Hecke algebras for {{\rm GL}_n} over local fields

We study the local Hecke algebra \({\mathcal{H}_{G}(K)}\) for \({G = {\rm GL}_{n}}\) and K a non-archimedean local field of characteristic zero. We show that for \({G = {\rm GL}_{2}}\) and any two such fields K and L, there is a Morita equivalence \({\mathcal{H}_{G}(K) \sim_{M} \mathcal{H}_{G}(L)}\), by using the Bernstein decomposition of the Hecke algebra and determining the intertwining algebras that yield the Bernstein blocks up to Morita equivalence. By contrast, we prove that for \({G = {\rm GL}_{n}}\), there is an algebra isomorphism \({\mathcal{H}_{G}(K) \cong \mathcal{H}_{G}(L)}\) which is an isometry for the induced \({L^1}\)-norm if and only if there is a field isomorphism \({K \cong L}\).     

Aspects of the Segre variety $${\mathcal{S}_{1,1,1}(2)}$$

We consider various aspects of the Segre variety \({\mathcal{S}:=\mathcal{S} _{1,1,1}(2)}\) in PG(7, 2), whose stabilizer group \({\mathcal{G}_{\mathcal{S}}<{\rm GL}(8,2)}\) has the structure \({\mathcal{N}\rtimes{\rm Sym}(3),}\) where \({\mathcal{N} :={\rm GL}(2,2)\times{\rm GL}(2,2)\times{\rm GL} (2,2).}\) In particular we prove that \({\mathcal{S}}\) determines a distinguished Z ₃-subgroup \({\mathcal{Z}<{\rm GL}(8,2)}\) such that \({A\mathcal{Z}A^{-1}=\mathcal{Z},}\) for all \({A\in\mathcal{G}_{\mathcal{S}},}\) and in consequence \({\mathcal{S}}\) determines a \({\mathcal{G}_{\mathcal{S}}}\)-invariant spread of 85 lines in PG(7, 2). Furthermore we see that Segre varieties \({\mathcal{S}_{1,1,1}(2)}\) in PG(7, 2) come along in triplets \({\{\mathcal{S},\mathcal{S}^{\prime},\mathcal{S}^{\prime\prime}\}}\) which share the same distinguished Z ₃-subgroup \({\mathcal{Z}<{\rm GL}(8,2).}\) We conclude by determining all fifteen \({\mathcal{G}_{\mathcal{S}}}\)-invariant polynomial functions on PG(7, 2) which have degree < 8, and their relation to the five \({\mathcal{G}_{\mathcal{S}}}\)-orbits of points in PG(7, 2).     

The power graph on the conjugacy classes of a finite group

A digraph \({\overrightarrow{\mathcal{Pc}}(G)}\) is said to be the directed power graph on the conjugacy classes of a group G, if its vertices are the non-trivial conjugacy classes of G, and there is an arc from vertex C to C′ if and only if \({C \neq C'}\) and \({C \subseteqq {C'}^{m}}\) for some positive integer \({m > 0}\). Moreover, the simple graph \({\mathcal{Pc}(G)}\) is said to be the (undirected) power graph on the conjugacy classes of a group G if its vertices are the conjugacy classes of G and two distinct vertices C and C′ are adjacent in \({\mathcal{Pc}(G)}\) if one is a subset of a power of the other. In this paper, we find some connections between algebraic properties of some groups and properties of the associated graph.     

Depth,Stanley depth,and regularity of ideals associated to graphs

Let \({\mathbb{K}}\) be a field and \({S=\mathbb{K}[x_1,\dots,x_n]}\) be the polynomial ring in n variables over \({\mathbb{K}}\). Let G be a graph with n vertices. Assume that \({I=I(G)}\) is the edge ideal of G and \({J=J(G)}\) is its cover ideal. We prove that \({{\rm sdepth}(J)\geq n-\nu_{o}(G)}\) and \({{\rm sdepth}(S/J)\geq n-\nu_{o}(G)-1}\), where \({\nu_{o}(G)}\) is the ordered matching number of G. We also prove the inequalities \({{\rmsdepth}(J^k)\geq {\rm depth}(J^k)}\) and \({{\rm sdepth}(S/J^k)\geq {\rmdepth}(S/J^k)}\), for every integer \({k\gg 0}\), when G is a bipartite graph. Moreover, we provide an elementary proof for the known inequality reg\({(S/I)\leq \nu_{o}(G)}\).     

Infinite dimensional multiplicity free spaces III: matrix coefficients and regular functions

In earlier papers we studied direct limits \({(G,\,K) = \varinjlim\, (G_n,K_n)}\) of two types of Gelfand pairs. The first type was that in which the G _n /K _n are compact Riemannian symmetric spaces. The second type was that in which \({G_n = N_n\rtimes K_n}\) with N _n nilpotent, in other words pairs (G _n , K _n ) for which G _n /K _n is a commutative nilmanifold. In each we worked out a method inspired by the Frobenius–Schur Orthogonality Relations to define isometric injections \({\zeta_{m,n}: L^2(G_n/K_n) \hookrightarrow L^2(G_m/K_m)}\) for m ≧ n and prove that the left regular representation of G on the Hilbert space direct limit \({L^2(G/K) := \varinjlim L^2(G_n/K_n)}\) is multiplicity-free. This left open questions concerning the nature of the elements of L ²(G/K). Here we define spaces \({\mathcal{A}(G_n/K_n)}\) of regular functions on G _n /K _n and injections \({\nu_{m,n} : \mathcal{A}(G_n/K_n) \to \mathcal{A}(G_m/K_m)}\) for m ≧ n related to restriction by \({\nu_{m,n}(f)|_{G_n/K_n} = f}\). Thus the direct limit \({\mathcal{A}(G/K) := \varinjlim \{\mathcal{A}(G_n/K_n), \nu_{m,n}\}}\) sits as a particular G-submodule of the much larger inverse limit \({\varprojlim \{\mathcal{A}(G_n/K_n), {\rm restriction}\}}\). Further, we define a pre Hilbert space structure on \({\mathcal{A}(G/K)}\) derived from that of L ²(G/K). This allows an interpretation of L ²(G/K) as the Hilbert space completion of the concretely defined function space \({\mathcal{A}(G/K)}\), and also defines a G-invariant inner product on \({\mathcal{A}(G/K)}\) for which the left regular representation of G is multiplicity-free.     

Some remarks on energy inequalities for harmonic maps with potential

We call the \({\delta}\)-vector of an integral convex polytope of dimension d flat if the \({\delta}\)-vector is of the form \({(1,0,\ldots,0,a,\ldots,a,0,\ldots,0)}\), where \({a \geq 1}\). In this paper, we give the complete characterization of possible flat \({\delta}\)-vectors. Moreover, for an integral convex polytope \({\mathcal{P}\subset \mathbb{R}^N}\) of dimension d, we let \({i(\mathcal{P},n)=|n\mathcal{P}\cap \mathbb{Z}^N|}\) and \({i^*(\mathcal{P},n)=|n(\mathcal{P} {\setminus}\partial \mathcal{P})\cap \mathbb{Z}^N|}\). By this characterization, we show that for any \({d \geq 1}\) and for any \({k,\ell \geq 0}\) with \({k+\ell \leq d-1}\), there exist integral convex polytopes \({\mathcal{P}}\) and \({\mathcal{Q}}\) of dimension d such that (i) For \({t=1,\ldots,k}\), we have \({i(\mathcal{P},t)=i(\mathcal{Q},t),}\) (ii) For \({t=1,\ldots,\ell}\), we have \({i^*(\mathcal{P},t)=i^*(\mathcal{Q},t)}\), and (iii) \({i(\mathcal{P},k+1) \neq i(\mathcal{Q},k+1)}\) and \({i^*(\mathcal{P},\ell+1)\neq i^*(\mathcal{Q},\ell+1)}\).     

Uniform in bandwidth exact rates for a class of kernel estimators

Given an i.i.d sample (Y _i , Z _i ), taking values in \({\mathbb{R}^{d'}\times\mathbb{R}^d}\), we consider a collection Nadarya–Watson kernel estimators of the conditional expectations \({\mathbb{E}( <\,c_g(z),g(Y)>+d_g(z)\mid Z=z)}\), where z belongs to a compact set \({H\subset \mathbb{R}^d}\), g a Borel function on \({\mathbb{R}^{d'}}\) and c _g (·), d _g (·) are continuous functions on \({\mathbb{R}^d}\). Given two bandwidth sequences \({h_n<\mathfrak{h}_n}\) fulfilling mild conditions, we obtain an exact and explicit almost sure limit bounds for the deviations of these estimators around their expectations, uniformly in \({g\in\mathcal{G},\;z\in H}\) and \({h_n\le h\le \mathfrak{h}_n}\) under mild conditions on the density f _Z , the class \({\mathcal{G}}\), the kernel K and the functions c _g (·), d _g (·). We apply this result to prove that smoothed empirical likelihood can be used to build confidence intervals for conditional probabilities \({\mathbb{P}( Y\in C\mid Z=z)}\), that hold uniformly in \({z\in H,\; C\in \mathcal{C},\; h\in [h_n,\mathfrak{h}_n]}\). Here \({\mathcal{C}}\) is a Vapnik–Chervonenkis class of sets.     

Arithmetic Groups,Base Change,and Representation Growth

Consider an arithmetic group \({\mathbf{G}(O_S)}\), where \({\mathbf{G}}\) is an affine group scheme with connected, simply connected absolutely almost simple generic fiber, defined over the ring of S-integers O _S of a number field K with respect to a finite set of places S. For each \({n \in \mathbb{N}}\), let \({R_n(\mathbf{G}(O_S))}\) denote the number of irreducible complex representations of \({\mathbf{G}(O_S)}\) of dimension at most n. The degree of representation growth \({\alpha(\mathbf{G}(O_S)) = \lim_{n \rightarrow\infty}\log R_n(\mathbf{G}(O_S)) / \log n}\) is finite if and only if \({\mathbf{G}(O_S)}\) has the weak Congruence Subgroup Property. We establish that for every \({\mathbf{G}(O_S)}\) with the weak Congruence Subgroup Property the invariant \({\alpha(\mathbf{G}(O_S))}\) is already determined by the absolute root system of \({\mathbf{G}}\). To show this we demonstrate that the abscissae of convergence of the representation zeta functions of such groups are invariant under base extensions \({K{\subset}L}\). We deduce from our result a variant of a conjecture of Larsen and Lubotzky regarding the representation growth of irreducible lattices in higher rank semi-simple groups. In particular, this reduces Larsen and Lubotzky’s conjecture to Serre’s conjecture on the weak Congruence Subgroup Property, which it refines.     

Integral means of weighted Hardy–Bloch functions

Let \({\mathcal{B}^\omega(p, q, B_d)}\) denote the \({\omega}\)-weighted Hardy–Bloch space on the unit ball B _d of \({\mathbb{C}^d}\), \({d\ge 1}\). For \({2< p,q < \infty}\) and \({f\in \mathcal{B}^\omega(p, q, B_d)}\), we obtain sharp estimates on the growth of the p-integral means M _p (f, r) as \({r\to 1-}\).     

Conformal embeddings of affine vertex algebras in minimal <Emphasis Type="Italic">W</Emphasis>-algebras II: decompositions

We present methods for computing the explicit decomposition of the minimal simple affine W-algebra \({W_k(\mathfrak{g}, \theta)}\) as a module for its maximal affine subalgebra \({\mathscr{V}_k(\mathfrak{g}^{\natural})}\) at a conformal level k, that is, whenever the Virasoro vectors of \({W_k(\mathfrak{g}, \theta)}\) and \({\mathscr{V}_k(\mathfrak{g}^\natural)}\) coincide. A particular emphasis is given on the application of affine fusion rules to the determination of branching rules. In almost all cases when \({\mathfrak{g}^{\natural}}\) is a semisimple Lie algebra, we show that, for a suitable conformal level k, \({W_k(\mathfrak{g}, \theta)}\) is isomorphic to an extension of \({\mathscr{V}_k(\mathfrak{g}^{\natural})}\) by its simple module. We are able to prove that in certain cases \({W_k(\mathfrak{g}, \theta)}\) is a simple current extension of \({\mathscr{V}_k(\mathfrak{g}^{\natural})}\). In order to analyze more complicated non simple current extensions at conformal levels, we present an explicit realization of the simple W-algebra \({W_{k}(\mathit{sl}(4), \theta)}\) at k = ?8/3. We prove, as conjectured in [3], that \({W_{k}(\mathit{sl}(4), \theta)}\) is isomorphic to the vertex algebra \({\mathscr{R}^{(3)}}\), and construct infinitely many singular vectors using screening operators. We also construct a new family of simple current modules for the vertex algebra \({V_k (\mathit{sl}(n))}\) at certain admissible levels and for \({V_k (\mathit{sl}(m \vert n)), m\ne n, m,n\geq 1}\) at arbitrary levels.     

Disposition p-groups

For any prime p and positive integers c, d there is up to isomorphism a unique p-group \({G_{d}^{c}(p)}\) of least order having any (finite) p-group G with rank \({d(G) \le d}\) and Frattini class \({c_{p}(G) \le c}\) as epimorphic image. Here \({c_{p}(G) = n}\) is the least positive integer such that G has a central series of length n with all factors being elementary. This “disposition” p-group \({G_{d}^{c}(p)}\) has been examined quite intensively in the literature, sometimes controversially. The objective of this paper is to present a summary of the known facts, and to add some new results. For instance we show that for \({G = G_{d}^{c}(p)}\) the centralizer \({C_{G}(x) = \langle Z(G), x \rangle}\) whenever \({x \in G}\) is outside the Frattini subgroup, and that for odd p and \({d \ge 2}\) the group \({E = G_{d}^{c+1}(p)/(G_{d}^{c+1}(p))^{p^{c}}}\) is a distinguished Schur cover of G with \({E/Z(E) \cong G}\). We also have a fibre product construction of \({G_{d}^{c+1}(p)}\) in terms of \({G = G_{d}^{c}(p)}\) which might be of interest for Galois theory.     

<Emphasis Type="Italic">k</Emphasis>-Extreme Points in Symmetric Spaces of Measurable Operators

Let \({\mathcal{M}}\) be a semifinite von Neumann algebra with a faithful, normal, semifinite trace \({\tau}\) and E be a strongly symmetric Banach function space on \({[0,\tau({\bf 1}))}\) . We show that an operator x in the unit sphere of \({E(\mathcal{M}, \tau)}\) is k-extreme, \({k \in {\mathbb{N}}}\) , whenever its singular value function \({\mu(x)}\) is k-extreme and one of the following conditions hold (i) \({\mu(\infty, x) = \lim_{t\to\infty}\mu(t, x) = 0}\) or (ii) \({n(x)\mathcal{M}n(x^*) = 0}\) and \({|x| \geq \mu(\infty, x)s(x)}\) , where n(x) and s(x) are null and support projections of x, respectively. The converse is true whenever \({\mathcal{M}}\) is non-atomic. The global k-rotundity property follows, that is if \({\mathcal{M}}\) is non-atomic then E is k-rotund if and only if \(E(\mathcal{M}, \tau)\) is k-rotund. As a consequence of the noncommutative results we obtain that f is a k-extreme point of the unit ball of the strongly symmetric function space E if and only if its decreasing rearrangement \({\mu(f)}\) is k-extreme and \({|f| \geq \mu(\infty,f)}\) . We conclude with the corollary on orbits Ω(g) and Ω′(g). We get that f is a k-extreme point of the orbit \({\Omega(g),\,g \in L_1 + L_{\infty}}\) , or \({\Omega'(g),\,g \in L_1[0, \alpha),\,\alpha < \infty}\) , if and only if \({\mu(f) = \mu(g)}\) and \({|f| \geq \mu(\infty, f)}\) . From this we obtain a characterization of k-extreme points in Marcinkiewicz spaces.     

Spectral properties of continuous representations of topological groups

Let \({\mathcal{L}(X)}\) be the algebra of all bounded operators on a Banach space X. \({\theta:G\rightarrow \mathcal{L}(X)}\) denotes a strongly continuous representation of a topological abelian group G on X. Set \({\sigma^1(\theta(g)):=\{\lambda/|\lambda|,\lambda\in\sigma(\theta(g))\}}\), where σ(θ(g)) is the spectrum of θ(g) and \({\Sigma:=\{g\in G/\enskip\text{there is no} \enskip P\in \mathcal{P}/P\subseteq \sigma^1(\theta(g))\}}\), where \({\mathcal{P}}\) is the set of regular polygons of \({\mathbb{T}}\) (we call polygon in \({\mathbb{T}}\) the image by a rotation of a closed subgroup of \({\mathbb{T}}\), the unit circle of \({\mathbb{C}}\)). We prove here that if G is a locally compact and second countable abelian group, then θ is uniformly continuous if and only if Σ is non-meager.     

Mutually prime sequences of inner functions and the ranks of subspaces over the bidisk

Let \({\{\varphi_n(z)\}_{n\ge0}}\) be a sequence of inner functions satisfying that \({\zeta_n(z):=\varphi_n(z)/\varphi_{n+1}(z)\in H^\infty(z)}\) for every n ≥ 0 and \({\{\varphi_n(z)\}_{n\ge0}}\) have no nonconstant common inner divisors. Associated with it, we have a Rudin type invariant subspace \({\mathcal{M}}\) of \({H^2(\mathbb{D}^2)}\) . We write \({\mathcal{N}= H^2(\mathbb{D}^2)\ominus\mathcal{M}}\) . If \({\{\zeta_n(z)\}_{n\ge0}}\) ia a mutually prime sequence, then we shall prove that \({rank_{\{T^\ast_z,T^\ast_w\}} \mathcal{N}=1}\) and \({rank_{\{\mathcal{F}^\ast_z\}}(\mathcal{M}\ominus w\mathcal{M})=1}\) , where \({\mathcal{F}_z}\) is the fringe operator on \({\mathcal{M}\ominus w\mathcal{M}}\) .     

The Proper Dissipative Extensions of a Dual Pair

Let A and \({(-\widetilde{A})}\) be dissipative operators on a Hilbert space \({\mathcal{H}}\) and let \({(A,\widetilde{A})}\) form a dual pair, i.e. \({A \subset \widetilde{A}^*}\), resp. \({\widetilde{A} \subset A^*}\). We present a method of determining the proper dissipative extensions \({\widehat{A}}\) of this dual pair, i.e. \({A\subset \widehat{A} \subset\widetilde{A}^*}\) provided that \({\mathcal{D}(A)\cap\mathcal{D}(\widetilde{A})}\) is dense in \({\mathcal{H}}\). Applications to symmetric operators, symmetric operators perturbed by a relatively bounded dissipative operator and more singular differential operators are discussed. Finally, we investigate the stability of the numerical range of the different dissipative extensions.     

The Katchalski–Lewis transversal problem for regular polygons

If every k-membered subfamily of a family of plane convex bodies has a line transversal, then we say that this family has property T(k). We say that a family \({\mathcal{F}}\) has property \({T-m}\), if there exists a subfamily \({\mathcal{G} \subset \mathcal{F}}\) with \({|\mathcal{F} - \mathcal{G}| \le m}\) admitting a line transversal. Heppes [7] posed the problem whether there exists a convex body K in the plane such that if \({\mathcal{F}}\) is a finite T(3)-family of disjoint translates of K, then m = 3 is the smallest value for which \({\mathcal{F}}\) has property \({T-m}\). In this paper, we study this open problem in terms of finite T(3)-families of pairwise disjoint translates of a regular 2n-gon \({(n \ge 5)}\). We find out that, for \({5 \le n \le 34}\), the family has property \({T - 3}\) ; for \({n \ge 35}\), the family has property \({T - 2}\).     

Maximum Principle for H-Surfaces in the Unit Cone and Dirichlet’s Problem for their Equation in Central Projection

In the unit cone\({\mathcal{C} := \{(x, y, z)} \in {\mathbb R}^{3} : {x}^{2} + {y}^{2} < {z}^{2}, {z} > {0}\}\) we establish a geometric maximum principle for H-surfaces, where its mean curvature \({H = H(x, y, z)}\) is optimally bounded. Consequently, these surfaces cannot touch the conical boundary \({\partial \mathcal{C}}\) at interior points and have to approach \({\partial \mathcal{C}}\) transversally. By a nonlinear continuity method, we then solve the Dirichlet problem of the H-surface equation in central projection for Jordan-domains \({\Omega}\) which are strictly convex in the following sense: On its whole boundary \({\partial \mathcal{C}(\Omega)}\) their associate cone \({\mathcal{C}(\Omega) := \{(rx, ry, r) \in {\mathbb R}^{3} : (x, y) \in \Omega, r \in (0,+\infty)}\}\) admits rotated unit cones \({O \circ \mathcal{C}}\) as solids of support, where \({O \in {\mathbb R}^{3\times3}}\) represents a rotation in the Euclidean space. Thus we construct the unique H-surface with one-to-one central projection onto these domains \({\Omega}\) bounding a given Jordan-contour \({\Gamma \subset \mathcal{C} \backslash \{0\}}\) with one-toone central projection.     

A Beurling-Blecher-Labuschagne Theorem for Noncommutative Hardy Spaces Associated with Semifinite von Neumann Algebras

We prove a Beurling-Blecher-Labuschagne theorem for \({H^\infty}\)-invariant spaces of \({L^p(\mathcal{M},\tau)}\) when \({0 < p \leq\infty}\), using Arveson’s non-commutative Hardy space \({H^\infty}\) in relation to a von Neumann algebra \({\mathcal{M}}\) with a semifinite, faithful, normal tracial weight \({\tau}\). Using the main result, we are able to completely characterize all \({H^\infty}\)-invariant subspaces of \({L^p(\mathcal{M} \rtimes_\alpha \mathbb{Z},\tau)}\), where \({\mathcal{M} \rtimes_\alpha \mathbb{Z} }\) is a crossed product of a semifinite von Neumann algebra \({\mathcal{M}}\) by the integer group \({\mathbb{Z}}\), and \({H^\infty}\) is a non-selfadjoint crossed product of \({\mathcal{M}}\) by \({\mathbb{Z}^+}\). As an example, we characterize all \({H^\infty}\)-invariant subspaces of the Schatten p-class \({S^p(\mathcal{H})}\), where \({H^\infty}\) is the lower triangular subalgebra of \({B(\mathcal{H})}\), for each \({0 < p \leq\infty}\).     

On a Class of Weakly Coupled Systems of Elliptic Operators with Unbounded Coefficients

We consider a class of weakly coupled systems of elliptic operators \({\mathcal{A}}\) with unbounded coefficients defined in \({\mathbb{R}^N}\). We prove that a semigroup (T(t))t ≥ 0 of bounded linear operators can be associated with \({\mathcal{A}}\), in a natural way, in the space of all bounded and continuous functions. We prove a compactness property of the semigroup as well as some uniform estimates on the derivatives of the function T(t)f, when f belongs to some spaces of Hölder continuous functions, which are the key tools to prove some optimal Schauder estimates for the solution to some nonhomogeneous elliptic equations and Cauchy problems associated with the operator \({\mathcal{A}}\). Under suitable additional conditions, we then prove that the restriction of the semigroup to the subspace of smooth and compactly supported functions extends by a strongly continuous semigroup to L ^p -spaces over \({\mathbb{R}^N}\), related to the Lebesgue measure, when \({p \in [1,\infty)}\). We also provide sufficient conditions for this semigroup to be analytic when \({p \in [1,\infty)}\). Finally, we prove some L ^p ?L ^q -estimates.