International Journal of Mathematics

If G is a reductive Lie group of Harish-Chandra class, H is a symmetric subgroup, and π is a discrete series representation of G, the authors give a condition on the pair (G, H) which guarantees that the direct integral decomposition of π|_H contains each irreducible representation of H with finite multiplicity. In addition, if G is a reductive Lie group of Harish-Chandra class, and H ⊂ G is a closed, reductive subgroup of Harish-Chandra class, the authors show that the multiplicity function in the direct integral decomposition of π|_H is constant along "continuous parameters". In obtaining these results, the authors develop a new technique for studying multiplicities in the restriction π|_H via convolution with Harish-Chandra characters. This technique has the advantage of being useful for studying the continuous spectrum as well as the discrete spectrum.

We classify real hypersurfaces with isometric Reeb flow in the complex quadrics Q^m = SO_m+2/SO_mSO₂, m ≥ 3. We show that m is even, say m = 2k, and any such hypersurface is an open part of a tube around a k-dimensional complex projective space ℂP^k which is embedded canonically in Q^2k as a totally geodesic complex submanifold. As a consequence, we get the non-existence of real hypersurfaces with isometric Reeb flow in odd-dimensional complex quadrics Q^2k+1, k ≥ 1. To our knowledge the odd-dimensional complex quadrics are the first examples of homogeneous Kähler manifolds which do not admit a real hypersurface with isometric Reeb flow.

We discuss positive closed currents and Fubini–Study currents on orbifolds, as well as Bergman kernels of singular Hermitian orbifold line bundles. We prove that the Fubini–Study currents associated to high powers of a semipositive singular line bundle converge weakly to the curvature current on the set where the curvature is strictly positive, generalizing a well-known theorem of Tian. We include applications to the asymptotic distribution of zeros of random holomorphic sections.

Given a 4-manifold X and an imbedding T² × B² ⊂ X, we describe an algorithm X ↦ X_p,q for drawing the handlebody of the 4-manifold obtained from X by (p, q)-logarithmic transforms along the parallel tori. By using this algorithm for relatively prime (p, q), we obtain a simple handle picture of the Dolgachev surface E(1)_p,q, from that we deduce that the exotic copy of differs from the original one by a codimension zero simply connected Stein submanifold M_p,q. This gives examples of infinitely many small Stein manifolds which are exotic copies of each other (rel boundaries). Also by using the description of S² × S² as a union of two cusps glued along their boundaries, and by using this algorithm, we show that a pair of log transforms along the tori in these cusps gives back S² × S² or .

The paper discusses nonautonomous discrete Lotka–Volterra type n-species competitive systems with pure-delays and feedback controls. New sufficient conditions for which a part of the n-species remains permanent and others is driven to extinction are established by using the method of multiple discrete Lyapunov functionals and introducing new analysis technique. Our results show that the feedback controls cannot influence the permanence of species.

We compute Gromov–Witten invariants of any genus for Del Pezzo surfaces of degree ≥ 2. The genus zero invariants have been computed a long ago [P. Di Francesco and C. Itzykson, Quantum intersection rings, in The Moduli Space of Curves, eds. R. Dijkgraaf et al., Progress in Mathematics, Vol. 129 (Birkhäuser, Boston, 1995), pp. 81–148; L. Göttsche and R. Pandharipande, The quantum cohomology of blow-ups of ℙ² and enumerative geometry, J. Differential Geom.48(1) (1998) 61–90], Gromov–Witten invariants of any genus for Del Pezzo surfaces of degree ≥ 3 have been found by Vakil [Counting curves on rational surfaces, Manuscripta Math.102 (2000) 53–84]. We solve the problem in two steps: (1) we consider curves on , the plane blown up at one point, which have given degree, genus, and prescribed multiplicities at fixed generic points on a conic that avoids the blown-up point; then we obtain a Caporaso–Harris type formula counting such curves subject to arbitrary additional tangency conditions with respect to the chosen conic; as a result we count curves of any given divisor class and genus on a surface of type , the plane blown up at six points on a given conic and at one more point outside the conic; (2) in the next step, we express the Gromov–Witten invariants of via enumerative invariants of , using Vakil's extension of the Abramovich–Bertram formula.

We study homothetic vector fields on Randers manifolds of projectively isotropic flag curvature and use them to construct new Randers metrics of scalar flag curvature.

We consider the partition function of a general vertex operator algebra V on a genus two Riemann surface formed by sewing together two tori. We consider the non-trivial degeneration limit where one torus is pinched down to a Riemann sphere and show that the genus one partition function on the degenerate torus is recovered up to an explicit universal V-independent multiplicative factor raised to the power of the central charge.

The commutator length of a Hamiltonian diffeomorphism f ∈ Ham(M,ω) of a closed symplectic manifold (M,ω) is by definition the minimal k such that f can be written as a product of k commutators in Ham(M,ω). We introduce a new invariant for Hamiltonian diffeomorphisms, called the k₊-area, which measures the "distance", in a certain sense, to the subspace of all products of k commutators. Therefore, this invariant can be seen as the obstruction to writing a given Hamiltonian diffeomorphism as a product of k commutators. We also consider an infinitesimal version of the commutator problem: what is the obstruction to writing a Hamiltonian vector field as a linear combination of k Lie brackets of Hamiltonian vector fields? A natural problem related to this question is to describe explicitly, for every fixed k, the set of linear combinations of k such Lie brackets. The problem can be obviously reformulated in terms of Hamiltonians and Poisson brackets. For a given Morse function f on a symplectic Riemann surface M (verifying a weak genericity condition) we describe the linear space of commutators of the form {f, g}, with .

We initiate a study of inner amenability for a locally compact quantum group 𝔾 in the sense of Kustermans–Vaes. We show that all amenable and co-amenable locally compact quantum groups are inner amenable. We then show that inner amenability of 𝔾 is equivalent to the existence of certain functionals associated to characters on L¹(𝔾). For co-amenable locally compact quantum groups, we introduce and study strict inner amenability and its relation to the extension of the co-unit ϵ from C₀(𝔾) to L^∞(𝔾). We then obtain a number of equivalent statements describing strict inner amenability of 𝔾 and the existence of certain means on subspaces of L^∞(𝔾) such as LUC(𝔾), RUC(𝔾) and UC(𝔾). Finally, we offer a characterization of strict inner amenability in terms of a fixed point property for L¹(𝔾)-modules.