Eigenfunctions and Eigenvalues on Surfaces of Revolution

Higher-order asymptotic formula for the eigenvalues of Sturm–Liouville problem with n turning points

We consider Sobolev problems with nonlocal boundary conditions associated with an action of a compact Lie group. We find a natural conditions of ellipticity of such problems, obtain the corresponding finiteness theorem, and give the index formula.

Introduction. 1. Hilbert's fifth problem and proper actions of Lie groups S. Illman. 2. Equivariant algebraic vector bundles over representations - a survey M. Masuda. 3. G-manifolds and G-vector bundles in algebraic, semialgebraic, and definable categories T. Kawakami. 4. Geometry of Finite Topological and equivariant finite topological spaces S. Kono, F. Ushitaki. 5. On the theory of homotopy representations. A survey I. Nagasaki. 6. Manifolds as fixed point sets of smooth compact Lie group actions K. Pawalowski. 7. Surgery and homotopy theory in study of the transformation groups Y. Kitada. 8. Kervaire's obstructions of free actions of finite cyclic groups on homotopy spheres Y. Kitada. 9. The Burnside ring revisited M. Morimoto. 10. Multiplicative stabilization and transformation groups S. Kwasik, R. Schultz. 11. Symmetries on manifolds, deformations and rational homotopy: a survey M. Raussen. 12. Rigidity of codimension one locally free actions of solvable Lie groups N. Tsuchiya, A. Yamakawa. 13. Smooth actions of non-compact semi-simple Lie groups F. Uchida, K. Mukoyama. 14. Hamiltonian group actions and equivariant indices T. Takakura. 15. Controlled methods in equivariant topology, a survey E. Pedersen. Index.

The theory of compact Lie transformation groups is a large and well researched area of mathematics, in which topological methods are widely used and which requires a separate survey. In this chapter we only touch upon certain problems of this theory. In §1 we shall survey the classical results of Gleason, Montgomery, Mostow, Palais, Samelson and other authors concerning the structure of of orbits of actions of compact Lie groups. We emphasize the differentiable aspects of the theory and a number of results can be formulated in a more general setting — for proper actions of not necessarily compact Lie groups. §2 is devoted to properties of differentiable invariants and almost-invariants of compact Lie groups. In §3 we consider complexifications of homogeneous spaces of compact Lie groups and factorizations of reductive algebraic groups. Surveys of many questions not considered here can be found in (Bourbaki 1982), (Bredon 1972), (Hsiang 1975), (Janich 1968), (Palais 1960), (Schultz 1984).

Let M M be a cusped hyperbolic 3 3 -manifold, e.g. a knot complement. Thurston showed that the space of deformations of its fundamental group in P G L ( 2 , C ) \mathrm {PGL}(2,\mathbf {C}) (up to conjugation) is of complex dimension the number ν \nu of cusps near the hyperbolic representation. It seems natural to ask whether some representations remain discrete after deformation. The answer is generically not. A simple reason for it lies inside the cusps: the degeneracy of the peripheral representation (i.e. representations of fundamental groups of the ν \nu peripheral tori). They indeed generically become non-discrete, except for a countable set. This last set corresponds to hyperbolic Dehn surgeries on M M , for which the peripheral representation is no more faithful. We work here in the framework of P G L ( n , C ) \mathrm {PGL}(n,\mathbf {C}) . The hyperbolic structure lifts, via the n n -dimensional irreducible representation, to a representation ρ g e o m \rho _{\mathrm {geom}} . We know from the work of Menal-Ferrer and Porti that the space of deformations of ρ geom \rho _{\textrm {geom}} has complex dimension ( n − 1 ) ν (n-1)\nu . We prove here that, unlike the P G L ( 2 ) \mathrm {PGL}(2) -case, the generic behaviour becomes the discreteness (and faithfulness) of the peripheral representation: in a neighbourhood of the geometric representation, the non-discrete peripheral representations are contained in a real analytic subvariety of codimension ≥ 1 \geq 1 .

Given a pair $(M,X)$, where $X$ is a smooth submanifold in a closed smooth manifold $M$, we study the operation, which takes each operator $D$ on the ambient manifold to a certain operator on the submanifold. The latter operator is called the trace of $D$. More precisely, we study traces of operators, associated with actions of compact Lie groups on $M$. We show that traces of such operators are localized at special submanifolds in $X$ and study the structure of the traces on these submanifolds.

The coupled magnon—Phonon modes in magnetic crystals with shubnikov symmetry

Let R be a quantum solvable algebra. It is proved that every prime ideal I of R which is stable with respect to the quantum adjoint action is completely prime, and Fract(R/I) is isomorphic to the skew field of fractions of an algebra of twisted polynomials. We study the correspondence between symplectic leaves and irreducible representations. A conjecture of De Concini-Kac--Procesi on the dimension of irreducible representations is proved for sufficiently large 1.

Slice theorem and orbit type stratification in infinite dimensions

A projection operator technique for determining selection rules

Let [Formula: see text] be a [Formula: see text]th root of unity where [Formula: see text] is odd. Let [Formula: see text] denote the quantum group with large center corresponding to the Lie algebra [Formula: see text] with generators [Formula: see text], and [Formula: see text]. A semicyclic representation of [Formula: see text] is an [Formula: see text]-dimensional irreducible representation [Formula: see text], so that [Formula: see text] with [Formula: see text], [Formula: see text] and [Formula: see text]. We construct a tangle functor for framed homogeneous tangles colored with semicyclic representations, and prove that for [Formula: see text]-tangles coming from knots, the invariant defined by the tangle functor coincides with Kashaev’s invariant.

Flavour symmetries appropriate for describing a neutrino spectrum with degenerate solar pair and a third massive or massless neutrino are discussed. We demand that the required residual symmetries of the leptonic mass matrices be subgroups of some discrete symmetry group $G_f$. $G_f$ can be a subgroup of SU(3) if the third neutrino is massive and we derive general results on the mixing angle predictions for various discrete subgroups of SU(3). The main results are: (a) All the SU(3) subgroups of type C fail in simultaneously giving correct $\theta_{13}$ and $\theta_{23}$. (b) All the groups of type D can predict a relation $\cos^2\theta_{13} \sin^2\theta_{23}=\frac{1}{3}$ among the mixing angles which appears to be a good zeroth order approximation. Among these, various $\Delta(6n^2)$ groups with $n\geq 8$ can simultaneously lead also to $\sin^2 \theta_{13}$ in agreement with global fit at 3$\sigma$. (c) The group $\Sigma(168)\cong PSL(2,7)$ predicts near to the best fit value for $\theta_{13}$ and $\theta_{23}$ within the 1$\sigma$ range. All discrete subgroups of U(3) with order $<512$ and having three dimensional irreducible representation are considered as possible $G_f$ when the third neutrino is massless. Only seven of them are shown to be viable and three of these can correctly predict $\theta_{13}$ and/or $\theta_{23}$. The solar angle remains undetermined at the leading order in all the cases due to degeneracy in the masses. A class of general perturbations which can correctly reproduce all the observables are discussed in the context of several groups which offer good leading order predictions.

The charge conjugation C and the parity P are assumed to generate a finite group without assuming any apriori commutation or anti-commutation relation. The group then generated is the di-hedral group D/sub n/ (n being even) which has two dimensional irreducible representations. This implies that we could have in principle, elementary particles with parity and charge-conjugation doublet structure.

Higher order asymptotic distribution of the eigenvalues of nondefinite Sturm–Liouville problems with one turning point

Suppose a locally compact group G acts on a separable metrizable space M, preserving a finite measure #. Then there is an associated unitary representation of G on LZ(M,/~)o, the square integrable functions orthogonal to the constants. This representation will in general be far from irreducible, and in fact, when G is not compact need not have any irreducible subrepresentations. Thus, in the decomposition of LZ(G)o into irreducible constituents, one not only requires a direct integral rather than a direct sum, but the associated spectral measure on the unitary dual G (say for type I groups) will, except in special circumstances, have no atoms. The point of this paper is to give natural and general conditions, mostly involving just rq(M), under which there will be an infinite atomic spectrum. (I.e., the spectral measure has infinitely many atoms.) The basic example of actions in which the spectrum is purely atomic, i.e., when the representation is a direct sum of irreducible representations, is the case of M = G/F, where F c G is a cocompact discrete subgroup. In this case, assuming 9 G to be simply connected, we have 7tl(M ) = F. The first results of this paper show that when G is a simple Lie group with R-rank(G)=> 2, we can deduce that any action of G on a compact manifold M with fundamental group isomorphic to such a F, and satisfying a natural hypothesis of either a geometric or ergodic theoretic nature, must also have an infinite atomic spectrum. In fact, under certain natural hypotheses the same is true as long as there is a surjection of hi(M) onto F. To formulate this more precisely, let F c Ad(G) be a cocompact lattice, and let Pr be the unitary representation of Ad(G) on L2(Ad(G)/F)o . We recall that an action of a group on a manifold (or more generally on a standard space in the sense of [4]) is called engaging if there is no loss of ergodicity in passing to finite covers. (See I'4] for discussion.)