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

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.

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