Volume 37, Issue 6

We construct Weil representations of twisted loop groups of type A n ( 2 ) {A_{n}^{(2)}} over local fields. We prove that the associated cover of the twisted loop group is the two fold metaplectic cover of the affine Kac–Moody group of type A n ( 2 ) {A_{n}^{(2)}} given by Patnaik and Puskas.

Let 𝐆 {\mathbf{G}} denote an affine Kac–Moody group, and G its points over the local field 𝔽 q ⁢ ( ( s ) ) {\mathbb{F}_{q}((s))} . We establish a local Birkhoff decomposition for a subset of G in terms of a pair of subgroups roughly of the form 𝐆 ⁢ ( 𝔽 q ⁢ [ [ s ] ] ) {\mathbf{G}(\mathbb{F}_{q}[[s]])} and 𝐆 ⁢ ( 𝔽 q ⁢ [ s - 1 ] ) {\mathbf{G}(\mathbb{F}_{q}[s^{-1}])} . Our techniques are global-to-local and use the reduction theory for loop groups due to H. Garland. Building on these ideas, we establish the finiteness of a set whose cardinality is related to spherical R -polynomials in D. Muthiah’s conjectural double-affine Kazhdan–Lusztig theory.

We prove the uniform boundedness of oscillatory singular integrals with singular kernels | x | - n ⁢ Ω ⁢ ( x | x | ) {|x|^{-n}\Omega(\frac{x}{|x|})} and rational phases of the form P ⁢ ( x ) + 1 Q ⁢ ( x ) {P(x)+\frac{1}{Q(x)}} for arbitrary real-valued polynomials P and Q . Our main result shows that the condition Q ⁢ ( 0 ) = 0 {Q(0)=0} imposed in [M. Folch-Gabayet and J. Wright, An estimation for a family of oscillatory integrals, Studia Math. 154 2003, 1, 89–97] is superfluous, which answers a question left open in that paper. As a secondary improvement of existing results, we also extend the space for Ω ⁢ ( ⋅ ) {\Omega(\,\cdot\,)} from L ⁢ log ⁡ L ⁢ ( 𝕊 n - 1 ) {L\log L({\mathbb{S}}^{n-1})} to the strictly larger space H 1 ⁢ ( 𝕊 n - 1 ) {H^{1}({\mathbb{S}}^{n-1})} .

In this paper we will study the action of 𝔽 q n 2 × 2 {\mathbb{F}_{q^{n}}^{2\times 2}} on the graph of an 𝔽 q {{{\mathbb{F}}_{q}}} -linear function of 𝔽 q n {{{\mathbb{F}}_{q^{n}}}} to itself. In particular, we will see that, under certain combinatorial assumptions, its stabilizer (together with the sum and product of matrices) is a field. We will also give some examples where this is not the case. We will also connect such a stabilizer to the right idealizer of the rank-metric code defined by the linear function, and give some structural results in the case where the polynomials are partially scattered.

We consider the Cauchy problem of the critical nonlocal heat equation u t - Δ ⁢ u = ( | x | - μ ∗ | u | 2 μ ∗ ) ⁢ | u | 2 μ ∗ - 2 ⁢ u   in  ⁢ ℝ N × ( 0 , T max ) , u_{t}-\Delta u=(|x|^{-\mu}\ast|u|^{2^{\ast}_{\mu}})|u|^{2^{\ast}_{\mu}-2}u% \quad\text{in }\mathbb{R}^{N}\times(0,T_{\max}), where N ≥ 3 {N\geq 3} , 0 < μ < min ⁡ { 4 , N } {0<\mu<\min\{4,N\}} and 2 μ ∗ = 2 ⁢ N - μ N - 2 {2^{\ast}_{\mu}=\frac{2N-\mu}{N-2}} is the critical exponent in the sense of the Hardy–Littlewood–Sobolev inequality. The aim of this paper is to study the behaviour of the solutions in time. More precisely, we prove the existence and decay of global solutions and blow-up in finite time. Furthermore, the global uniform bound of the global solutions in H ˙ 1 ⁢ ( ℝ N ) {\dot{H}^{1}(\mathbb{R}^{N})} and the asymptotic behaviour of them are given.

By a recent result of Juhász and van Mill, a locally compact topological group whose dense subspaces are all separable is metrizable. In this note we investigate the following question: Is every locally compact group having all dense subgroups separable also metrizable? We give an example to show the answer is negative for locally compact abelian groups, thereby showing that one cannot directly generalize the assertion by replacing “subspaces” with “subgroups”. On the other hand, we prove that the answer is positive for compact groups which are either connected or algebraically abelian; and for locally compact groups containing only separable subgroups. As an application, we obtain a necessary condition for metrizability of pronilpotent groups.

We study the main open parts of the Kawaguchi–Silverman conjecture, asserting that for a birational self-map f of a smooth projective variety X defined over ℚ ¯ {{\overline{\mathbb{Q}}}} , the arithmetic degree α f ⁢ ( x ) {\alpha_{f}(x)} exists and coincides with the first dynamical degree δ f {\delta_{f}} for any ℚ ¯ {\overline{{\mathbb{Q}}}} -point x of X with a Zariski dense orbit. Among other results, we show that this holds when X has Kodaira dimension zero and irregularity q ⁢ ( X ) ≥ dim ⁡ X - 1 {q(X)\geq\dim X-1} or X is an irregular threefold (modulo one possible exception). We also study the existence of Zariski dense orbits, with explicit examples.

We give a result estimating the dimension of the Lie algebra of Killing vector fields on an irreducible non-trivial gradient Ricci soliton. Then we study the structure of this manifold when the maximal dimension is attained. There are local and global implications.

We extend a classical theorem of P. Hall that claims that if the index of every maximal subgroup of a finite group G is a prime or the square of a prime, then G is solvable. Precisely, we prove that if one allows, in addition, the possibility that every maximal subgroup of G is nilpotent instead of having prime or squared-prime index, then G continues to be solvable. Likewise, we obtain the solvability of G when we assume that every proper non-maximal subgroup of G lies in some subgroup of index prime or squared prime.

In this paper we give a trace formula for Hecke operators acting on the cohomology of a Fuchsian group of finite covolume Γ with coefficients in a module V . The proof is based on constructing an operator whose trace on V equals the Lefschetz number of the Hecke correspondence on cohomology, generalizing the operator introduced together with Don Zagier for the modular group.

Standard modules over quasi-hereditary algebras are by definition certain quotients of projective modules. In this article, we study when they can be realized as submodules of projective modules, a property enjoyed by Verma modules in the BGG category 𝒪 {\mathcal{O}} for a finite-dimensional complex semisimple Lie algebra and by quantum Weyl modules over quantized Schur algebras S ⁢ ( n , r ) {S(n,r)} with n ≥ r {n\geq r} .

In this paper, we explore the behaviour of the fixed subgroups of endomorphisms of free-abelian times free (FATF) groups. We exhibit an algorithm which, given a finitely generated subgroup ℋ {\mathcal{H}} of a FATF group 𝒢 {\mathcal{G}} , decides whether ℋ {\mathcal{H}} is the fixed subgroup of some (finite) family of endomorphisms of 𝒢 {\mathcal{G}} and, in the affirmative case, finds such a family. The algorithm combines both combinatorial and algebraic methods.

With the method of mollification and the simple Kuznetsov trace formula, we study the central 𝐿-values of GL 2 \mathrm{GL}_{2} Maass forms of cubic level and establish a positive-proportional non-vanishing result in the spectral aspect in short intervals.

We study “partition Eisenstein series”, extensions of the Eisenstein series G 2 ⁢ k ⁢ ( τ ) {G_{2k}(\tau)} , defined by λ = ( 1 m 1 , 2 m 2 , … , k m k ) ⊢ k ↦ G λ ⁢ ( τ ) := G 2 ⁢ ( τ ) m 1 ⁢ G 4 ⁢ ( τ ) m 2 ⁢ ⋯ ⁢ G 2 ⁢ k ⁢ ( τ ) m k . \lambda=(1^{m_{1}},2^{m_{2}},\dots,k^{m_{k}})\vdash k\mapsto G_{\lambda}(\tau)% :=G_{2}(\tau)^{m_{1}}G_{4}(\tau)^{m_{2}}\cdots G_{2k}(\tau)^{m_{k}}. For functions ϕ : 𝒫 ↦ ℂ {\phi:\mathcal{P}\mapsto\mathbb{C}} on partitions, the weight 2 ⁢ k {2k} partition Eisenstein trace is the quasimodular form Tr k ⁡ ( ϕ ; τ ) := ∑ λ ⊢ k ϕ ⁢ ( λ ) ⁢ G λ ⁢ ( τ ) . \operatorname{Tr}_{k}(\phi;\tau):=\sum_{\lambda\vdash k}\phi(\lambda)G_{% \lambda}(\tau). These traces give explicit formulas for some well-known generating functions, such as the k th elementary symmetric functions of the inverse points of 2-dimensional complex lattices ℤ ⊕ ℤ ⁢ τ {\mathbb{Z}\oplus\mathbb{Z}\tau} , as well as the 2 ⁢ k {2k} th power moments of the Andrews–Garvan crank function. To underscore the ubiquity of such traces, we show that their generalizations give the Taylor coefficients of generic Jacobi forms with torsional divisor.

In this paper, the Gabor system based on the unitary dual of the Heisenberg group H n \mathbb{H}^{n} is introduced and a sufficient condition is obtained for the Gabor system to be a Bessel sequence for L 2 ⁢ ( R ∗ , B 2 ; d ⁢ κ ) L^{2}(\mathbb{R}^{*},{\mathcal{B}_{2}};d\kappa) using the Schrödinger representation of H n \mathbb{H}^{n} , where B 2 \mathcal{B}_{2} denotes the class of Hilbert–Schmidt operators on L 2 ⁢ ( R n ) L^{2}(\mathbb{R}^{n}) and d ⁢ κ d\kappa denotes the Haar measure on R ∗ \mathbb{R}^{*} . Further, a necessary and sufficient condition is provided for the Gabor system to be an orthonormal system, a Parseval frame sequence, a frame sequence and a Riesz sequence.

Let 𝐹 be a linear combination (of a certain form) of N ≥ 1 N\geq 1 Dirichlet 𝐿-functions attached to even (or odd) primitive characters. Selberg proved that a positive proportion of non-trivial zeros of 𝐹 lie on the critical line. Our work here is to provide an explicit lower bound for this proportion. In particular, we show that the lower bound 2.16 × 10 − 6 / ( N ⁢ log ⁡ N ) 2.16\times 10^{-6}/(N\log N) is admissible for large 𝑁.