EXOTIC SYMMETRIC SPACES OF HIGHER LEVEL: SPRINGER CORRESPONDENCE FOR COMPLEX REFLECTION GROUPS

Let G?=?GL(V) for a 2n-dimensional vector space V, and θ an involutive automorphism of G such that H?=?G ^θ ???Sp(V). Let Open image in new window be the set of unipotent elements g ∈ G such that θ(g)?=?g ^?1. For any integer r?≥?2, we consider the variety Open image in new window , on which H acts diagonally. Let Open image in new window be a complex reflection group. In this paper, generalizing the known result for r?=?2, we show that there exists a natural bijective correspondence (Springer correspondence) between the set of irreducible representations of W _n,r and a certain set of H-equivariant simple perverse sheaves on Open image in new window . We also consider a similar problem for Open image in new window , on which G acts diagonally, where G?=?GL(V) for a finite-dimensional vector space V.     

Weighted Solyanik Estimates for the Hardy–Littlewood Maximal Operator and Embedding of into

Let Open image in new window denote a weight in Open image in new window which belongs to the Muckenhoupt class Open image in new window and let Open image in new window denote the uncentered Hardy–Littlewood maximal operator defined with respect to the measure Open image in new window . The sharp Tauberian constant of Open image in new window with respect to Open image in new window , denoted by Open image in new window , is defined by

Open image in new window

In this paper, we show that the Solyanik estimate

$$\begin{aligned} \lim _{\alpha \rightarrow 1^-}\mathsf{C}_{w}(\alpha ) = 1 \end{aligned}$$

holds. Following the classical theme of weighted norm inequalities we also consider the sharp Tauberian constants defined with respect to the usual uncentered Hardy–Littlewood maximal operator Open image in new window and a weight Open image in new window :

Open image in new window

We show that we have Open image in new window if and only if Open image in new window . As a corollary of our methods we obtain a quantitative embedding of Open image in new window into Open image in new window .

     

ON WEIGHT MODULES OF ALGEBRAS OF TWISTED DIFFERENTIAL OPERATORS ON THE PROJECTIVE SPACE

We classify blocks of categories of weight and generalized weight modules of algebras of twisted differential operators on ? ⁿ . Necessary and sufficient conditions for these blocks to be tame and proofs that some of the blocks are Koszul are provided. We also establish equivalences of categories between these blocks and categories of bounded and generalized bounded weight Open image in new window -modules in the cases of nonintegral and singular central character.     

SINGULAR CONFORMALLY INVARIANT TRILINEAR FORMS,I THE MULTIPLICITY ONE THEOREM

A normalized holomorphic family (depending on Open image in new window ∈ ?³) of conformally invariant trilinear forms on the sphere is studied. Its zero set Z is described. For Open image in new window ? Z, the multiplicity of the space of conformally invariant trilinear forms is shown to be 1.     

COTANGENT BUNDLE TO THE GRASSMANN VARIETY

We show that there is an affine Schubert variety in the infinite-dimensional partial Flag variety (associated to the two-step parabolic subgroup of the Kac–Moody group Open image in new window , corresponding to omitting α ₀; α _d ) which is a natural compactification of the cotangent bundle to the Grassmann variety.     

Asymptotic Normality of Some Graph Sequences

For a simple finite graph G denote by Open image in new window the number of ways of partitioning the vertex set of G into k non-empty independent sets (that is, into classes that span no edges of G). If \(E_n\) is the graph on n vertices with no edges then Open image in new window coincides with Open image in new window , the ordinary Stirling number of the second kind, and so we refer to Open image in new window as a graph Stirling number. Harper showed that the sequence of Stirling numbers of the second kind, and thus the graph Stirling sequence of \(E_n\), is asymptotically normal—essentially, as n grows, the histogram of Open image in new window , suitably normalized, approaches the density function of the standard normal distribution. In light of Harper’s result, it is natural to ask for which sequences \((G_n)_{n \ge 0}\) of graphs is there asymptotic normality of Open image in new window . Thanh and Galvin conjectured that if for each n, \(G_n\) is acyclic and has n vertices, then asymptotic normality occurs, and they gave a proof under the added condition that \(G_n\) has no more than \(o(\sqrt{n/\log n})\) components. Here we settle Thanh and Galvin’s conjecture in the affirmative, and significantly extend it, replacing “acyclic” in their conjecture with “co-chromatic with a quasi-threshold graph, and with negligible chromatic number”. Our proof combines old work of Navon and recent work of Engbers, Galvin and Hilyard on the normal order problem in the Weyl algebra, and work of Kahn on the matching polynomial of a graph.     

Strongly Gauduchon Spaces

The authors define strongly Gauduchon spaces and the class■■, which are generalization of strongly Gauduchon manifolds in complex spaces. Comparing with the case of Kahlerian, the strongly Gauduchon space and the class■are similar to the Kahler space and the Fujiki class■■ respectively. Some properties about these complex spaces are obtained. Moreover, the relations between the strongly Gauduchon spaces and the class■■ are studied.     

Dualities and algebras with a near-unanimity term

Let P be a finite relational structure that admits a (k +  1)-ary nearunanimity polymorphism. Then the NU Duality Theorem tells us that the algebra, whose operations are the polymorphisms of P, is dualisable with a dualising alter ego given by. We show that a more efficient alter ego can be obtained by using obstructions, as introduced by Zádori. We show that in the case that P is an ordered set (and therefore is an order-primal algebra), the duality that we obtain is strong. We close the paper by showing that if P is a finite fence, then our duality is optimal.     

Commuting variety of Witt algebra

Let \(\mathfrak{g} = W_1 \) be the Witt algebra over an algebraically closed field k of characteristic p > 3; and let Open image in new window be the commuting variety of g. In contrast with the case of classical Lie algebras of P. Levy [J. Algebra, 2002, 250: 473–484], we show that the variety Open image in new window is reducible, and not equidimensional. Irreducible components of Open image in new window and their dimensions are precisely given. As a consequence, the variety Open image in new window is not normal.     

Power boundedness in the Fourier and Fourier–Stieltjes algebras on homogeneous spaces

We study power boundedness in the Fourier and Fourier–Stieltjes algebras, Open image in new window and Open image in new window of a homogeneous space Open image in new window The main results characterizes when all elements with spectral radius at most one, in any of these algebras, are power bounded.     

Positive definite and Gram tensor complementarity problems

Given a continuous function Open image in new window and Open image in new window , the non-linear complementarity problem \(\text{ NCP }(g,q)\) is to find a vector Open image in new window such that

$$\begin{aligned} x \ge 0,~~y:=g(x) +q\ge 0~~\text{ and }~~x^Ty=0. \end{aligned}$$

We say that g has the Globally Uniquely Solvable (\(\text{ GUS }\))-property if \(\text{ NCP }(g,q)\) has a unique solution for all Open image in new window and C-property if \(\mathrm{NCP}(g,q)\) has a convex solution set for all Open image in new window . In this paper, we find a class of non-linear functions that have the \(\text{ GUS }\)-property and C-property. These functions are constructed by some special tensors which are positive semidefinite. We call these tensors as Gram tensors.

     

Geodesic mapping onto Kählerian spaces of the first kind

In the present paper a generalized Kählerian space Open image in new window of the first kind is considered as a generalized Riemannian space \(\mathbb{G}\mathbb{R}_N \) with almost complex structure F _i ^h that is covariantly constant with respect to the first kind of covariant derivative.Using a non-symmetric metric tensor we find necessary and sufficient conditions for geodesic mappings f: Open image in new window with respect to the four kinds of covariant derivatives. These conditions have the form of a closed system of partial differential equations in covariant derivatives with respect to unknown components of the metric tensor and the complex structure of the Kählerian space Open image in new window .     

QUOTIENTS OF CONIC BUNDLES

Let k be an arbitrary field of characteristic zero. In this paper we study quotients of k-rational conic bundles over Open image in new window by finite groups of automorphisms. We construct smooth minimal models for such quotients. We show that any quotient is birationally equivalent to a quotient of other k-rational conic bundle cyclic group \( {\mathrm{\mathfrak{C}}}_{2^k} \) of order 2 ^k , dihedral group \( {\mathfrak{D}}_{2^k} \) of order 2 ^k , alternating group \( {\mathfrak{A}}_4 \) of degree 4, symmetric group \( {\mathfrak{S}}_4 \) of degree 4 or alternating group \( {\mathfrak{A}}_5 \) of degree 5 effectively acting on the base of the conic bundle. Also we construct infinitely many examples of such quotients which are not k-birationally equivalent to each other.     

Categorification of tensor powers of the vector representation of $$U_q({\mathfrak {gl}}(1|1))$$

We consider the monoidal subcategory of finite-dimensional representations of \(U_q(\mathfrak {gl}(1|1))\) generated by the vector representation, and we provide a diagram calculus for the intertwining operators, which allows to compute explicitly the canonical basis. We construct then a categorification of these representations and of the action of both \(U_q(\mathfrak {gl}(1|1))\) and the intertwining operators using subquotient categories of the BGG category Open image in new window .     

Invariant and stationary measures for the action on Moduli space

We prove some ergodic-theoretic rigidity properties of the action of Open image in new window on moduli space. In particular, we show that any ergodic measure invariant under the action of the upper triangular subgroup of Open image in new window is supported on an invariant affine submanifold.The main theorems are inspired by the results of several authors on unipotent flows on homogeneous spaces, and in particular by Ratner’s seminal work.     

A weird relation between two cardinals

For a set M, let \({\text {seq}}(M)\) denote the set of all finite sequences which can be formed with elements of M, and let \([M]^2\) denote the set of all 2-element subsets of M. Furthermore, for a set A, let Open image in new window denote the cardinality of A. It will be shown that the following statement is consistent with Zermelo–Fraenkel Set Theory \(\textsf {ZF}\): There exists a set M such that Open image in new window and no function Open image in new window is finite-to-one.     

Some remarks on the ring

Some properties like factoriality, seminormality and being a Krull domain, … are studied on power series rings , and over a commutative ring A. If \(\mathbb{X}\) is an uncountable set, there is an other sub-ring of that stands strictly between and , we denote it by . In this paper, we study properties mentioned before on the ring .     

Non-Jumping Numbers for 4-Uniform Hypergraphs

Let r≥2 be an integer. A real number α ∈ [0,1) is a jump for r if for any Open image in new window >0 and any integer m, m≥r, any r-uniform graph with n>n₀( Open image in new window ,m) vertices and at least Open image in new window edges contains a subgraph with m vertices and at least Open image in new window edges, where c=c(α) does not depend on Open image in new window and m. It follows from a theorem of Erd?s, Stone and Simonovits that every α ∈ [0,1) is a jump for r=2. Erd?s asked whether the same is true for r≥3. Frankl and Rödl gave a negative answer by showing that Open image in new window is not a jump for r if r≥3 and l>2r. Following a similar approach, we give several sequences of non-jumping numbers generalizing the above result for r=4.     

Integration with Respect to Fractional Local Time with Hurst Index 1/2 < H < 1

Let Open image in new window be the weighted local time of fractional Brownian motion B ^H with Hurst index 1/2?H?Open image in new window As an application, we investigate the weighted quadratic covariation \([f\big(B^H\big),B^H]^{(W)}\) defined by

$ \left[f\big(B^H\big),B^H\right]^{(W)}_t:=\lim_{n\to \infty}2H\sum_{k=0}^{n-1} k^{2H-1}\left\{f\big(B^H_{t_{k+1}}\big)-f\big(B^H_{t_{k}}\big)\right\} \left(B^H_{t_{k+1}}-B^H_{t_{k}}\right), $

where the limit is uniform in probability and t _k ?=?kt/n. We show that it exists and

Open image in new window

provided f is of bounded p-variation with \(1\leq p<\frac{2H}{1-H}\). Moreover, we extend this result to the time-dependent case. These allow us to write the fractional Itô formula for new classes of functions.

     

Fixed points of <Emphasis Type="Italic">n</Emphasis>-valued maps on surfaces and the Wecken property—a configuration space approach

In this paper, we explore the fixed point theory of n-valued maps using configuration spaces and braid groups, focusing on two fundamental problems, the Wecken property, and the computation of the Nielsen number. We show that the projective plane (resp. the 2-sphere S²) has the Wecken property for n-valued maps for all n ∈ ? (resp. all n ≥ 3). In the case n = 2 and S², we prove a partial result about the Wecken property. We then describe the Nielsen number of a non-split n-valued map Open image in new window of an orientable, compact manifold without boundary in terms of the Nielsen coincidence numbers of a certain finite covering q: X? → X with a subset of the coordinate maps of a lift of the n-valued split map Open image in new window .