Critères d'induction et de coinduction pour certains anneaux d'opérateurs différentiels

RésuméSoit L une algèbre de Lie opérant par dérivations sur un anneau local commutatif noethérien (R,M,K=R/M), et soit V l'anneau des opérateurs différentiels construits à partir de R et L. Posons L(M)={d∈L/d(M)⊂M} et L0 = L(M)/ML: L0 est une algèbre de Lie qui opère sur K par dérivations et l'on peut construire un anneau d'opérateurs différentiels sur K à l'aide de L0, noté V0. Grâce au (V0 − V)-bimodule V/MV on définit l'induction (resp. la coinduction) de V0 à V par IndVv0=−⊗v0V/MV (resp. CoindVv0=Homv0(V/MV,−)) et on donne un critère pour qu'un V-module soit induit (resp. coinduit) à partir de V0. Ces résultats sont des analogues de ceux, établis par Mackey pour les groupes de Lie et par Blattner pour les algèbres de Lie, analogues, basés sur la notion de système d'imprimitivité.AbstractLet L be a Lie algebra acting by derivations on a commutative noetherian local ring (R,M,K=R/M) and let V be the ring of differential operators built on R and L. Defining L(M)={d∈L/d(M)⊂M} et L0 = L(M)/ML: L0 is a Lie algebra which acts on K by derivations, and we can construct a differential operators ring on K with L0, denoted by V0. With the help of the (V0 − V)-bimodule V/MV we define the induction (resp. coinduction) from V0 to V by IndVv0=−⊗v0V/MV and we give a criterion for a V-module to be induced (resp. coinduced) from V0. These results are similar to those established by Mackey for Lie groups and Blattner for Lie algebras, which are based on the notion of the system of imprimitivity

Alien Registration- Levasseur, John T. (Millinocket, Penobscot County)

https://digitalmaine.com/alien_docs/8051/thumbnail.jp

Modeling Porous Dust Grains with Ballistic Aggregates. II. Light Scattering Properties

We study the light scattering properties of random ballistic aggregates constructed in Shen et al. (Paper I). Using the discrete-dipole-approximation, we compute the scattering phase function and linear polarization for random aggregates with various sizes and porosities, and with two different compositions: 100% silicate and 50% silicate-50% graphite. We investigate the dependence of light scattering properties on wavelength, cluster size and porosity using these aggregate models. We find that while the shape of the phase function depends mainly on the size parameter of the aggregates, the linear polarization depends on both the size parameter and the porosity of the aggregates, with increasing degree of polarization as the porosity increases. Contrary to previous studies, we argue that monomer size has negligible effects on the light scattering properties of ballistic aggregates, as long as the constituent monomer is smaller than the incident wavelength up to 2*pi*a_0/lambda\sim 1.6 where a_0 is the monomer radius. Previous claims for such monomer size effects are in fact the combined effects of size parameter and porosity. Finally, we present aggregate models that can reproduce the phase function and polarization of scattered light from the AU Mic debris disk and from cometary dust, including the negative polarization observed for comets at scattering angles 160<theta<180 deg. These aggregates have moderate porosities, P\sim 0.6, and are of sub-micron-size for the debris disk case, or micron-size for the comet case.Comment: Submitted to ApJ. Scattering properties can be downloaded at http://www.astro.princeton.edu/~draine/SDJ2009.html Target geometries are at http://www.astro.princeton.edu/~draine/agglom.htm

Primitive Ideals of Cq(SL(n))

AbstractThe primitive ideals of the quantum group Cq[SL(n)] are classified in the case where q is a non-zero complex number which is not a root of unity. It is shown that the orbits in Prim Cq[SL(n)] under the action of the character group H ≅ (C*)n−1 are parameterized naturally by W × W where W is the associated Weyl group. It is shown that there is a natural one-to-one correspondence between primitive ideals of Cq[SL(n)] and symplectic leaves of the associated Poisson algebraic group SL(n, C)

The minimal nilpotent orbit, the Joseph ideal, and differential operators

AbstractFix a simple complex Lie algebra g, not of type G2, F4, or E8. Let Ōmin denote the Zariski closure of the minimal non-zero nilpotent orbit in g, and let g = n+ ⊕ h ⊕ n− be a triangular decomposition. We proveTHEOREM. (1) If g is not of type An then there exists an irreducible component X̄ of Ōmin ∩ n+ such that U(g)/Jo = D(X̄), where Jo is the Joseph ideal and D(X̄) denotes the ring of differential operators on X̄. (2) If g is of type An then for n − 2 of the n irreducible components X̄i of Ōmin∩ n+ there exist (distinct) maximal ideals Ji of U(g) such that U(g)/Ji= D(X̄i)

Quantum Hamiltonian Reduction for Polar Representations

Let $G$ be a reductive complex Lie group with Lie algebra $\mathfrak{g}$ and suppose that $V$ is a polar $G$-representation. We prove the existence of a radial parts map $\mathrm{rad}: \mathcal{D}(V)^G\to A_{\kappa}$ from the $G$-invariant differential operators on $V$ to the spherical subalgebra $A_{\kappa}$ of a rational Cherednik algebra. Under mild hypotheses $\mathrm{rad}$ is shown to be surjective. If $V$ is a symmetric space, then $\mathrm{rad}$ is always surjective, and we determine exactly when $A_{\kappa}$ is a simple ring. When $A_{\kappa}$ is simple, we also show that the kernel of $\mathrm{rad}$ is $\left(\mathcal{D}(V)\tau(\mathfrak{g}\right)^G$, where $\tau:\mathfrak{g}\to \mathcal{D}(V)$ is the differential of the $G$-action.Comment: 59 pages; minor typos and references update

Interactions entre les structures d'échappement et les structures à grande échelle dans l'écoulement turbulent des rivières à lit de graviers

Dans les rivières graveleuses, il est établi que les structures d'échappement formées dans la zone de recirculation à l'aval d'amas de galets génèrent d'intenses échanges turbulents. Le mécanisme responsable de l'échappement demeure par contre mal connu. Peu d'études sur la dynamique des structures d'échappement ont été réalisées dans des écoulements où le nombre de Reynolds est élevé comme c'est le cas en rivières. De plus, les connaissances actuelles ne tiennent pas compte des découvertes récentes sur la turbulence en rivière à lit de graviers où on a observé des structures de forte et de faible vitesse occupant toute la profondeur de l'écoulement et pouvant durer plusieurs secondes. Ces structures à grande échelle devraient jouer un rôle sur le mécanisme d'échappement étant donné l'influence de la vitesse ambiante sur la dynamique de la zone de recirculation. Nous rapportons les résultats de deux expériences originales sur les liens dynamiques entre les structures à grande échelle et le mécanisme d'échappement en aval d'un amas de galets. La première expérience repose sur l'analyse de corrélations croisées entre des séries de vitesses obtenues au sommet et à l'aval proximal d'un amas de galets. Les résultats montrent que les fortes fluctuations dans le sens de l'écoulement au sommet de l'obstacle sont liées, quelques instants plus tard, à de fortes fluctuations vers l'amont dans la zone de recirculation. La seconde expérience utilise la visualisation des structures d'échappement et la mesure simultanée des vitesses de l'écoulement. L'analyse combinée des images vidéo et de séries de vitesse suggère une relation entre le passage des structures à grande échelle et les manifestations de l'échappement. Ces résultats nous permettent de présenter un modèle où, lors du passage d'un front de haute vitesse, une structure d'échappement se développe et prend de l'expansion vers le lit et vers la surface en se propageant vers l'aval alors que, lors du passage d'un front de faible vitesse, elle s'élève vers la surface de manière plus cohérente. Cette étude propose un nouveau mécanisme d'échappement et révèle le rôle que joue la structure de l'écoulement ambiant sur le développement de structures dans les cours d'eau à lit graveleux.The flow structure in a gravel-bed river is closely related to the presence of protruding clasts and of pebble clusters. It is well known that shedding motions from the lee side of large clasts and clusters are a recurrent process that explains the strong exchanges of momentum in river flows. However, shedding has yet to be fully characterised for high Reynolds number flows such as those found in gravel-bed rivers. Moreover, our current understanding of shedding mechanisms does not include the recent discovery that large-scale flow structures in the form of high- and low-speed wedges occupy the entire flow depth over a gravel-bed river. From two original experiments, this paper investigates the influence of these wedges on the nature of shedding in the lee of a pebble cluster. The interactions between the large-scale wedges and shedding may be a key element for understanding flow organisation at the river reach scale. The first experiment provides an analysis of the space-time correlation of velocity time series obtained downstream from a pebble cluster in a natural river. Two pairs of one-minute time series were sampled. The first series of each pair was located in the region of flow separation downstream from the obstacle whereas the second was located at its crest. Results show that a significant negative correlation occurs with a negative time lag for the downstream velocity component. This reveals that a strong downstream velocity vector at the crest of the obstacle is followed 1 to 4 seconds later by a strong upstream velocity vector in the region of flow separation. The strength of the recirculation motion responds to the velocity fluctuations above the cluster. This is a crucial process in the development of vortex shedding. The second experiment aimed at visualising the shedding motion downstream from an obstacle. An underwater camera was used to obtain images of fluid motion in the lee of a pebble cluster while three electromagnetic current meters measured streamwise and vertical velocity fluctuations along a vertical profile downstream from the obstacle. A white tracer was injected in the region of flow separation to depict the development of flow structures that are shed into the flow. Despite the high Reynolds number of the flow, we have obtained good quality images revealing the presence of different modes of vortex shedding initiated in the region of flow separation. From the velocity records, it was possible to identify the large-scale flow wedges and to show that the type of vortex shedding is controlled by high- and low-speed wedges.Based on these results, we propose a model having two steps: when a high-speed wedge approaches the pebble cluster, the shedding motion develops vertically both towards the water surface and towards the bed as the structures convect downstream; when a low-speed wedge passes, the shedding motion advects mainly towards the surface and it conserves a stronger coherence. This response of the shedding motion to the type of flow wedge is a recurrent and fundamental phenomenon. The results and the model presented herein shed light on the complex nature of vortex shedding in flows at high Reynolds number such as those found in rivers

GAIA accuracy on radial velocities assessed from a synthetic spectra database

Spectrograph aboard the GAIA satellite operates in the near-IR, in the 8490-- 8740 \AA window accessible also from the ground. The most important parameter yet to be determined is the spectral resolution. Realistic estimates of the zodiacal light background are obtained and a total of $2\times 10^5$ correlation runs are used to study the accuracy of radial velocity measured by the spectrograph as a function of resolution, magnitude of the target, its spectral type and luminosity class. Accuracy better than 2 km/s is achievable for bright stars if a high enough dispersion is chosen. Radial velocity error of 5 km/s is at $V=17.5$ for Cepheids and at 17.7 for horizontal branch stars. Even for very faint objects, with spectra dominated by background and readout noise, the optimal dispersion is still in the 0.25 / 0.75 \AA/pix range. This is also true for complicated cases such as spectroscopic binaries or if information other than radial velocity, i.e. abundances of individual elements or stellar rotation velocity, is sought after. The results can be scaled to assess performance of future ground based instruments.Comment: accepted for publication in Astronomy and Astrophysics, 8 pages, 4 figure