# Spherical inversion for a small $K$-type on the split real Lie group of type $G_2$

## TL;DR

This paper derives explicit formulas for the Harish-Chandra c-function and spherical inversion for a small K-type on the split real Lie group of type G2, advancing harmonic analysis on this group.

## Contribution

It provides the first explicit formulas for the Harish-Chandra c-function and spherical inversion for a small K-type on G2, a significant step in understanding harmonic analysis on this group.

## Key findings

- Explicit formula for the Harish-Chandra c-function for the small K-type on G2.
- Explicit formula for spherical inversion for this small K-type.
- Enhances understanding of harmonic analysis on split real Lie groups of type G2.

## Abstract

We give an explicit formula for the Harish-Chandra $c$-function for a small $K$-type on a split real Lie group of type $G_2$. As an application we give an explicit formula for spherical inversion for this small $K$-type.