After reviewing geometric quantisation of linear bosonic and fermionic systems, we study the holonomy of the projectively flat connection on the bundle of Hilbert spaces over the space of compatible complex structures and relate it to the Maslov index and its various generalisations. We also consider bosonic and fermionic harmonic oscillators parametrised by compatible complex structures and compare Berry’s phase with the above holonomy.

Let $K$ be a complex reductive algebraic group and $V$ a representation of $K$. Let $S$ denote the ring of polynomials on $V$. Assume that the action of $K$ on $S$ is multiplicity-free. If $\lambda $ denotes the isomorphism class of an irreducible representation of $K$, let $\rho \lambda :K\to GL({{V}_{\lambda }})$ denote the corresponding irreducible representation and ${{S}_{\lambda }}$ the $\lambda $-isotypic component of $S$. Write ${{S}_{\lambda }}.{{S}_{\mu }}$ for the subspace of $S$ spanned by products of ${{S}_{\lambda }}$ and ${{S}_{\mu }}$. If ${{V}_{v}}$ occurs as an irreducible constituent of ${{V}_{\lambda }}\otimes {{V}_{\mu }}$, is it true that ${{S}_{v}}\subseteq {{S}_{\lambda }}.{{S}_{\mu }}?$ In this paper, the authors investigate this question for representations arising in the context of Hermitian symmetric pairs. It is shown that the answer is yes in some cases and, using an earlier result of Ruitenburg, that in the remaining classical cases, the answer is yes provided that a conjecture of Stanley on the multiplication of Jack polynomials is true. It is also shown how the conjecture connects multiplication in the ring $S$ to the usual Littlewood–Richardson rule.