Classification of Unitary Representations for E6(-14) via Root Embedding and $\theta$-Projection: A Rank-2 Reduction Framework with Sp(4)

TL;DR

This paper introduces a rank-2 reduction framework for classifying unitary representations of E6(-14) by embedding root systems and using $ heta$-projection, significantly simplifying the complex classification process.

Contribution

It develops a novel dimensional reduction method combining root embedding, $ heta$-projection, and Langlands parameters for E6(-14), enabling efficient classification of its unitary representations.

Findings

Reduced classification problem from 6D to 2D via root embedding

Verified rationality of $ heta$-projection algebraically and analytically

Proved injectivity and type preservation of representation correspondence

Abstract

The classification of unitary representations for the non-compact real form E6(-14) of the exceptional Lie group E6 has long been hindered by computational bottlenecks due to its complex root system (72 roots) and large Weyl group (order 51840). This paper proposes a dimensional reduction method integrating root subsystem embedding, $\theta$-weight projection, and Langlands parameterization: by embedding the non-compact root subsystem of E6(-14) into the C2 root system of Sp(4), combined with weight space quotient (eliminating compact root redundancy), the problem is reduced from 6 dimensions to 2; the rationality of $\theta$-projection is verified from both algebraic and analytic perspectives, including analytic realization on representation spaces, redundancy of compact weight sublattices, and compatibility with analytic continuation of Langlands parameters; based on Langlands parameter restriction, the injectivity of representation correspondence and type preservation for discrete series, principal series, and complementary series are proven. This framework provides an algorithmic classification tool for E6(-14) unitary representations, which can be generalized to higher-rank exceptional groups.