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

Subhajit Jana; Paul D. Nelson

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

Number Theory 2020-10-29 v2 Representation Theory

Authors: Subhajit Jana , Paul D. Nelson

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.

Keywords

gaudin algebras and representation theory

Cite

@article{arxiv.1911.01880,
  title  = {Analytic newvectors for $\mathrm{GL}_n(\mathbb{R})$},
  author = {Subhajit Jana and Paul D. Nelson},
  journal= {arXiv preprint arXiv:1911.01880},
  year   = {2020}
}

Comments

59 pages. New: Theorems on non-existence (Theorem 2), uniqueness (Theorem 6), and trace estimates (Theorem 8)

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

Abstract

Keywords

Cite

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