arXiv subjects

Gabriele Barbieri

Publications and source records attributed to Gabriele Barbieri.

3 recordsLinked to original sources

arXiv2026

Pseudo--K\"ahler induction on Lie groups with entire Grauert tubes

Given a closed subgroup $H\subseteq G$ and a Hamiltonian $H$-space $Y$, one can construct an induced Hamiltonian $G$-space. In this paper we investigate this construction in the setting where $Y$ is endowed with a compatible complex structure. Our aim is to develop symplectic induction in this framework and to establish the corresponding K\"ahler analogues of the classical results. The main idea is to replace the cotangent bundle $T^*G$, appearing in the standard construction of $\operatorname{Ind}$, with the Grauert tube of $G$. We focus on Lie groups admitting a globally defined Grauert tube and study the resulting pseudo--K\"ahler induction procedure.

math.SG↗

arXiv2024

Remarks on Diffeological Frobenius Reciprocity

A recent paper [R22] established "Frobenius reciprocity" as a bijection $t$ between certain symplectically reduced spaces (which need not be manifolds), and conjectured: 1{\deg}) $t$ is a diffeomorphism when these spaces are endowed with their natural subquotient diffeologies, 2{\deg}) $t$ respects the reduced diffeological $2$-forms they may (or might not) carry. In this paper, we prove both this conjecture and a similar one on prequantum reduction, and also give new sufficient conditions for the reduced forms to exist. We stop short of proving that they always exist.

math.SG↗

arXiv2023

Diffeological Symplectic Frobenius Reciprocity

In this note we prove that the symplectic Frobenius Reciprocity established in the paper "Symplectic Induction, Prequantum Induction and Prequantum Multiplicities" as a set bijection is indeed a diffeological diffeomorphism, as conjectured by its authors Ratiu and Ziegler. The same holds in the prequantum space context.

math.SG↗