Rm Sp Research Articles

Uniform distribution of orbits of lattices on spaces of frames

We study distribution of orbits of a lattice $\Gamma\subseteq\SL(n,\mathbb{R})$ in the the space $\mathcal{V}_{n,l}$ of $l$-frames in $\mathbb{R}^n$ ($1\le l\le n-1$). Examples of dense $\Gamma$-orbits are known from the work of Dani, Raghavan, and Veech. We show that dense orbits of $\Gamma$ are uniformly distributed in $\mathcal{V}_{n,l}$ with respect to an explicitly described measure. We also establish an analogous result for lattices in ${\rm Sp}(n,\mathbb{R})$ which act on the space of isotropic $n$-frames.

New Frequency-Averaged Approximations to the Equations of Radiative Heat Transfer

We develop new direction- and frequency-averaged approximations to the equations of radiative heat transfer in glass for optically thick, diffusive regimes. These approximations, which are based on the $\rm SP_N$ approach given in [E. Larsen, G. Thommes, A. Klar, M. Seaid, and T. Gotz, J. Comput. Phys. 83 (2002), pp. 652--675], represent asymptotic corrections to the familiar Rosseland, or equilibrium diffusion, approximation. Numerical results for realistic problems in the simulation of radiative heat transfer in glass cooling confirm the accuracy and efficiency of the new approximations.

BROUÉ'S CONJECTURE FOR THE HALL–JANKO GROUP AND ITS DOUBLE COVER

Broué's abelian defect conjecture suggests a deep link between the module categories of a block of a group algebra and its Brauer correspondent, viz. that they should be derived equivalent. We are able to verify Broué's conjecture for the Hall–Janko group, even its double cover $2.J_2$, as well as for $U_3(4)$ and ${\rm Sp}_4(4)$. In fact we verify Rickard's refinement to Broué's conjecture and show that the derived equivalence can be chosen to be a splendid equivalence for these examples.2000 Mathematical Subject Classification: 20C20, 20C34.

ALGÈBRES DE HECKE ET SÉRIES PRINCIPALES GÉNÉRALISÉES DE Sp4(F)

The aim of this work is to expand Bushnell and Kutzko's theory of $G$-covers [Proc. London Math. Soc. 77 (1998) 582–634] up to a full description of the generalized principal series of the $p$-adic group ${\rm Sp}_4(F)$, with $p$ odd.We start with a Levi component $M$ of a maximal parabolic subgroup $P$ of $G = {\rm Sp}_4(F)$ and an explicit type $(J_M, \tau_M)$ for the inertial class $S$ in $M$ of a supercuspidal representation of $M$. We compute the Hecke algebra of a $G$-cover $(J, \tau)$ of $(J_M, \tau_M)$ constructed in our previous work [Ann. Inst. Fourier 49 (1999) 1805–1851]: it is a convolution algebra on a Coxeter group (namely, the affine Weyl group of either $U(1,1)(F)$, in the case of the Siegel parabolic, or ${\rm SL}_2(F)$), described explicitly by generators and relations.From this and Bushnell and Kutzko's work we derive the structure of the parabolically induced representations ${\rm ind}_P^G \pi$, for $\pi$ in $S$, and we find their discrete series subrepresentations if any, thus recovering, through the theory of $G$-covers, results previously obtained by Shahidi using different methods.The paper is written in French.2000 Mathematical Subject Classification: 22E50, 11F70.

Uniform pointwise bounds for matrix coefficients of unitary representations and applications to Kazhdan constants

Let $k$ be a local field of characteristic not $2$, and let $G$ be the group of $k$-rational points of a connected reductive linear algebraic group defined over $k$ with a simple derived group of $k$-rank at least $2$. We construct new uniform pointwise bounds for the matrix coefficients of all infinite-dimensional irreducible unitary representations of $G$. These bounds turn out to be optimal for ${\rm SL}\sb n(k), n\geq 3$, and ${\rm Sp}\sb {2n}(k),n\geq 2$. As an application, we discuss a simple method of calculating Kazhdan constants for various compact subsets of semisimple $G$.

Hysteretic behavior of the vortex lattice at the onset of the second peak for theHgBa2CuO4+δsuperconductor

By means of local Hall probe ac and dc permeability measurements we investigated the phase diagram of vortex matter for the HgBa$_2$CuO$_{4+\delta }$ superconductor in the regime near the critical temperature. The second peak line, $H_{\rm sp}$, in contrast to what is usually assumed, doesn't terminate at the critical temperature. Our local ac permeability measurements revealed pronounced hysteretic behavior and thermomagnetic history effects near the onset of the second peak, giving evidence for a phase transition of vortex matter from an ordered qausilattice state to a disordered glass.

Toda versus Pfaff lattice and related polynomials

The Pfaff lattice was introduced by us in the context of a Lie algebra splitting of ${\rm gl}(\infty)$ into ${\rm sp}(\infty)$ and an algebra of lower-triangular matrices. The Pfaff lattice is equivalent to a set of bilinear identities for the wave functions, which yield the existence of a sequence of "$\tau$-functions". The latter satisfy their own set of bilinear identities, which moreover characterize them. In the semi-infinite case, the $\tau$-functions are Pfaffians, in the same way that for the Toda lattice the $\tau$-functions are Hänkel determinants; interesting examples occur in the theory of random matrices, where one considers symmetric and symplectic matrix integrals for the Pfaff lattice and Hermitian matrix integrals for the Toda lattice. There is a striking parallel between the Pfaff lattice and the Toda lattice, and even more striking, there is a map from one to the other, mapping skew-orthogonal to orthogonal polynomials. In particular, we exhibit two maps, dual to each other, mapping Hermitian matrix integrals to either symmetric matrix integrals or symplectic matrix integrals.

A certain class of Poincaré series on {${\rm Sp}\sb n$}. {II}

A certain class of Poincaré series on {${\rm Sp}\sb n$}. {II}

Errata: Principal series Whittaker functions on {${\rm Sp}(2;{\bf R})$}, II

Errata: Principal series Whittaker functions on {${\rm Sp}(2;{\bf R})$}, II

Principal series Whittaker functions on ${\rm Sp}(2;{\bf R})$, II

Principal series Whittaker functions on ${\rm Sp}(2;{\bf R})$, II

Spherical Functions of the Principal Series Representations of $Sp(2, \mathbf R)$ as Hypergeometric Functions of $\mathrm C_2$-Type

In this article we determine explicitly the systems of differential equations satisfied by spherical functions with non-trivial K-typQs of the principal series and the generalized principal series representations of Sp(2, R). Then we obtain series expansions and integral formulas of spherical functions of the generalized principal series representation. We shall define spherical functions. Let G be a real reductive Lie group and K be maximal compact subgroup of G, Po=MoA0No be a parabolic subgroup of G. Let Hn be an admissible representation of G and (r, Vr), (#, Vri) be irreducible representations of K which is contained in HK. We call elements of Horru( Vr, C7(K\G))^ C?(K\G)®KV? = C~,r(K\G/K) spherical functions of type-(#, r), where CTM(K\G) is the space of smooth sections of the homogeneous vector bundle over K\G associated to V? and V? is the contragredient representation of VT. Let 0E:Horri(ai K}(Hn, CTM(K\G}} and i^ Horrid Vr, HTC then °i is a spherical function attached to H*. There are many studies on the system of differential equations satisfied by spherical functions for 1-dimensional K-typzs. Moreover they are generalized as the Weyl group invariant commuting differential operators with continuous parameters, which are introduced by generalizing root multiplicities (cf. [DGl], [DG2], [HI], [HO], [Ko], [OO], [Opl], [Op2], [Os], [OS], [Sh]). On the other hand, there are few studies for vector-valued spherical functions. Besides, spherical functions are rarely calculated in explicit forms except for rank one cases. Therefore it is interesting to study vector-valued spherical functions of higher rank Lie group explicitly.

Smooth ${\rm Sp}(2,{\bf R})$-actions on the $4$-sphere

Smooth ${\rm Sp}(2,{\bf R})$-actions on the $4$-sphere

Further Results on T-Coloring and Frequency Assignment Problems

A graph coloring problem (called T-coloring) is investigated in which integers are assigned to the vertices of a graph G, under the constraint that the absolute value of the difference between integers assigned to adjacent vertices does not belong to a forbidden set (called the T-set). The T-coloring problem has applications in the radio frequency channel assignment problem. T-sets of the form $\{0,s,23,\ldots,ks\}\cup S$, where $s,k\geq 1$ and $S\supseteq\{s+1,s+2,\ldots,ks-1\}$, are considered. This set has been named the k-multiple of s set. Values of interest are the minimum cardinality, $\chi_T(G)$, and the minimum span ${\rm sp}_T(G)$, of the set of assigned integers, over all possible T-colorings of G. A greedy algorithm, which T-colors perfectly orderable graphs in $\chi_T(G)$ colors with span ${\rm sp}_T(G)$ (for a k-multiple of s set), is given.

An explicit integral representation of Whittaker functions on ${\rm Sp}(2;{\bf R})$ for the large discrete series representations

An explicit integral representation of Whittaker functions on ${\rm Sp}(2;{\bf R})$ for the large discrete series representations

Cohomology of the symplectic group ${\rm Sp}(4,{\bf Z})$. II. Computations at the prime $2$.

Cohomology of the symplectic group ${\rm Sp}(4,{\bf Z})$. II. Computations at the prime $2$.

Irreducibility of certain degenerate principal series representations of ${\rm Sp}(n,\,{\bf R})$

An irreducibility theorem is proved for degenerate principal series representations of ${\text {Sp}}(n,{\mathbf {R}})$ induced from unitary characters of a certain maximal parabolic subgroup.

Smooth ${\rm Sp}(n)$-actions on cohomology quaternion projective spaces

Smooth ${\rm Sp}(n)$-actions on cohomology quaternion projective spaces

Viscosity of heparin as a function of dielectric constant and of desulfation.

AbstractThe effect of dielectric constant (D) of the solvent on the viscosity of heparin was examined using the relation \documentclass{article}\pagestyle{empty}\begin{document}$ \eta _{{\rm sp}} /c = [\eta ]_\infty (1 + k/\sqrt c) $\end{document}, where [η]∞ is the shielded intrinsic viscosity obtained by extrapolating \documentclass{article}\pagestyle{empty}\begin{document}$ \eta _{{\rm sp}} /c\,{\rm vs}{\rm . }\,1/\sqrt c ) $\end{document} to infinite concentration, and k is an interaction parameter independent of the dielectric constant of the solvent. This equation was previously reported by the authors9 for describing the reduced viscosities of strong polyelectrolytes in salt‐free polar solvents. It was found that the [η]∞ of heparin increases linearly with increasing dielectric constant of the solvent whereas the k values were, within experimental error, independent of D in the range 54.7 < D < 93.2 examined. Graded hydrolysis of heparin from its acid form (heparinic acid) at 57°C resulted in samples of varying degree of desulfation with corresponding decrease in biological activity. It was found that both [η]∞ and k decrease with increasing desulfation.

Errata: ``A note on manifolds whose holonomy group is a subgroup of $s{\rm Sp}(n)\cdot {\rm Sp}(1)''$.

Errata: ``A note on manifolds whose holonomy group is a subgroup of $s{\rm Sp}(n)\cdot {\rm Sp}(1)''$.

A note on manifolds whose holonomy group is a subgroup of ${\rm Sp}(n)\cdot {\rm Sp}(1)$.

A note on manifolds whose holonomy group is a subgroup of ${\rm Sp}(n)\cdot {\rm Sp}(1)$.