Products of longitudinal pseudodifferential operators on flag varieties

TL;DR

This paper studies the algebra of operators on flag varieties related to simple roots of SL(n,C), showing how products of certain operators are compact, using noncommutative harmonic analysis and representation theory.

Contribution

It introduces a new algebra of operators on flag varieties and analyzes their products, revealing conditions under which these products are compact, based on representation-theoretic properties.

Findings

Product of fibrewise smoothing operators is compact if the union of their associated sets of roots is full.

The algebra contains longitudinal pseudodifferential operators of negative order.

Uses noncommutative harmonic analysis and a representation theoretic property of SU(n) subgroups.

Abstract

Associated to each set $S$ of simple roots for $SL(n,\mathbb{C})$ is an equivariant fibration $X\to X_S$ of the space $X$ of complete flags of $\mathbb{C}^n$. To each such fibration we associate an algebra $J_S$ of operators on $L^2(X)$ which contains, in particular, the longitudinal pseudodifferential operators of negative order tangent to the fibres. These form a lattice of operator ideals whose common intersection is the compact operators. As a consequence, the product of fibrewise smoothing operators (for instance) along the fibres of two such fibrations, $X\to X_S$ and $X\to X_T$, is a compact operator if $S\cup T$ is the full set of simple roots. The construction uses noncommutative harmonic analysis, and hinges upon a representation theoretic property of subgroups of SU(n), which may be described as `essential orthogonality of subrepresentations'.