# Dimensions of Automorphic Representations, $L$-Functions and Liftings

## TL;DR

This paper explores the limits of the Rankin-Selberg method for automorphic L-functions, proposing an extended dimension equation to include more cases and connecting it to theta liftings and new integral constructions.

## Contribution

It introduces an extended dimension equation that encompasses additional Rankin-Selberg integrals and relates these to theta liftings and new lifting frameworks.

## Key findings

- Extended dimension equation includes more Rankin-Selberg integrals.
- Connections established between integrals, theta liftings, and new lifts.
- Illustration of a new natural lift example.

## Abstract

There are many Rankin-Selberg integrals representing Langlands $L$-functions, and it is not apparent what the limits of the Rankin-Selberg method are. The Dimension Equation is an equality satisfied by many such integrals that suggests a priority for further investigations. However there are also Rankin-Selberg integrals that do not satisfy this equation. Here we propose an extension and reformulation of the dimension equation that includes many additional cases. We explain some of these cases, including the new doubling integrals of the authors, Cai and Kaplan. We then show how this same equation can be used to understand theta liftings, and how doubling integrals fit into a lifting framework. We give an example of a new type of lift that is natural from this point of view.