  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'.
