[Paper Review] On the unramified spherical automorphic spectrum

Maurizio Martino, Heiermann, Volker+1 more|arXiv (Cornell University)|Dec 29, 2015

Advanced Algebra and Geometry74 references3 citations

TL;DR

This paper establishes a precise spectral decomposition of the unramified spherical automorphic spectrum for a reductive group over a number field, using normalized Eisenstein series and residue distributions from graded affine Hecke algebras. It proves that in the split case with trivial character, the normalized spectrum exhausts the entire spherical automorphic spectrum.

ABSTRACT

For an unramified connected reductive group $G$ defined over a number field $F$, consider the part of the spherical automorphic spectrum with cuspidal support $[T,\mathcal{O}(χ)]$, where $T$ is a maximal torus and $χ$ is an unramified automorphic character. We define a normalization of the Eisenstein series and we give the precise spectral decomposition of the closure of the subspace spanned by the normalized pseudo-Eiseinstein series. The proof uses residue distributions which were introduced by the third author (in joint work with G. Heckman) in the study of graded affine Hecke algebras, which is an ingredient of a purely local nature. In the case when $G$ is split and $χ$ is the trivial character, we show that the normalized spectrum is in fact the whole spherical automorphic spectrum. The necessary argument to conclude the result in the split case are based on combinatorial results proved in [DMHO].

Motivation & Objective

• To provide a complete spectral decomposition of the unramified spherical automorphic spectrum for a reductive group over a number field.
• To define and analyze a normalization of Eisenstein series that ensures convergence and spectral purity in the automorphic setting.
• To show that in the split case with trivial character, the normalized spectrum coincides with the full spherical automorphic spectrum.
• To establish the role of residue distributions—originally developed for graded affine Hecke algebras—as a key local tool in the global spectral decomposition.
• To unify local harmonic analysis techniques with global automorphic representation theory via Arthur parameters and truncation methods.

Proposed method

• Introduces a normalization of pseudo-Eisenstein series using Paley-Wiener coefficients to ensure convergence and spectral orthogonality.
• Applies residue distributions from the theory of graded affine Hecke algebras (developed with G. Heckman) to analyze poles and residues of Eisenstein series.
• Employs a cascade of contour shifts in the complex plane, tracking residues at critical points via iterated residue integrals.
• Uses truncation techniques with parameter $Τ$ to control growth and define orthogonal projections onto finite spectral subspaces.
• Relies on the Arthur parameter formalism and the $L$-group structure to parametrize automorphic representations and relate them to spectral measures.
• Applies combinatorial results from [DMHO] to verify the non-degeneracy of residue kernels and the absence of spurious poles in the spectral decomposition.

Experimental results

Research questions

• RQ1 How can the unramified spherical automorphic spectrum be decomposed into irreducible components using normalized Eisenstein series?
• RQ2 What is the precise role of residue distributions in the spectral decomposition of automorphic forms?
• RQ3 Under what conditions does the normalized spectrum coincide with the full spherical automorphic spectrum?
• RQ4 How do the local structures of graded affine Hecke algebras contribute to the global spectral decomposition?
• RQ5 What is the relationship between the Arthur parameter $φ_{\chi}$ and the spectral measure in the decomposition of $L^2(G(F)\backslash G(\mathbb{A}_F), \xi)$?

Key findings

• The closure of the subspace spanned by normalized pseudo-Eisenstein series admits a spectral decomposition via iterated residue integrals over critical orbits.
• In the split case with trivial character, the normalized spectrum is isomorphic to the entire spherical automorphic spectrum, i.e., $L^2(G(F)\backslash G(\mathbb{A}_F), \xi)_{[T,\mathcal{O}(1)]}^K$ is completely decomposed by the normalized Eisenstein series.
• The spectral inner product $(\theta_\phi, \theta_\psi)$ is expressed as a sum over $W$-orbits of residual spaces, involving the $A_0$-transforms of $r\phi$ and $r\psi$.
• The residue kernels $K^{M,sym}_X(\theta)$ and $K^{M,sym}_Y(\theta)$ are shown to be holomorphic on the tempered loci $L^{\textup{temp}}$, enabling the limit $\mathfrak{T} \to \infty$.
• The truncation process $q_{\mathfrak{T}}$ converges to the orthogonal projection onto the spectral subspace, validating the use of truncated integrals in the spectral formula.
• The final spectral decomposition is expressed as a sum over $W\backslash\mathcal{L}$ of integrals involving $|W| \int_{L^{\textup{temp}}}(r(-\lambda)r(\lambda))^{-1} A_0(r\psi)(\lambda) \overline{A_0(r\phi)(\lambda)} d\nu_L(\lambda)$, which is the main spectral identity.