We investigate (twisted) rings of differential operators on the resolution of singularities of a particular irreducible component of the (Zarisky) closure of the minimal orbit $bar O_{mathrm{min}}$ of $mathfrak{sp}_{2n}$, intersected with the Borel s
ubalgebra $mathfrak n_+$ of $mathfrak{sp}_{2n}$, using toric geometry and show that they are homomorphic images of a subalgebra of the Universal Enveloping Algebra (UEA) of $mathfrak{sp}_{2n}$, which contains the maximal parabolic subalgebra $mathfrak p$ determining the minimal nilpotent orbit. Further, using Fourier transforms on Weyl algebras, we show that (twisted) rings of well-suited weighted projective spaces are obtained from the same subalgebra. Finally, investigating this subalgebra from the representation-theoretical point of view, we find new primitive ideals and rediscover old ones for the UEA of $mathfrak{sp}_{2n}$ coming from the aforementioned resolution of singularities.
In this short note, we investigate some features of the space $Inject{d}{m}$ of linear injective maps from $bbR^d$ into $bbR^m$; in particular, we discuss in detail its relationship with the Stiefel manifold $V_{m,d}$, viewed, in this context, as the
set of orthonormal systems of $d$ vectors in $bbR^m$. Finally, we show that the Stiefel manifold $V_{m,d}$ is a deformation retract of $Inject{d}{m}$. One possible application of this remarkable fact lies in the study of perturbative invariants of higher-dimensional (long) knots in $bbR^m$: in fact, the existence of the aforementioned deformation retraction is the key tool for showing a vanishing lemma for configuration space integrals {`a} la Bott--Taubes (see cite{BT} for the 3-dimensional results and cite{CR1}, cite{C} for a first glimpse into higher-dimensional knot invariants).
The aim of this paper is to review and discuss in detail local aspects of principal bundles with groupoid structure. Many results, in particular from the second and third section, are already known to some extents, but, due to the lack of a ``unified
point of view on the subject, I decided nonetheless to (re)define all the main concepts and write all proofs; however, some results are reformulated in a more elegant way, using the division map and the generalized conjugation of a Lie groupoid. In the same framework, I discuss later generalized groupoids and Morita equivalences from a local point of view; in particular, I found a (so far as I know) new characterization of generalized morphisms coming from nonabelian ech cohomology, which allows one to view generalized morphisms as a generalization of classical descent data. I found also a factorization formula for the division map, which is the crucial point in the local formulation of Morita equivalences.
Motivated by the computations done in cite{C1}, where I introduced and discussed what I called the groupoid of generalized gauge transformations, viewed as a groupoid over the objects of the category $mathsf{Bun}_{G,M}$ of principal $G$-bundles over
a given manifold $M$, I develop in this paper the same ideas for the more general case of {em principal $calG$-bundles or principal bundles with structure groupoid $calG$}, where now $calG$ is a Lie groupoid in the sense of cite{Moer2}. Most of the concepts introduced in cite{C1} can be translated almost verbatim in the framework of principal bundles with structure groupoid $calG$; in particular, the key r�le for the construction of generalized gauge transformations is again played by (the equivalent in the framework of principal bundles with groupoid structure of) the division map $f_P$. Of great importance are also the generalized conjugation in a groupoid and the concept of (twisted) equivariant maps between groupoid-spaces.
The motivation for this paper stems cite{CR} from the need to construct explicit isomorphisms of (possibly nontrivial) principal $G$-bundles on the space of loops or, more generally, of paths in some manifold $M$, over which I consider a fixed princi
pal bundle $P$; the aforementioned bundles are then pull-backs of $P$ w.r.t. evaluation maps at different points. The explicit construction of these isomorphisms between pulled-back bundles relies on the notion of {em parallel transport}. I introduce and discuss extensively at this point the notion of {em generalized gauge transformation between (a priori) distinct principal $G$-bundles over the same base $M$}; one can see immediately that the parallel transport can be viewed as a generalized gauge transformation for two special kind of bundles on the space of loops or paths; at this point, it is possible to generalize the previous arguments for more general pulled-back bundles. Finally, I discuss how flatness of the reference connection, w.r.t. which I consider holonomy and parallel transport, is related to horizontality of the associated generalized gauge transformation.
