# Functorial transfer of Cohomological Representations from $SP(4,\mathbb{R})$ to $GL(5,\mathbb{R})$

## TL;DR

This paper investigates how cohomological properties of certain unitary representations of $Sp(4,\mathbb{R})$ transfer to $GL(5,\mathbb{R})$ via Langlands functoriality, including cases with trivial and non-trivial coefficients.

## Contribution

It demonstrates the functorial transfer of cohomological representations from $Sp(4,\mathbb{R})$ to $GL(5,\mathbb{R})$ and analyzes the preservation of cohomological properties.

## Key findings

- Cohomological representations transfer under the inclusion from $SO(5,\mathbb{C})$ to $GL(5,\mathbb{C})$
- The transfer preserves cohomological properties for trivial coefficients
- Extension to non-trivial coefficients is also established

## Abstract

Let $G=Sp(4,\mathbb{R})$ and let $\pi$ be an irreducible, unitary representation of $G$ which is cohomological with respect to trivial coefficients. Using the inclusion from $SO(5,\mathbb{C})$ to $GL(5,\mathbb{C})$, we transfer $\pi$ to an irreducible representation $\iota(\pi)$ of $GL(5,\mathbb{R})$ and determine how the property of being cohomological behaves under Langlands functoriality. We also consider representations which are cohomological with respect to non-trivial coefficients.