Levi Factors and Admissible Automorphisms

Meng-Kiat Chuah,Rita Fioresi

doi:10.1007/s10468-020-10024-8

Meng-Kiat Chuah, Rita Fioresi

Open Access

https://doi.org/10.1007/s10468-020-10024-8

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Abstract

Let mathfrak {g} be a complex simple Lie algebra. We consider subalgebras mathfrak {m} which are Levi factors of parabolic subalgebras of mathfrak {g}, or equivalently mathfrak {m} is the centralizer of its center. We introduced the notion of admissible systems on finite order mathfrak {g}-automorphisms \U0001d703, and show that \U0001d703 has admissible systems if and only if its fixed point set is a Levi factor. We then use the extended Dynkin diagrams to characterize such automorphisms, and look for automorphisms of minimal order.

Highlights

Let g be a finite dimensional complex simple Lie algebra
M is called a Levi factor of parabolic subalgebra, and Kostant [9] studied the restriction of its root system to the center of m with the Borel-de
Our work establishes a correspondence between certain gradings of g and the gautomorphisms with admissible positive systems

Summary

Introduction

Let g be a finite dimensional complex simple Lie algebra. An involution θ on g leads to the decomposition g = k + p, consisting of the ±1-eigenspaces of θ. It is well-known that k has a 1-dimensional center if and only if there exists a positive system of roots such that p = p+ + p−, and k acts irreducibly on p± Such systems are called admissible and they play a key role in the construction of the holomorphic Harish-Chandra representations of a simple real group (see [2, 6]). Our work establishes a correspondence between certain gradings of g and the gautomorphisms with admissible positive systems The existence of such simple systems in the classical theory (see [6]) is crucial in the construction of the Harish-Chandra representations of a given real form GR of G, the connected group with g = Lie(G), when rank k = rank g. The above theorems allow us to study Levi factors as fixed point sets m = gθ , where the Kac diagrams of θ are relatively prime.

Levi Factors

Admissible Automorphisms

Kac Diagrams

Automorphisms of Minimal Orders

Examples

Harish-Chandra