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Analytic newvectors for $\mathrm{GL}_n(\mathbb{R})$

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arxiv 1911.01880 v2 pith:XVCAKHMY submitted 2019-11-05 math.NT math.RT

Subhajit Jana , Paul D. Nelson This is my paper · ORCID
classification math.NTmath.RT
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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.

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  1. On the spectral aspect density hypothesis and application

    math.NT 2025-04 accept novelty 8.0 of 10

    For n >= 4, the paper establishes Sarnak's density hypothesis in the spectral aspect for GL_n(Z) cuspidal representations and uses it to prove the Diophantine exponent of the SL_n(Z[1/p])-action is optimal (kappa = 1).

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