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

Abstract

We relate the analytic conductor of a generic irreducible representation of GL n (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 PGL n (Z)\PGL n (R) ordered by analytic conductor.

Conductor zeta function for the GL(2) universal family

“…A definition of archimedean conductor as suggested above lends itself quite naturally to use in the Arthur-Selberg trace formula (or other types of trace formulae, such as that of Kuznetsov), as has been powerfully demonstrated in the recent thesis of Jana [19] and subsequent outgrowths. In this usage, one takes as an archimedean test function a smoothened characteristic function of K 1 (X, τ 0 ), see [20,Section 8]. The latter can be viewed as a bump function about the origin in the Kirillov model of π; it is the analytic new vector in the title of [20].…”

“…In this usage, one takes as an archimedean test function a smoothened characteristic function of K 1 (X, τ 0 ), see [20,Section 8]. The latter can be viewed as a bump function about the origin in the Kirillov model of π; it is the analytic new vector in the title of [20].…”

“…In this section we discuss some issues regarding smoothing and spectral inversion of test functions, and comment extensively on the recent work of Jana-Nelson [20]. Since this appendix is purely expository and comparative, we shall work in the more general setting of the group G = GL n over Q.…”

“…Archimedean conductor via approximate invariance. In [20], Jana and Nelson observe that the archimedean conductor of a generic irreducible π ∈ G(R) ∧,gen , as defined by Iwaniec-Sarnak, can be captured by its approximate invariance properties. More precisely, for X 1 and τ ∈ (0, 1), we denote…”

“…The Weyl laws for Maass forms are deduced from a sharp Tauberian theorem that we state and prove in Appendix B. A discussion and comparison with the alternative approach of using the theory of analytic newvectors, due to Jana and Nelson [20], is given in Appendix C.…”