[Paper Review] Holomorphic extension of representations: (I) automorphic functions

Bernhard Kroetz, Robert J. Stanton|ArXiv.org|Oct 7, 2002

Advanced Algebra and Geometry13 references4 citations

TL;DR

This paper constructs a maximal $K_C$-$G$ double coset domain in the complexification $G_C$ of a real semisimple Lie group $G$, proving that the action of $G$ on $K$-finite vectors of any irreducible unitary representation extends holomorphically to this domain. It establishes sharp estimates for singularities and boundary behavior of holomorphically extended matrix coefficients, enabling $L^∞$ bounds on automorphic functions and uniform decay rate estimates for Fourier coefficients of Maaß forms and Rankin-Selberg products.

ABSTRACT

Let G be a connected, real, semisimple Lie group contained in its complexification G_C, and let K be a maximal compact subgroup of G. We construct a K_C-G double coset domain in G_C, and we show that the action of G on the K-finite vectors of any irreducible unitary representation of G has a holomorphic extension to this domain. For the resultant holomorphic extension of K-finite matrix coefficients we obtain estimates of the singularities at the boundary, as well as majorant/minorant estimates along the boundary. We obtain L^\infty bounds on holomorphically extended automorphic functions on G/K in terms of Sobolev norms, and we use these to estimate the Fourier coefficients of combinations of automorphic functions in a number of cases, e.g. of triple products of Maass forms.

Motivation & Objective

• To construct a universal, maximal $K_C$-$G$ double coset domain in $G_C$ to which the action of $G$ on $K$-finite vectors of any irreducible unitary representation extends holomorphically.
• To derive precise estimates for the singularities of holomorphically extended $K$-finite matrix coefficients at the boundary of the domain.
• To establish $L^∞$ bounds on automorphic functions on $G/K$ in terms of Sobolev norms for applications to Fourier coefficient decay.
• To provide a uniform, representation-theoretic proof of decay rate estimates for Fourier coefficients of Maaß forms and Rankin-Selberg products across real rank-one and higher-rank groups.
• To verify a uniform version of Sarnak's conjecture on exponential decay rates for Fourier coefficients of combinations of Maaß forms.

Proposed method

• Constructs a $K_C$-$G$ double coset domain in $G_C$ using the Iwasawa decomposition and complexification of the Lie algebra $\mathfrak{g}$, leveraging the Cartan decomposition $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}$ and root space decomposition.
• Uses the Iwasawa decomposition $G = KAN$ and its complexification to define the domain $\Omega = K_C A_C^1 N_C \cap G_C$, showing it is maximal in certain directions.
• Applies analytic continuation techniques to extend the action of $G$ on $K$-finite vectors holomorphically to $\Omega$, relying on holomorphic dependence of $\kappa(g), a(g), n(g)$ on $g \in G$.
• Derives majorant/minorant estimates along the boundary of the domain by analyzing the behavior of $a_z k_\theta$ in $K_C A_C^1 N_C$ using complex analytic functions $z' = \sqrt{z^2 + \sin^2\theta(1/z^2 - z^2)}$.
• Establishes $L^\infty$ bounds on automorphic functions via Sobolev norm estimates, using the holomorphic extension of matrix coefficients.
• Applies these bounds to estimate Fourier coefficients of products of automorphic forms, including triple products of Maaß forms and Rankin-Selberg convolutions.

Experimental results

Research questions

• RQ1 Can a universal, maximal domain in $G_C$ be constructed such that the action of $G$ on $K$-finite vectors of any irreducible unitary representation extends holomorphically to it?
• RQ2 What are the precise boundary singularity and growth estimates for the holomorphically extended $K$-finite matrix coefficients of unitary representations?
• RQ3 How can $L^\infty$ bounds on automorphic functions on $G/K$ be derived from Sobolev norms using holomorphic extension?
• RQ4 What is the uniform decay rate of Fourier coefficients of combinations of Maaß forms across real rank-one groups, and can it be proven via representation-theoretic methods?
• RQ5 Can the conjecture of Sarnak on exponential decay of Fourier coefficients of Maaß form products be uniformly verified using holomorphic extension techniques?

Key findings

• The action of $G$ on $K$-finite vectors of any irreducible unitary representation extends holomorphically to a maximal $K_C$-$G$ double coset domain $\Omega \subset G_C$, which is shown to be maximal in certain directions.
• The holomorphically extended $K$-finite matrix coefficients admit precise majorant and minorant estimates along the boundary of $\Omega$, with singularities controlled by the complex analytic structure of the domain.
• The paper establishes $L^\infty$ bounds on automorphic functions on $G/K$ in terms of Sobolev norms, using the holomorphic extension of matrix coefficients.
• It provides a uniform proof of the conjectured exponential decay rate for Fourier coefficients of triple products of Maaß forms for real rank-one groups.
• For $G = \mathrm{SL}(n,\mathbb{R})$, the method yields estimates for the decay rate of Fourier coefficients of Rankin-Selberg products of Maaß forms.
• A conceptually simple proof is given for Good's results on the growth rate of Fourier coefficients of Rankin-Selberg products for co-finite volume lattices in $\mathrm{SL}(2,\mathbb{R})$.