Extremal Problems for Mappings with Generalized Parametric Representation in $${{\mathbb {C}}}^{n}$$

In this paper we are concerned with the family \(\widetilde{S}^t_A(\mathbb {B}^n)\) (\(t\ge 0\)) of normalized biholomorphic mappings on the Euclidean unit ball \(\mathbb {B}^n\) in \({\mathbb {C}}^n\) that can be embedded in normal Loewner chains whose normalizations are given by time-dependent operators \(A\in \widetilde{\mathcal {A}}\), where \(\widetilde{\mathcal {A}}\) is a family of measurable mappings from \([0,\infty )\) into \(L({\mathbb {C}}^n)\) which satisfy certain natural assumptions. In particular, we consider extreme points and support points associated with the compact family \(\widetilde{S}^t_A(\mathbb {B}^n)\), where \(A\in \widetilde{\mathcal {A}}\). We prove that if \(f(z,t)=V(t)^{-1}z+\cdots \) is a normal Loewner chain such that \(V(s)f(\cdot ,s)\in \mathrm{ex}\,\widetilde{S}^s_A(\mathbb {B}^n)\) (resp. \(V(s)f(\cdot ,s)\in \mathrm{supp}\,\widetilde{S}^s_A(\mathbb {B}^n)\)), then \(V(t)f(\cdot ,t)\in \mathrm{ex}\, \widetilde{S}^t_A(\mathbb {B}^n)\), for all \(t\ge s\) (resp. \(V(t)f(\cdot ,t)\in \mathrm{supp}\,\widetilde{S}^t_A(\mathbb {B}^n)\), for all \(t\ge s\)), where V(t) is the unique solution on \([0,\infty )\) of the initial value problem: \(\frac{d V}{d t}(t)=-A(t)V(t)\), a.e. \(t\ge 0\), \(V(0)=I_n\). Also, we obtain an example of a bounded support point for the family \(\widetilde{S}_A^t(\mathbb {B}^2)\), where \(A\in \widetilde{\mathcal {A}}\) is a certain time-dependent operator. We also consider the notion of a reachable family with respect to time-dependent linear operators \(A\in \widetilde{\mathcal {A}}\), and obtain characterizations of extreme/support points associated with these families of bounded biholomorphic mappings on \(\mathbb {B}^n\). Useful examples and applications yield that the study of the family \(\widetilde{S}^t_A(\mathbb {B}^n)\) for time-dependent operators \(A\in \widetilde{\mathcal {A}}\) is basically different from that in the case of constant time-dependent linear operators.     

Bavrin’s Type Factorization of the Temljakov Operator for Holomorphic Functions in Circular Domains of $$\mathbb {C}^{n}$$

The paper concerns investigations of holomorphic functions of several complex variables with a factorization of their Temljakov transform. Firstly, there were considered some inclusions between the families \(\mathcal {C}_{\mathcal {G}},\mathcal {M}_{\mathcal {G}},\mathcal {N}_{\mathcal {G}},\mathcal {R}_{\mathcal {G}},\mathcal {V}_{\mathcal {G}}\) of such holomorphic functions on complete n-circular domain \(\mathcal {G}\) of \(\mathbb {C}^{n}\) in some papers of Bavrin, Fukui, Higuchi, Michiwaki. A motivation of our investigations is a condensation of the mentioned inclusions by some new families of Bavrin’s type. Hence we consider some families \(\mathcal {K}_{ \mathcal {G}}^{k},k\ge 2,\) of holomorphic functions f :  \(\mathcal {G}\rightarrow \mathbb {C},f(0)=1,\) defined also by a factorization of \( \mathcal {L}f\) onto factors from \(\mathcal {C}_{\mathcal {G}}\) and \(\mathcal {M} _{\mathcal {G}}.\) We present some interesting properties and extremal problems on \(\mathcal {K}_{\mathcal {G}}^{k}\).     

Positive Systems of Kostant Roots

Let \(\mathfrak {g}\) be a simple complex Lie algebra and let \(\mathfrak {t} \subset \mathfrak {g}\) be a toral subalgebra of \(\mathfrak {g}\). As a \(\mathfrak {t}\)-module \(\mathfrak {g}\) decomposes as

$$\mathfrak{g} = \mathfrak{s} \oplus \left( \oplus_{\nu \in \mathcal{R}}~ \mathfrak{g}^{\nu}\right)$$

where \(\mathfrak {s} \subset \mathfrak {g}\) is the reductive part of a parabolic subalgebra of \(\mathfrak {g}\) and \(\mathcal {R}\) is the Kostant root system associated to \(\mathfrak {t}\). When \(\mathfrak {t}\) is a Cartan subalgebra of \(\mathfrak {g}\) the decomposition above is nothing but the root decomposition of \(\mathfrak {g}\) with respect to \(\mathfrak {t}\); in general the properties of \(\mathcal {R}\) resemble the properties of usual root systems. In this note we study the following problem: “Given a subset \(\mathcal {S} \subset \mathcal {R}\), is there a parabolic subalgebra \(\mathfrak {p}\) of \(\mathfrak {g}\) containing \(\mathcal {M} = \oplus _{\nu \in \mathcal {S}} \mathfrak {g}^{\nu }\) and whose reductive part equals \(\mathfrak {s}\)?”. Our main results is that, for a classical simple Lie algebra \(\mathfrak {g}\) and a saturated \(\mathcal {S} \subset \mathcal {R}\), the condition \((\text {Sym}^{\cdot }(\mathcal {M}))^{\mathfrak {s}} = \mathbb {C}\) is necessary and sufficient for the existence of such a \(\mathfrak {p}\). In contrast, we show that this statement is no longer true for the exceptional Lie algebras F₄,E₆,E₇, and E₈. Finally, we discuss the problem in the case when \(\mathcal {S}\) is not saturated.

     

Largest initial segments pointwise fixed by automorphisms of models of set theory

Given a model \(\mathcal {M}\) of set theory, and a nontrivial automorphism j of \(\mathcal {M}\), let \(\mathcal {I}_{\mathrm {fix}}(j)\) be the submodel of \(\mathcal {M}\) whose universe consists of elements m of \(\mathcal {M}\) such that \(j(x)=x\) for every x in the transitive closure of m (where the transitive closure of m is computed within \(\mathcal {M}\)). Here we study the class \(\mathcal {C}\) of structures of the form \(\mathcal {I}_{\mathrm {fix}}(j)\), where the ambient model \(\mathcal {M}\) satisfies a frugal yet robust fragment of \(\mathrm {ZFC}\) known as \(\mathrm {MOST}\), and \(j(m)=m\) whenever m is a finite ordinal in the sense of \(\mathcal {M}.\) Our main achievement is the calculation of the theory of \(\mathcal {C}\) as precisely \(\mathrm {MOST+\Delta }_{0}^{\mathcal {P}}\)-\(\mathrm {Collection}\). The following theorems encapsulate our principal results: Theorem A. Every structure in \(\mathcal {C}\) satisfies \(\mathrm {MOST+\Delta }_{0}^{\mathcal {P}}\)-\(\mathrm { Collection}\). Theorem B. Each of the following three conditions is sufficient for a countable structure \(\mathcal {N}\) to be in \(\mathcal {C}\):(a) \(\mathcal {N}\) is a transitive model of \(\mathrm {MOST+\Delta }_{0}^{\mathcal {P}}\)-\(\mathrm {Collection}\).(b) \(\mathcal {N}\) is a recursively saturated model of \(\mathrm {MOST+\Delta }_{0}^{\mathcal {P}}\)-\(\mathrm {Collection}\).(c) \(\mathcal {N}\) is a model of \(\mathrm {ZFC}\). Theorem C. Suppose \(\mathcal {M}\) is a countable recursively saturated model of \(\mathrm {ZFC}\) and I is a proper initial segment of \(\mathrm {Ord}^{\mathcal {M}}\) that is closed under exponentiation and contains \(\omega ^\mathcal {M}\) . There is a group embedding \(j\longmapsto \check{j}\) from \(\mathrm {Aut}(\mathbb {Q})\) into \(\mathrm {Aut}(\mathcal {M})\) such that I is the longest initial segment of \(\mathrm {Ord}^{\mathcal {M}}\) that is pointwise fixed by \(\check{j}\) for every nontrivial \(j\in \mathrm {Aut}(\mathbb {Q}).\) In Theorem C, \(\mathrm {Aut}(X)\) is the group of automorphisms of the structure X, and \(\mathbb {Q}\) is the ordered set of rationals.     

Inflation of Finite Lattices Along All-or-nothing Sets

We introduce a new generalization of Alan Day’s doubling construction. For ordered sets \(\mathcal {L}\) and \(\mathcal {K}\) and a subset \(E \subseteq \ \leq _{\mathcal {L}}\) we define the ordered set \(\mathcal {L} \star _{E} \mathcal {K}\) arising from inflation of \(\mathcal {L}\) along E by \(\mathcal {K}\). Under the restriction that \(\mathcal {L}\) and \(\mathcal {K}\) are finite lattices, we find those subsets \(E \subseteq \ \leq _{\mathcal {L}}\) such that the ordered set \(\mathcal {L} \star _{E} \mathcal {K}\) is a lattice. Finite lattices that can be constructed in this way are classified in terms of their congruence lattices.A finite lattice is binary cut-through codable if and only if there exists a 0?1 spanning chain \(\left \{\theta _{i}\colon 0 \leq i \leq n \right \}\) in \(Con(\mathcal {L})\) such that the cardinality of the largest block of ?? _i /?? _i?1 is 2 for every i with 1≤i≤n. These are exactly the lattices that can be constructed by inflation from the 1-element lattice using only the 2-element lattice. We investigate the structure of binary cut-through codable lattices and describe an infinite class of lattices that generate binary cut-through codable varieties.     

A Law of the Iterated Logarithm for Directed Last Passage Percolation

Let \({\widetilde{H}}_N\), \(N \ge 1\), be the point-to-point last passage times of directed percolation on rectangles \([(1,1), ([\gamma N], N)]\) in \({\mathbb {N}}\times {\mathbb {N}}\) over exponential or geometric independent random variables, rescaled to converge to the Tracy–Widom distribution. It is proved that for some \(\alpha _{\sup } >0\),

$$\begin{aligned} \alpha _{\sup } \, \le \, \limsup _{N \rightarrow \infty } \frac{{\widetilde{H}}_N}{(\log \log N)^{2/3}} \, \le \, \Big ( \frac{3}{4} \Big )^{2/3} \end{aligned}$$

with probability one, and that \(\alpha _{\sup } = \big ( \frac{3}{4} \big )^{2/3}\) provided a commonly believed tail bound holds. The result is in contrast with the normalization \((\log N)^{2/3}\) for the largest eigenvalue of a GUE matrix recently put forward by E. Paquette and O. Zeitouni. The proof relies on sharp tail bounds and superadditivity, close to the standard law of the iterated logarithm. A weaker result on the liminf with speed \((\log \log N)^{1/3}\) is also discussed.

     

Almost sure limit theorems of extremes of complete and incomplete samples of stationary sequences

Let \((X_{n}^{\ast})\) be an independent identically distributed random sequence. Let \(M_{n}^{\ast}\) and \(m_{n}^{\ast}\) denote, respectively, the maximum and minimum of \(\{X_{1}^{\ast},\cdots,X_{n}^{\ast}\}\). Suppose that some of the random variables \(X_1^{\ast},X_2^{\ast},\cdots\) can be observed and let \(\widetilde{M}_n^{\ast}\) and \(\widetilde{m}_n^{\ast}\) denote, respectively, the maximum and minimum of the observed random variables from the set \(\{X_1^{\ast},\cdots,X_n^{\ast}\}\). In this paper, we consider the asymptotic joint limiting distribution and the almost sure limit theorems related to the random vector \((\widetilde{M}_n^{\ast}, \widetilde{m}_n^{\ast}, M_n^{\ast}, m_n^{\ast})\). The results are extended to weakly dependent stationary Gaussian sequences.     

On the image,characterization, and automatic continuity of $$\varvec{(\sigma }, \varvec{\tau }$$ )-derivations

The first main theorem of this paper asserts that any \((\sigma , \tau )\)-derivation d, under certain conditions, either is a \(\sigma \)-derivation or is a scalar multiple of (\(\sigma - \tau \)), i.e. \(d = \lambda (\sigma - \tau )\) for some \(\lambda \in \mathbb {C} \backslash \{0\}\). By using this characterization, we achieve a result concerning the automatic continuity of \((\sigma , \tau \))-derivations on Banach algebras which reads as follows. Let \(\mathcal {A}\) be a unital, commutative, semi-simple Banach algebra, and let \(\sigma , \tau : \mathcal {A} \rightarrow \mathcal {A}\) be two distinct endomorphisms such that \(\varphi \sigma (\mathbf e )\) and \(\varphi \tau (\mathbf e )\) are non-zero complex numbers for all \(\varphi \in \Phi _\mathcal {A}\). If \(d : \mathcal {A} \rightarrow \mathcal {A}\) is a \((\sigma , \tau )\)-derivation such that \(\varphi d\) is a non-zero linear functional for every \(\varphi \in \Phi _\mathcal {A}\), then d is automatically continuous. As another objective of this research, we prove that if \(\mathfrak {M}\) is a commutative von Neumann algebra and \(\sigma :\mathfrak {M} \rightarrow \mathfrak {M}\) is an endomorphism, then every Jordan \(\sigma \)-derivation \(d:\mathfrak {M} \rightarrow \mathfrak {M}\) is identically zero.     

Centralizing traces with automorphisms on triangular algebras

Let \({\mathcal{T}}\) be a triangular algebra over a commutative ring \({\mathcal{R}}\), \({\xi}\) be an automorphism of \({\mathcal{T}}\) and \({\mathcal{Z}_{\xi}(\mathcal{T})}\) be the \({\xi}\)-center of \({\mathcal{T}}\). Suppose that \({\mathfrak{q}\colon \mathcal{T}\times \mathcal{T}\longrightarrow \mathcal{T}}\) is an \({\mathcal{R}}\)-bilinear mapping and that \({\mathfrak{T}_{\mathfrak{q}}\colon \mathcal{T}\longrightarrow \mathcal{T}}\) is a trace of \({\mathfrak{q}}\). The aim of this article is to describe the form of \({\mathfrak{T}_{\mathfrak{q}}}\) satisfying the commuting condition \({[\mathfrak{T}_{\mathfrak{q}}(x), x]_{\xi}=0}\) (resp. the centralizing condition \({[\mathfrak{T}_{\mathfrak{q}}(x), x]_{\xi}\in \mathcal{Z}_\xi(\mathcal{T})}\)) for all \({x\in \mathcal{T}}\). More precisely, we will consider the question of when \({\mathfrak{T}_{\mathfrak{q}}}\) satisfying the previous condition has the so-called proper form.     

Malcev Products of Monoids and Varieties of Bands

As a generalization of completely regular semigroups, which can be written as \({\mathcal{(G \circ RB) \circ S}}\) where \({\mathcal{G}}\), \({\mathcal{RB}}\) and \({\mathcal S}\) are the varieties of groups, rectangular bands and semilattices, respectively, we have replaced \({\mathcal G}\) by the class \({\mathcal M}\) of monoids. This calls for finding the structure of such semigroups, and, as a first step, characterizations.     

Symmetric Pairs and Self-Adjoint Extensions of Operators,with Applications to Energy Networks

We provide a streamlined construction of the Friedrichs extension of a densely-defined self-adjoint and semibounded operator A on a Hilbert space \(\mathcal {H}\), by means of a symmetric pair of operators. A symmetric pair is comprised of densely defined operators \(J: \mathcal {H}_1 \rightarrow \mathcal {H}_2\) and \(K: \mathcal {H}_2 \rightarrow \mathcal {H}_1\) which are compatible in a certain sense. With the appropriate definitions of \(\mathcal {H}_1\) and J in terms of A and \(\mathcal {H}\), we show that \((\textit{JJ}^\star )^{-1}\) is the Friedrichs extension of A. Furthermore, we use related ideas (including the notion of unbounded containment) to construct a generalization of the construction of the Krein extension of A as laid out in a previous paper of the authors. These results are applied to the study of the graph Laplacian on infinite networks, in relation to the Hilbert spaces \(\ell ^2(G)\) and \(\mathcal {H}_{\mathcal {E}}\) (the energy space).     

Absolute continuity and singularity of Palm measures of the Ginibre point process

We prove a dichotomy between absolute continuity and singularity of the Ginibre point process \(\mathsf {G}\) and its reduced Palm measures \(\{\mathsf {G}_{\mathbf {x}}, \mathbf {x} \in \mathbb {C}^{\ell }, \ell = 0,1,2\ldots \}\), namely, reduced Palm measures \(\mathsf {G}_{\mathbf {x}}\) and \(\mathsf {G}_{\mathbf {y}}\) for \(\mathbf {x} \in \mathbb {C}^{\ell }\) and \(\mathbf {y} \in \mathbb {C}^{n}\) are mutually absolutely continuous if and only if \(\ell = n\); they are singular each other if and only if \(\ell \not = n\). Furthermore, we give an explicit expression of the Radon–Nikodym density \(d\mathsf {G}_{\mathbf {x}}/d \mathsf {G}_{\mathbf {y}}\) for \(\mathbf {x}, \mathbf {y} \in \mathbb {C}^{\ell }\).     

Representations of groups with CAT(0) fixed point property

We show that certain representations over fields with positive characteristic of groups having CAT\((0)\) fixed point property \(\mathrm{F}\mathcal {B}_{\widetilde{A}_n}\) have finite image. In particular, we obtain rigidity results for representations of the following groups: the special linear group over \({\mathbb {Z}}\), \({\mathrm{SL}}_k({\mathbb {Z}})\), the special automorphism group of a free group, \(\mathrm{SAut}(F_k)\), the mapping class group of a closed orientable surface, \(\mathrm{Mod}(\Sigma _g)\), and many other groups. In the case of characteristic zero, we show that low dimensional complex representations of groups having CAT\((0)\) fixed point property \(\mathrm{F}\mathcal {B}_{\widetilde{A}_n}\) have finite image if they always have compact closure.     

An extension of Wilf’s conjecture to affine semigroups

Let \(\mathcal {C}\subset \mathbb {Q}^p_+\) be a rational cone. An affine semigroup \(S\subset \mathcal {C}\) is a \(\mathcal {C}\)-semigroup whenever \((\mathcal {C}\setminus S)\cap \mathbb {N}^p\) has only a finite number of elements. In this work, we study the tree of \(\mathcal {C}\)-semigroups, give a method to generate it and study the \(\mathcal {C}\)-semigroups with minimal embedding dimension. We extend Wilf’s conjecture for numerical semigroups to \(\mathcal {C}\)-semigroups and give some families of \(\mathcal {C}\)-semigroups fulfilling the extended conjecture. Other conjectures formulated for numerical semigroups are also studied for \(\mathcal {C}\)-semigroups.     

The Finite Hankel Transform Operator: Some Explicit and Local Estimates of the Eigenfunctions and Eigenvalues Decay Rates

For fixed real numbers \(c>0,\)\(\alpha >-\frac{1}{2},\) the finite Hankel transform operator, denoted by \(\mathcal {H}_c^{\alpha }\) is given by the integral operator defined on \(L^2(0,1)\) with kernel \(K_{\alpha }(x,y)= \sqrt{c xy} J_{\alpha }(cxy).\) To the operator \(\mathcal {H}_c^{\alpha },\) we associate a positive, self-adjoint compact integral operator \(\mathcal Q_c^{\alpha }=c\, \mathcal {H}_c^{\alpha }\, \mathcal {H}_c^{\alpha }.\) Note that the integral operators \(\mathcal {H}_c^{\alpha }\) and \(\mathcal Q_c^{\alpha }\) commute with a Sturm-Liouville differential operator \(\mathcal D_c^{\alpha }.\) In this paper, we first give some useful estimates and bounds of the eigenfunctions \(\varphi ^{(\alpha )}_{n,c}\) of \(\mathcal H_c^{\alpha }\) or \(\mathcal Q_c^{\alpha }.\) These estimates and bounds are obtained by using some special techniques from the theory of Sturm-Liouville operators, that we apply to the differential operator \(\mathcal D_c^{\alpha }.\) If \((\mu _{n,\alpha }(c))_n\) and \(\lambda _{n,\alpha }(c)=c\, |\mu _{n,\alpha }(c)|^2\) denote the infinite and countable sequence of the eigenvalues of the operators \(\mathcal {H}_c^{(\alpha )}\) and \(\mathcal Q_c^{\alpha },\) arranged in the decreasing order of their magnitude, then we show an unexpected result that for a given integer \(n\ge 0,\)\(\lambda _{n,\alpha }(c)\) is decreasing with respect to the parameter \(\alpha .\) As a consequence, we show that for \(\alpha \ge \frac{1}{2},\) the \(\lambda _{n,\alpha }(c)\) and the \(\mu _{n,\alpha }(c)\) have a super-exponential decay rate. Also, we give a lower decay rate of these eigenvalues. As it will be seen, the previous results are essential tools for the analysis of a spectral approximation scheme based on the eigenfunctions of the finite Hankel transform operator. Some numerical examples will be provided to illustrate the results of this work.     

Branching Rules for Finite-Dimensional $\mathcal {U}_{q}(\mathfrak {su}(3))$-Representations with Respect to a Right Coideal Subalgebra

We consider the quantum symmetric pair \((\mathcal {U}_{q}(\mathfrak {su}(3)), \mathcal {B})\) where \(\mathcal {B}\) is a right coideal subalgebra. We prove that all finite-dimensional irreducible representations of \(\mathcal {B}\) are weight representations and are characterised by their highest weight and dimension. We show that the restriction of a finite-dimensional irreducible representation of \(\mathcal {U}_{q}(\mathfrak {su}(3))\) to \(\mathcal {B}\) decomposes multiplicity free into irreducible representations of \(\mathcal {B}\). Furthermore we give explicit expressions for the highest weight vectors in this decomposition in terms of dual q-Krawtchouk polynomials.     

Conformally Covariant Differential Operators for the Diagonal Action of $$O(p,\,q)$$ on Real Quadrics

Let \(X=G/P\) be a real projective quadric, where \(G=O(p,\,q)\) and P is a parabolic subgroup of G. Let \((\pi _{\lambda ,\epsilon },\, \mathcal H_{\lambda ,\epsilon })_{ (\lambda ,\epsilon )\in {\mathbb {C}}\times \{\pm \}}\) be the family of (smooth) representations of G induced from the characters of P. For \((\lambda ,\, \epsilon ),\, (\mu ,\, \eta )\in {\mathbb {C}}\times \{\pm \},\) a differential operator \(\mathbf D_{(\mu ,\eta )}^\mathrm{reg}\) on \(X\times X,\) acting G-covariantly from \({\mathcal {H}}_{\lambda ,\epsilon } \otimes {\mathcal {H}}_{\mu , \eta }\) into \({\mathcal {H}}_{\lambda +1,-\epsilon } \otimes {\mathcal {H}}_{\mu +1, -\eta }\) is constructed.     

Tempered representations of <Emphasis Type="Italic">p</Emphasis>-adic groups: special idempotents and topology

Let \(G=G(k)\) be a connected reductive group over a p-adic field k. The smooth (and tempered) complex representations of G can be considered as the nondegenerate modules over the Hecke algebra \({\mathcal {H}}={\mathcal {H}}(G)\) and the Schwartz algebra \({\mathcal {S}}={\mathcal {S}}(G)\) forming abelian categories \({\mathcal {M}}(G)\) and \({\mathcal {M}}^t(G)\), respectively. Idempotents \(e\in {\mathcal {H}}\) or \({\mathcal {S}}\) define full subcategories \({\mathcal {M}}_e(G)= \{V : {\mathcal {H}}eV=V\}\) and \({\mathcal {M}}_e^t(G)= \{V : {\mathcal {S}}eV=V\}\). Such an e is said to be special (in \({\mathcal {H}}\) or \({\mathcal {S}}\)) if the corresponding subcategory is abelian. Parallel to Bernstein’s result for \(e\in {\mathcal {H}}\) we will prove that, for special \(e \in {\mathcal {S}}\), \({\mathcal {M}}_e^t(G) = \prod _{\Theta \in \theta _e} {\mathcal {M}}^t(\Theta )\) is a finite direct product of component categories \({\mathcal {M}}^t(\Theta )\), now referring to connected components of the center of \({\mathcal {S}}\). A special \(e\in {\mathcal {H}}\) will be also special in \({\mathcal {S}}\), but idempotents \(e\in {\mathcal {H}}\) not being special can become special in \({\mathcal {S}}\). To obtain conditions we consider the sets \(\mathrm{Irr}^t(G) \subset \mathrm{Irr}(G)\) of (tempered) smooth irreducible representations of G, and we view \(\mathrm{Irr}(G)\) as a topological space for the Jacobson topology defined by the algebra \({\mathcal {H}}\). We use this topology to introduce a preorder on the connected components of \(\mathrm{Irr}^t(G)\). Then we prove that, for an idempotent \(e \in {\mathcal {H}}\) which becomes special in \({\mathcal {S}}\), its support \(\theta _e\) must be saturated with respect to that preorder. We further analyze the above decomposition of \({\mathcal {M}}_e^t(G)\) in the case where G is k-split with connected center and where \(e = e_J \in {\mathcal {H}}\) is the Iwahori idempotent. Here we can use work of Kazhdan and Lusztig to relate our preorder on the support \(\theta _{e_J}\) to the reverse of the natural partial order on the unipotent classes in G. We finish by explicitly computing the case \(G=GL_n\), where \(\theta _{e_J}\) identifies with the set of partitions of n. Surprisingly our preorder (which is a partial order now) is strictly coarser than the reverse of the dominance order on partitions.     

Infra-nilmanifolds modeled on the group of uni-triangular matrices

Let \({\mathcal {N}}_m\) be the group of \(m\times m\) upper triangular real matrices with all the diagonal entries 1. Then it is an \((m-1)\)-step nilpotent Lie group, diffeomorphic to \({\mathbb {R}}^{\frac{1}{2} m(m-1)}\). It contains all the integer matrices as a lattice \(\Gamma _m\). The automorphism group of \({\mathcal {N}}_m \ (m\ge 4)\) turns out to be extremely small. In fact, \(\mathrm {Aut}({\mathcal {N}})=\mathcal {I} \rtimes \mathrm {Out}({\mathcal {N}})\), where \(\mathcal {I}\) is a connected, simply connected nilpotent Lie group, and \(\mathrm {Out}({\mathcal {N}})={{\tilde{K}}}={(\mathbb {R}^*)^{m-1}\rtimes \mathbb {Z}_2}\). With a nice left-invariant Riemannian metric on \({\mathcal {N}}\), the isometry group is \(\mathrm {Isom}({\mathcal {N}})= {\mathcal {N}} \rtimes K\), where \(K={(\mathbb {Z}_2)^{m-1}\rtimes \mathbb {Z}_2}\subset {{\tilde{K}}}\) is a maximal compact subgroup of \(\mathrm {Aut}({\mathcal {N}})\). We prove that, for odd \(m\ge 4\), there is no infra-nilmanifold which is essentially covered by the nilmanifold \(\Gamma _m\backslash {\mathcal {N}}_m\). For \(m=2n\ge 4\) (even), there is a unique infra-nilmanifold which is essentially (and doubly) covered by the nilmanifold \(\Gamma _m\backslash {\mathcal {N}}_m\).     

The canonical projection associated with certain possibly infinite generalized iterated function systems as a fixed point

In this paper, influenced by the ideas from Mihail (Fixed Point Theory Appl 2015:15, 2015), we associate to every generalized iterated function system \(\mathcal {F}\) (of order m) an operator \(H_{\mathcal {F}}:\mathcal {C} ^{m}\rightarrow \mathcal {C}\), where \(\mathcal {C}\) stands for the space of continuous functions from the shift space on the metric space corresponding to the system. We provide sufficient conditions (on the constitutive functions of \(\mathcal {F}\)) for the operator \(H_{\mathcal {F}}\) to be continuous, contraction, \(\varphi \)-contraction, Meir–Keeler or contractive. We also give sufficient condition under which \(H_{\mathcal {F}}\) has a unique fixed point \(\pi _{0}\). Moreover, we prove that, under these circumstances, the closure of the imagine of \(\pi _{0}\) is the attractor of \(\mathcal {F}\) and that \(\pi _{0}\) is the canonical projection associated with \(\mathcal {F}\). In this way we give a partial answer to the open problem raised on the last paragraph of the above-mentioned Mihail’s paper.