Algebraic and analytic properties of invariant differential operators on a homogeneous space of complexity $1$

TL;DR

This paper investigates the algebraic and analytic structure of invariant differential operators on the homogeneous space SL_3(R)/A, revealing generators, relations, the center, and self-adjoint operators, including non-central examples.

Contribution

It characterizes the algebra of invariant differential operators on SL_3(R)/A, identifies its center, and finds new self-adjoint operators outside the center, a first in this context.

Findings

Described generators and relations of the algebra

Identified the center and symmetric elements as essentially self-adjoint

Found new self-adjoint operators not in the center

Abstract

Denote by $SL_3(\mathbb R)$ the special linear group of degree 3 over the real numbers, $A$ the subgroup consisting of the diagonal matrices with positive entries. In this paper, we study the algebraic and analytic properties of the invariant differential operators on the homogeneous space $SL_3(\mathbb R)/A$. Firstly, we specify the noncommutative algebra of invariant differential operators in terms of generators and their relations. Secondly, we describe the center of this algebra and prove that all of its symmetric elements are essentially self-adjoint. Thirdly, for the first time on homogeneous spaces, we identify several essentially self-adjoint invariant differential operators which do not lie in the center of the algebra of invariant differential operators.