Whittaker functions on ${\rm GL}_n$ via theta lifting

TL;DR

The paper develops a novel approach to explicit Whittaker functions on GL_n by exploiting explicit theta lifting for the dual pair (GL_n,GL_{n+1}). It proves two main results: (i) a nonzero GL_n×GL_{n+1}-equivariant theta-lift map realized by an absolutely convergent integral, and (ii) a propagation formula that expresses minimal K-type Whittaker functions on GL_{n+1} in terms of those on GL_n via joint-harmonics data. These results unify and reinterpret classical formulas (Shintani, Miyazaki) as instances of theta-propagation, and enable new explicit formulas for GL_n(C) together with computations of Asai local zeta integrals. The framework provides a practical method to inductively compute Whittaker functions and their local zeta integrals, with potential impact on automorphic L-functions and related arithmetic questions.

Abstract

In the literature, two main approaches have been used to establish explicit formulas or propagation formulas for Whittaker functions over Archimedean local fields: one based on Jacquet integrals, and the other on the analysis of systems of partial differential equations. In this paper, we introduce a third approach via explicit theta correspondence. As an example, we derive new cases of explicit formulas for Whittaker functions on ${\rm GL}_n(\mathbb{C})$ and compute the associated Asai local zeta integrals.

Theorems & Definitions (32)

• Theorem 1.1: Theorem \ref{['T:image']}
• Theorem 1.2: Theorem \ref{['T:main']}
• Remark 1.1
• Theorem 3.1: Mœ glin, Admas-Barbasch and Minguez
• Corollary 3.1
• proof
• Proposition 3.1
• Lemma 3.1
• proof
• Theorem 3.2