Branching problems in reproducing kernel spaces

A class of irreducible unitary representations in the discrete series of SUp,q is explicitly determined in a space of holomorphic functions of three complex matrices. The discrete series, which is the set of all square integrable representations, corresponds to a compact subgroup of SUp,q. The relevant algebraic properties of the group SUp,q are discussed in detail. For a degenerate irreducible unitary representation an explicit construction of the infinitesimal generators of the Lie algebra [Formula: see text] in terms of differential operators is given.

We construct a family of pairwise commuting operators such that the Macdonald symmetric functions of infinitely many variables x 1,x 2,? and of two parameters q,t are their eigenfunctions. These operators are defined as limits at N?? of renormalized Macdonald operators acting on symmetric polynomials in the variables x 1,?,x N . They are differential operators in terms of the power sum variables $p_{n}=x_{1}^{n}+x_{2}^{n}+\cdots$ and we compute their symbols by using the Macdonald reproducing kernel. We express these symbols in terms of the Hall---Littlewood symmetric functions of the variables x 1,x 2,?. Our result also yields elementary step operators for the Macdonald symmetric functions.

Part explanation of important recent work, and part introduction to some of the techniques of modern partial differential equations, this monograph is a self-contained exposition of the Neumann problem for the Cauchy-Riemann complex and certain of its applications. The authors prove the main existence and regularity theorems in detail, assuming only a knowledge of the basic theory of differentiable manifolds and operators on Hilbert space. They discuss applications to the theory of several complex variables, examine the associated complex on the boundary, and outline other techniques relevant to these problems. In an appendix they develop the functional analysis of differential operators in terms of Sobolev spaces, to the extent it is required for the monograph.

For classes of periodic functions defined by constraints imposed on the L1-norm of the result of action of differential operators with constant coefficients and real spectrum on these functions, we determine the exact values of the best L1-approximations by generalized splines from the classes considered.

For the numerical solution of linear variational problems involving elliptic partial differential operators in n ≥ 2 space dimensions, iterative solution schemes are indispensable on account of their problem size. Our guiding principle is to devise iterative solvers which are optimal in the number of arithmetic operations, i.e., which are of linear complexity in the total number of unknowns. For these algorithms, asymptotically optimal preconditioners are required. The class of preconditioners for which this can be shown are of multilevel type, requiring nested approximation spaces to approximate the solution of the system on a fine user-specified grid. For smooth solutions of standard second and fourth order elliptic PDEs (partial differential equations) in variational form, approximations based on tensor products of higher-order B-splines yield high accuracy.For such problem classes, this survey collects the main ingredients for multilevel preconditioners in terms of higher order B-splines. There are three types of multilevel preconditioners for which asymptotic optimality can be shown. Two of them, the so-called additive preconditioners, are specified for isogeometric analysis involving linear elliptic partial differential operators in terms of variants of the BPX (Bramble-Pasciak-Xu) preconditioner and wavelet preconditioners. The third class are the so-called multiplicative preconditioners, specifically, multigrid methods.An essential ingredient for all these multilevel preconditioners are intergrid operators which transform vectors or matrices between grids of different grid spacing. For higher order B-splines, these intergrid operators can be derived from their refinement relations. In addition to a presentation of the theoretical ingredients, the performance of the different preconditioners will be demonstrated by some numerical examples.KeywordsVariational InequalityRiesz BasisBiorthogonal WaveletElliptic PDEsSpline SpaceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Division of differential operators, intertwine relations and darboux transformations

In the present paper, we investigate the underlying Stokes complex structure of the Virtual Element Method for Stokes and Navier–Stokes introduced in previous papers by the same authors, restricting our attention to the two dimensional case. We introduce a Virtual Element space {varPhi }_h subset H^2({varOmega }) and prove that the triad {{varPhi }_h, {varvec{V}}_h, Q_h} (with {varvec{V}}_h and Q_h denoting the discrete velocity and pressure spaces) is an exact Stokes complex. Furthermore, we show the computability of the associated differential operators in terms of the adopted degrees of freedom and explore also a different discretization of the convective trilinear form. The theoretical findings are supported by numerical tests.

Let Vect(ℝ) be the Lie algebra of smooth vector fields on ℝ. The space of symbols Pol(T*ℝ) admits a non-trivial deformation (given by differential operators on weighted densities) as a Vect(ℝ)-module that becomes trivial once the action is restricted to (2) ⊂ Vect(ℝ). The deformations of Pol(T*ℝ), which become trivial once the action is restricted to (2) and such that the Vect(ℝ)-action on them is expressed in terms of differential operators, are classified by the elements of the weight basis of , where denotes the differential cohomology (i.e., we consider only cochains that are given by differential operators) and where D λ,μ = Homdiff(F λ, F μ) is the space of differential operators acting on weighted densities. The main result of this paper is computation of this cohomology. In addition to relative cohomology, we exhibit 2-cocycles spanning and (2).

For a locally compact group $G$, the dual space $\hat G$ is the set of unitary equivalence classes of irreducible unitary representations equipped with the hull-kernel topology. We prove three results about $\hat G$ in the case that $G$ is a semisimple Lie group: (1) the irreducible principal series forms a Hausdorff subspace of $\hat G$; (2) the “discrete series” of square-integrable representations does in fact inherit the discrete topology from $\hat G$; (3) the topology of the reduced dual ${\hat G_r}$, that is the support of the Plancherel measure, is computed explicitly for split-rank 1 groups.

ABSTRACTThe unitary irreducible representations on a Hilbert space of the universal covering group G of the 3 + 2 deSitter group are determined, using Harish-Chandra's theory of semi-simple Lie groups. These irreducible representations of G are reduced out with respect to representations of the universal covering group K of the direct product R2 × R3 of a two-dimensional and a three-dimensional rotation group. It is convenient to distinguish between two types of irreducible representations of G, called ‘irreducible case’ and ‘reducible case’ representations. The former are more numerous in a certain sense, while the latter may be regarded as special cases.Let (j, µ) be an irreducible unitary representation of K, where j denotes the (ray) representation of R3 involved (2j is a non-negative integer) and µ that of R2 (µ is a real number). Then in an ‘irreducible case’ representation of G, each (j, µ) that occurs at all occurs with multiplicity j + ½ if 2j is odd, or j + ½ ± ½ if 2j is even. In a ‘reducible case’ representation of G, the multiplicity of the (j, µ)'s is smaller, and may even be 1 for all (j, µ)'s which occur. (The analogous problem for the 4 + 1 deSitter group has been solved by L. H. Thomas by reducing out the irreducible representations of the whole group with respect to those of the four-dimensional rotation group R4. However, each irreducible representation of the 4 + 1 deSitter group contains any irreducible representation of R4 at most once, so that the more general Harish-Chandra theory is not then needed to carry out the classification.)In an irreducible representation of G, only those irreducible representations (j, µ) of K may occur for which the fractional part of µ is fixed throughout the representation of G. This fractional part of µ helps to characterize the representations of G, as do the two polynomials in the infinitesimal elements, one of degree 2 and the other of degree 4, which generate all those polynomials which have the property of commuting with all members of the Lie algebra of G.Representations of the inhomogeneous Lorentz group may be obtained from representations of G by the process of contraction, as defined by Inonu and Wigner, just as representations of the former group may in turn be contracted to obtain representations of the Galilei group.

Using a method based on geometrical properties of homogeneous spaces of rank one homeomorphic to coset spaces of Lie groups, a series of degenerate unitary irreducible representations of the noncompact symplectic group Sp(p, q) is investigated. The representation spaces for a discrete series determined by two integer numbers and a continuous series determined by one real and one integer parameter are given, the corresponding basis functions being formed by the linear combinations of eigenfunctions of the Laplace-Beltrami operator of the considered space. Explicit formulas for the action of generators of Sp(p, q) in these representations are obtained. The results provide a deeper insight into the structure of the two-parameter ``not most degenerate'' unitary representations.

We give a general bosonic construction of oscillator-like unitary irreducible representations (UIR) of non-compact groups whose coset spaces with respect to their maximal compact subgroups are Hermitian symmetric. With the exception of E7(7), they include all the non-compact invariance groups of extended supergravity theories in four dimensions. These representations have the remarkable property that each UIR is uniquely determined by an irreducible representation of the maximal compact subgroup. We study the connection between our construction, the Hermitian symmetric spaces and the Tits-Koecher construction of the Lie algebras of corresponding groups. We then give the bosonic construction of the Lie algebra ofE 7(7) in SU(8), SO(8) and U(7) bases and study its properties. Application of our method toE 7(7) leads to reducible unitary representations.

Introduction. In applying symmetry considerations to quantum mechanics, one is often forced to consider representations T of a symmetry group G by means of Hilbert space operators which may be either unitary or conjugate-unitary. Indeed, in some situations there is a well-determined subgroup N of G (necessarily normal and of index 2 in G) such that T, must be unitary for x E N and conjugate-unitary for x E G N. Such a representation T will be called a conjugating representation of G (relative to N). More generally one can consider projective conjugating representations T, in which the homomorphism relation TT, = T, is replaced by Tj, = A(x, y)Txy (the A(x, y) being complex scalars). These also are required in the applications to quantum mechanics. In ?1, by a small modification of Mackey's analysis, we shall classify the equivalence classes of irreducible projective conjugating representations of G in terms of the irreducible projective representations of N (provided N is of Type I). As an immediate application of this, we obtain in ?2 a classification of all the irreducible unitary representations of any Type I group acting in real or quaternionic Hilbert space, in terms of those acting in complex Hilbert space. In working out this classification for a given group G, the essential step is to subdivide the self-conjugate irreducible complex representations T of G into two classes-those of real type and those of quaternionic type-according as the conjugate-linear map setting up the equivalence of T with its conjugate has positive or negative square. Exactly the same results are obtained if, instead of unitary (not necessarily finite-dimensional) representations, we consider finite-dimensional (not necessarily unitary) representations. The remaining sections of this paper concern finite-dimensional representations of semisimple groups. Fix a connected semisimple Lie group G. The family G of all finite-dimensional (not necessarily unitary) irreducible complex representations of G is parametrized by the well-known Cartan-Weyl method of dominant weights. Which of the self-conjugate elements of Gi are of real type and which are of quaternionic type? In ??3 and 4 this question is answered in two steps: First, Theorem 5 reduces the general case to the case that G is compact. It turns out that, if G is not compact, each self-conjugate element T of G gives rise to a self-conjugate element

We explain that the set of new integrable systems, generalizing the Calogero family and implied by the study of WLZZ models, which was described in arXiv:2303.05273, is only the tip of the iceberg. We provide its wide generalization and explain that it is related to commutative subalgebras (Hamiltonians) of the W1+∞ algebra. We construct many such subalgebras and explain how they look in various representations. We start from the even simpler w∞ contraction, then proceed to the one-body representation in terms of differential operators on a circle, further generalizing to matrices and in their eigenvalues, in finally to the bosonic representation in terms of time-variables. Moreover, we explain that some of the subalgebras survive the β-deformation, an intermediate step from W1+∞ to the affine Yangian. The very explicit formulas for the corresponding Hamiltonians in these cases are provided. Integrable many-body systems generalizing the rational Calogero model arise in the representation in terms of eigenvalues. Each element of W1+∞ algebra gives rise to KP/Toda τ-functions. The hidden symmetry given by the families of commuting Hamiltonians is in charge of the special, (skew) hypergeometric τ-functions among these.

Local couplings, double insertions and the Weyl consistency condition