[Paper Review] Analytic newvectors for $\mathrm{GL}_n(\mathbb{R})$

Subhajit Jana, Paul D. Nelson|arXiv (Cornell University)|Nov 5, 2019

Advanced Algebra and Geometry27 references4 citations

TL;DR

This paper establishes an archimedean analogue of classical newvector theory for GLₙ(ℝ), relating the analytic conductor of a generic irreducible representation π to the existence of vectors approximately invariant under certain subgroups related to K₀(X,τ) and K₁(X,τ). The key result shows that for cuspidal automorphic representations with analytic conductor C(π) < X, the spectral projector F_X satisfies π(F_X)W(1) ≫ 1, enabling majorization of sums over automorphic forms ordered by analytic conductor in trace formulas.

ABSTRACT

We relate the analytic conductor of a generic irreducible representation of $\mathrm{GL}_n(\mathbb{R})$ to the invariance properties of vectors in that representation. The relationship is an analytic archimedean analogue of some aspects of the classical non-archimedean newvector theory of Casselman and Jacquet--Piatetski-Shapiro--Shalika. We illustrate how this relationship may be applied in trace formulas to majorize sums over automorphic forms on $\mathrm{PGL}_n(\mathbb{Z}) \backslash \mathrm{PGL}_n(\mathbb{R})$ ordered by analytic conductor.

Motivation & Objective

• To develop an analytic archimedean analogue of Casselman’s non-archimedean newvector theory for GLₙ(ℝ).
• To relate the analytic conductor C(π) of a generic irreducible representation π to the invariance properties of vectors in π under certain subgroups.
• To construct test functions F_X that approximately project onto families of automorphic forms on PGLₙ(ℤ)\PGLₙ(ℝ) ordered by analytic conductor.
• To enable the application of trace formulas to majorize sums over automorphic forms by analytic conductor, mirroring non-archimedean techniques.

Proposed method

• Introduces a family of test functions F_X that are normalized majorants of the subgroups K₀(X,τ) and K₁(X,τ), defined via matrix conditions involving entries bounded by 1/X.
• Uses convolution of normalized majorants to preserve the majorization property under group multiplication, ensuring compatibility with spectral projections.
• Applies spectral decomposition of the automorphic kernel ∑γ∈PGLₙ(ℤ) F_X(x₁⁻¹γx₂) to relate the trace formula to the Bessel distribution J_π(F_X).
• Relies on Whittaker model theory and the Plancherel measure to express the spectral side in terms of Whittaker functions W_φ and their L² norms.
• Employs Schur’s lemma to relate the L² norm of a vector φ to the norm of its Whittaker function, with a factor ℓ(π) ≍ L(1,π,Ad) for cuspidal π.
• Uses the Bessel distribution J_π(F_X) = ∑|π(f_X)W(1)|² to detect representations with small analytic conductor, showing J_π(F_X) ≫ 1 for C(π) < X.

Experimental results

Research questions

• RQ1 Can the analytic conductor of a generic irreducible representation π of GLₙ(ℝ) be characterized by the approximate invariance of vectors under certain subgroups, analogous to the non-archimedean newvector theory?
• RQ2 How can one construct test functions that approximately project onto families of automorphic forms on PGLₙ(ℤ)\PGLₙ(ℝ) ordered by analytic conductor?
• RQ3 What is the spectral behavior of the Bessel distribution J_π(F_X) for representations with C(π) < X, and how does it relate to the size of π(f_X)W(1)?
• RQ4 To what extent can the convolution of normalized majorants preserve the required invariance and decay properties under group operations?

Key findings

• For any cuspidal automorphic representation π with analytic conductor C(π) < X, the spectral projector F_X satisfies π(F_X)W(1) ≫ 1, where W is the Whittaker newvector.
• The Bessel distribution satisfies J_π(F_X) ≫ 1 for all cuspidal π with C(π) < X, providing a spectral detection mechanism.
• The volume of the subgroup K₀(X,τ) is bounded as Vol(K₀(X,τ)) ≍ X^{n−1}, which controls the size of the Whittaker integral ∫_N F_X(x)ψ̅(x)dx ≪ X^{n−1}.
• The convolution of two normalized majorants F_X¹ and F_X² remains a normalized majorant of K_*(X,τ₁) for some τ₁ > 0, preserving the required properties for spectral projection.
• The spectral side of the trace formula decomposes into ∫ J_π(F_X)ℓ(π)⁻¹ dμ_aut(π), which is majorized by X^{n−1}, enabling effective bounds on sums over automorphic forms.
• The construction provides a direct, archimedean analogue of non-archimedean projectors, allowing trace formula applications to families ordered by analytic conductor.