Spectral average of central values of automorphic L-functions for holomorphic cusp forms on SO_0(m,2) II

TL;DR

This paper investigates the distribution of Satake parameters associated with holomorphic cusp forms on orthogonal groups, linking central L-values and Whittaker-Bessel periods, under certain non-negativity assumptions.

Contribution

It establishes an equidistribution result connecting central L-values, Whittaker-Bessel periods, and Satake parameters for automorphic forms on SO_0(m,2).

Findings

Proves equidistribution of Satake parameters under specified conditions.

Links central L-values with automorphic representation parameters.

Provides a new perspective on the spectral average of automorphic L-values.

Abstract

Given a maximal even-integral lattice $\cL$ of signature $(m+, 2-)$ with an odd $m\geq 3$, we consider the holomorphic cusp forms $F$ of weight $l$ on the bounded symmetric domain of type IV of dimension $m$ with respect to the discriminant subgroup of the orthogonal group $O(\cL)$ defined by $\cL$. Under a non-negativity assumption on the central $L$-values, we prove an equidistribution result of Satake parameters in an ensemble constructed from the central values of standard $L$-functions and the square of the Whittaker-Bessel periods.