselman-Shalika formula for spherical Whittaker functions on tame central extensions of unramified reductive groups over non-archimedean local fields. By generalizing Chinta and Offen's approach, it derives an explicit formula involving a Weyl group action with coefficients tied to root system data and the residue field size $ q \equiv 1 \pmod{2n} $, showing that the metaplectic Whittaker function matches the $ p $-part of a Weyl group multiple Dirichlet series.

ABSTRACT

This paper studies spherical Whittaker functions for central extensions of reductive groups over local fields. We follow the development of Chinta and Offen to produce a metaplectic Casselman-Shalika formula for tame covers of all unramified groups.

Motivation & Objective

• To extend the classical Casselman-Shalika formula to metaplectic groups, i.e., central extensions of reductive groups by a finite cyclic group.
• To develop a uniform formula for spherical Whittaker functions on metaplectic covers of unramified reductive groups over non-archimedean local fields.
• To establish a precise connection between the metaplectic Whittaker function and the $ p $-part of Weyl group multiple Dirichlet series as constructed by Chinta and Gunnells.
• To clarify the action of the Weyl group on the metaplectic Whittaker function through explicit formulas involving root system invariants and $ q $-deformations.

Proposed method

• Adopts the framework of Chinta and Offen to generalize the Casselman-Shalika method to metaplectic covers of unramified groups.
• Uses a central extension $ \widetilde{G} $ of $ G $ by $ \mu_n $, with $ \mu_n \subset \mathbb{C}^\times $ via a fixed embedding $ \epsilon $, and considers genuine representations where $ \mu_n $ acts via $ \epsilon $.
• Employs a Weyl group action on the geometric cocharacter lattice $ X_*(S) $, with explicit formulas for the action of simple reflections derived from intertwining operators and cocycle data.
• Derives a metaplectic Gindikin-Karpelevic formula in Section 12 as a key intermediate step.
• Compares the metaplectic Whittaker function to the $ p $-part of the Weyl group multiple Dirichlet series via a change of variables and normalization to match the Chinta-Gunnells construction.
• Proves that two Weyl group actions—$ \circ_1 $ and $ \circ_2 $—are equivalent, establishing consistency with the standard construction in multiple Dirichlet series theory.

Experimental results

Research questions

• RQ1 How can the Casselman-Shalika formula be generalized to metaplectic groups, i.e., central extensions of reductive groups by $ \mu_n $?
• RQ2 What is the explicit form of the spherical Whittaker function on a metaplectic cover of an unramified group?
• RQ3 How does the Weyl group action on the Whittaker function differ from the classical case, and what are the correction terms?
• RQ4 To what extent does the metaplectic Whittaker function coincide with the $ p $-part of a Weyl group multiple Dirichlet series?

Key findings

• The metaplectic Casselman-Shalika formula (Theorem 8.1) expresses the Whittaker function on $ \widetilde{G}(F) $ as a sum over the Weyl group with coefficients involving $ q $-deformed terms and root system data.
• The action of a simple reflection $ s_\alpha $ on a monomial $ x^\lambda $ is given by a precise formula involving $ \tau^{1}_{\mu,\mu} $ and $ \tau^{2}_{\mu,s\mu+\alpha} $, which depend on the bilinear form $ B(\alpha,\lambda) $ and the quadratic form $ Q(\alpha) $.
• The formula for the Whittaker function at a torus element $ \varpi^\lambda $ is shown to equal $ \chi(\varpi^\lambda) N(\chi,\lambda) $, where $ N(\chi,\lambda) $ is the $ p $-part of the Weyl group multiple Dirichlet series from Chinta and Gunnells.
• The two Weyl group actions $ \circ_1 $ and $ \circ_2 $, defined via normalization and sign factors, are proven to be equivalent, validating the consistency of the metaplectic construction with the standard theory.
• The assumption $ q \equiv 1 \pmod{2n} $ is essential for simplifying the cocycle structure and trivializing the second twist in Weissman’s framework, though the author expects only sign changes under the weaker $ q \equiv 1 \pmod{n} $ condition.