Stepwise square integrable representations of nilpotent Lie groups

We study the conditions for a nilpotent Lie group to be foliated into subgroups that have square integrable (relative discrete series) unitary representations, that fit together to form a filtration by normal subgroups. Then we use that filtration to construct a class of “stepwise square integrable” representations on which Plancherel measure is concentrated. Further, we work out the character formulae for those stepwise square integrable representations, and we give an explicit Plancherel formula. Next, we use some structure theory to check that all these constructions and results apply to nilradicals of minimal parabolic subgroups of real reductive Lie groups. Finally, we develop multiplicity formulae for compact quotients $N/\varGamma $ where $\varGamma $ respects the filtration.     

Two-dimensional Toda field equations related to the exceptional algebra \mathfrak{g}_2 : Spectral properties of the lax operators

We analyze spectral properties of the Lax operator corresponding to the two-dimensional Toda field equations related to the algebra $\mathfrak{g}_2 $ . We construct two minimal sets of scattering data $\mathcal{T}_s $ , s = 1, 2, understanding the map between the potential and each of the sets $\mathcal{T}_s $ as a generalized Fourier transformation. We construct explicit recursion operators with special factorization properties.     

Universal Lagrangian bundles

The obstruction to construct a Lagrangian bundle over a fixed integral affine manifold was constructed by Dazord and Delzant (J Differ Geom 26:223–251, 1987) and shown to be given by ‘twisted’ cup products in Sepe (Differ Geom Appl 29(6): 787–800, 2011). This paper uses the topology of universal Lagrangian bundles, which classify Lagrangian bundles topologically [cf. Sepe in J Geom Phys 60:341–351, 2010], to reinterpret this obstruction as the vanishing of a differential on the second page of a Leray-Serre spectral sequence. Using this interpretation, it is shown that the obstruction of Dazord and Delzant depends on an important cohomological invariant of the integral affine structure on the base space, called the radiance obstruction, which was introduced by Goldman and Hirsch (Trans Am Math Soc 286(2):629–649, 1984). Some examples, related to non-degenerate singularities of completely integrable Hamiltonian systems, are discussed.     

Polytopal Affine Semigroups with Holes Deep Inside

Given a positive integer $k$ k , we construct a lattice $3$ 3 -simplex $P$ P with the following property: The affine semigroup $Q_P$ Q P associated to $P$ P is not normal, and every element $q \in \overline{Q}_P \setminus Q_P$ q ∈ Q ¯ P ? Q P has lattice distance at least $k$ k above every facet of $Q_P$ Q P .     

Sharp estimates for fully bubbling solutions of a SU(3) Toda system

In this paper, we obtain sharp estimates of fully bubbling solutions of SU(3) Toda system in a compact Riemann surface. In geometry, the SU(n?+?1) Toda system is related to holomorphic curves, harmonic maps or harmonic sequences of the Riemann surface to ${\mathbb{CP}^n}$ . In order to compute the Leray?CSchcuder degree for the Toda system, we have to obtain accurate approximations of the bubbling solutions. Our main goals in this paper are (i) to obtain a sharp convergence rate, (ii) to completely determine the locations, and (iii) to derive the ${\partial _z^2}$ condition, a unexpected and important geometric constraint.     

Relations among Henstock,McShane and Pettis integrals for multifunctions with compact convex values

Fremlin (Ill J Math 38:471–479, 1994) proved that a Banach space valued function is McShane integrable if and only if it is Henstock and Pettis integrable. In this paper we prove that the result remains valid also in case of multifunctions with compact convex values being subsets of an arbitrary Banach space (see Theorem 3.4). Di Piazza and Musia? (Monatsh Math 148:119–126, 2006) proved that if $X$ is a separable Banach space, then each Henstock integrable multifunction which takes as its values convex compact subsets of $X$ is a sum of a McShane integrable multifunction and a Henstock integrable function. Here we show that such a decomposition is true also in case of an arbitrary Banach space (see Theorem 3.3). We prove also that Henstock and McShane integrable multifunctions possess Henstock and McShane (respectively) integrable selections (see Theorem 3.1).     

A note on the reducibility of binary affine polynomials

Stickelberger–Swan Theorem is an important tool for determining parity of the number of irreducible factors of a given polynomial. Based on this theorem, we prove in this note that every affine polynomial A(x) over ${\mathbb{F}_2}$ with degree >1, where A(x) = L(x) + 1 and ${L(x)=\sum_{i=0}^{n}{x^{2^i}}}$ is a linearized polynomial over ${\mathbb{F}_2}$ , is reducible except x ² + x + 1 and x ⁴ + x + 1. We also give some explicit factors of some special affine pentanomials over ${\mathbb{F}_2}$ .     

The rank and generating set for inverse limits of wreath products of Abelian groups

We derive an exact formula for the topological rank d(W) of the inverse limit ${W = \ldots \wr A_2 \wr A_1}$ of iterated wreath products of arbitrary nontrivial finite Abelian groups. By using the language of automorphisms of a spherically homogeneous rooted tree, we construct and study a topological generating set for W with cardinality ${d(A_1) + \rho'}$ , where ${\rho'}$ is the topological rank of the profinite Abelian group ${A_2 \times A_3 \times \cdots}$ . In particular, if the group A ₁ is cyclic, this approach gives a minimal generating set for W.     

Monadic MV-algebras II: Monadic implicational subreducts

In this paper, we study the class of all monadic implicational subreducts, that is, the ${\{\rightarrow, \forall,1\}}$ -subreducts of the class of monadic MV-algebras. We prove that this class is an equational class, which we denote by ${\mathcal{ML}}$ , and we give an equational basis for this variety. An algebra in ${\mathcal{ML}}$ is called a monadic ?ukasiewicz implication algebra. We characterize the subdirectly irreducible members of ${\mathcal{ML}}$ and the congruences of every monadic ?ukasiewicz implication algebra by monadic filters. We prove that ${\mathcal{ML}}$ is generated by its finite members. Finally, we completely describe the lattice of subvarieties, and we give an equational basis for each proper subvariety.     

Extended Relativistic Toda Lattice,L-Orthogonal Polynomials and Associated Lax Pair

When a measure \(\varPsi(x)\) on the real line is subjected to the modification \(d\varPsi^{(t)}(x) = e^{-tx} d \varPsi(x)\), then the coefficients of the recurrence relation of the orthogonal polynomials in \(x\) with respect to the measure \(\varPsi^{(t)}(x)\) are known to satisfy the so-called Toda lattice formulas as functions of \(t\). In this paper we consider a modification of the form \(e^{-t(\mathfrak{p}x+ \mathfrak{q}/x)}\) of measures or, more generally, of moment functionals, associated with orthogonal L-polynomials and show that the coefficients of the recurrence relation of these L-orthogonal polynomials satisfy what we call an extended relativistic Toda lattice. Most importantly, we also establish the so called Lax pair representation associated with this extended relativistic Toda lattice. These results also cover the (ordinary) relativistic Toda lattice formulations considered in the literature by assuming either \(\mathfrak{p}=0\) or \(\mathfrak{q}=0\). However, as far as Lax pair representation is concern, no complete Lax pair representations were established before for the respective relativistic Toda lattice formulations. Some explicit examples of extended relativistic Toda lattice and Langmuir lattice are also presented. As further results, the lattice formulas that follow from the three term recurrence relations associated with kernel polynomials on the unit circle are also established.

     

On canonical metrics on Cartan–Hartogs domains

The Cartan–Hartogs domains are defined as a class of Hartogs type domains over irreducible bounded symmetric domains. The purpose of this paper is twofold. Firstly, for a Cartan–Hartogs domain \(\Omega ^{B^{d_0}}(\mu )\) endowed with the canonical metric \(g(\mu ),\) we obtain an explicit formula for the Bergman kernel of the weighted Hilbert space \(\mathcal {H}_{\alpha }\) of square integrable holomorphic functions on \(\left( \Omega ^{B^{d_0}}(\mu ), g(\mu )\right) \) with the weight \(\exp \{-\alpha \varphi \}\) (where \(\varphi \) is a globally defined Kähler potential for \(g(\mu )\) ) for \(\alpha >0\) , and, furthermore, we give an explicit expression of the Rawnsley’s \(\varepsilon \) -function expansion for \(\left( \Omega ^{B^{d_0}}(\mu ), g(\mu )\right) .\) Secondly, using the explicit expression of the Rawnsley’s \(\varepsilon \) -function expansion, we show that the coefficient \(a_2\) of the Rawnsley’s \(\varepsilon \) -function expansion for the Cartan–Hartogs domain \(\left( \Omega ^{B^{d_0}}(\mu ), g(\mu )\right) \) is constant on \(\Omega ^{B^{d_0}}(\mu )\) if and only if \(\left( \Omega ^{B^{d_0}}(\mu ), g(\mu )\right) \) is biholomorphically isometric to the complex hyperbolic space. So we give an affirmative answer to a conjecture raised by M. Zedda.     

Polytopes of Minimum Positive Semidefinite Rank

The positive semidefinite (psd) rank of a polytope is the smallest $k$ k for which the cone of $k \times k$ k × k real symmetric psd matrices admits an affine slice that projects onto the polytope. In this paper we show that the psd rank of a polytope is at least the dimension of the polytope plus one, and we characterize those polytopes whose psd rank equals this lower bound. We give several classes of polytopes that achieve the minimum possible psd rank including a complete characterization in dimensions two and three.     

Sharp regularity for certain nilpotent group actions on the interval

According to the classical Plante–Thurston Theorem, all nilpotent groups of $C^2$ -diffeomorphisms of the closed interval are Abelian. Using techniques coming from the works of Denjoy and Pixton, Farb and Franks constructed a faithful action by $C^1$ -diffeomorphisms of $[0,1]$ for every finitely-generated, torsion-free, non-Abelian nilpotent group. In this work, we give a version of this construction that is sharp in what concerns the Hölder regularity of the derivatives. Half of the proof relies on results on random paths on Heisenberg-like groups that are interesting by themselves.     

Geometric Satake, Springer correspondence and small representations

For a simply-connected simple algebraic group $G$ over $\mathbb C $ , we exhibit a subvariety of its affine Grassmannian that is closely related to the nilpotent cone of $G$ , generalizing a well-known fact about $GL_n$ . Using this variety, we construct a sheaf-theoretic functor that, when combined with the geometric Satake equivalence and the Springer correspondence, leads to a geometric explanation for a number of known facts (mostly due to Broer and Reeder) about small representations of the dual group.     

Geometry of Diffeomorphism Groups, Complete integrability and Geometric statistics

We study the geometry of the space of densities Dens(M), which is the quotient space Diff(M)/Diff _μ (M) of the diffeomorphism group of a compact manifold M by the subgroup of volume-preserving diffeomorphisms, endowed with a right-invariant homogeneous Sobolev ${\dot{H}^1}$ -metric. We construct an explicit isometry from this space to (a subset of) an infinite-dimensional sphere and show that the associated Euler–Arnold equation is a completely integrable system in any space dimension whose smooth solutions break down in finite time. We also show that the ${\dot{H}^1}$ -metric induces the Fisher–Rao metric on the space of probability distributions and its Riemannian distance is the spherical version of the Hellinger distance.     

Fundamental domains for properly discontinuous affine groups

We construct a fundamental region for the action on the \(2d+1\) -dimensional affine space of some free, discrete, properly discontinuous groups of affine transformations preserving a quadratic form of signature \((d+1, d)\) , where \(d\) is any odd positive integer.     

On Dynamical Adjoint Functor

We give an explicit formula relating the dynamical adjoint functor and dynamical twist over nonalbelian base to the invariant pairing on parabolic Verma modules. As an illustration, we give explicit $U\bigl(\mathfrak{sl}(n)\bigr)$ - and $U_\hbar\bigl(\mathfrak{sl}(n)\bigr)$ -invariant star product on projective spaces.