Holomorphic harmonic analysis on complex reductive groups

TL;DR

This paper introduces a holomorphic Fourier transform for functions on complex reductive groups, establishing foundational properties and demonstrating its independence from specific measures, with potential applications in harmonic analysis.

Contribution

It defines a new holomorphic Fourier transform on complex reductive groups and proves its key properties, including inversion and measure independence.

Findings

Holomorphic Fourier transform is well-defined and invertible.

$K$-admissible measures are plentiful and do not affect the transform.

The approach extends harmonic analysis tools to complex reductive groups.

Abstract

We define the holomorphic Fourier transform of holomorphic functions on complex reductive groups, prove some properties like the Fourier inversion formula, and give some applications. The definition of the holomorphic Fourier transform makes use of the notion of $K$-admissible measures. We prove that $K$-admissible measures are abundant, and the definition of holomorphic Fourier transform is independent of the choice of $K$-admissible measures.