Volume 31, Issue 5

We prove a version of the Selberg integral formula for local fields of characteristic zero.

We consider the Lie groups SU⁢(n,1){\mathrm{SU}(n,1)} and Sp⁢(n,1){\mathrm{Sp}(n,1)} that act as isometries of the complex and the quaternionic hyperbolic spaces, respectively. We classify pairs of semisimple elements in Sp⁢(n,1){\mathrm{Sp}(n,1)} and SU⁢(n,1){\mathrm{SU}(n,1)} up to conjugacy. This gives local parametrization of the representations ρ in Hom⁢(F2,G)/G{{\mathrm{Hom}}({\mathrm{F}}_{2},G)/G} such that both ρ⁢(x){\rho(x)} and ρ⁢(y){\rho(y)} are semisimple elements in G , where F2=〈x,y〉{{\mathrm{F}}_{2}=\langle x,y\rangle}, G=Sp⁢(n,1){G=\mathrm{Sp}(n,1)} or SU⁢(n,1){\mathrm{SU}(n,1)}. We use the PSp⁢(n,1){{\mathrm{PSp}}(n,1)}-configuration space M⁢(n,i,m-i){{\mathrm{M}}(n,i,m-i)} of ordered m -tuples of distinct points in 𝐇ℍn¯{\overline{{\mathbf{H}}_{{\mathbb{H}}}^{n}}}, where the first i points in an m -tuple are boundary points, to classify the semisimple pairs. Further, we also classify points on M⁢(n,i,m-i){{\mathrm{M}}(n,i,m-i)}.

In this article, we study the Kapustin–Witten equations on a closed, simply connected, four-dimensional manifold which were introduced by Kapustin and Witten. We use Taubes’ compactness theorem [C. H. Taubes, Compactness theorems for SL⁢(2;ℂ){\mathrm{SL}(2;\mathbb{C})} generalizations of the 4-dimensional anti-self dual equations, preprint 2014, https://arxiv.org/abs/1307.6447v4] to prove that if (A,ϕ){(A,\phi)} is a smooth solution to the Kapustin–Witten equations and the connection A is closed to a generic ASD connection A∞{A_{\infty}}, then (A,ϕ){(A,\phi)} must be a trivial solution. We also prove that the moduli space of the solutions to the Kapustin–Witten equations is non-connected if the connections on the compactification of moduli space of ASD connections are all generic . At last, we extend the results for the Kapustin–Witten equations to other equations on gauge theory such as the Hitchin–Simpson equations and the Vafa–Witten on a compact Kähler surface.

We solve the differentiability problem for the evolution map in Milnor’s infinite-dimensional setting. We first show that the evolution map of each Ck{C^{k}}-semiregular Lie group G (for k∈ℕ⊔{lip,∞}{k\in\mathbb{N}\sqcup\{\mathrm{lip},\infty\}}) admits a particular kind of sequentially continuity – called Mackey k-continuity. We then prove that this continuity property is strong enough to ensure differentiability of the evolution map. In particular, this drops any continuity presumptions made in this context so far. Remarkably, Mackey k-continuity arises directly from the regularity problem itself, which makes it particular among the continuity conditions traditionally considered. As an application of the introduced notions, we discuss the strong Trotter property in the sequentially and the Mackey continuous context. We furthermore conclude that if the Lie algebra of G is a Fréchet space, then G is Ck{C^{k}}-semiregular (for k∈ℕ⊔{∞}{k\in\mathbb{N}\sqcup\{\infty\}}) if and only if G is Ck{C^{k}}-regular.

We show a prime geodesic theorem for the group SL3⁢(ℤ){\mathrm{SL}_{3}(\mathbb{Z})} counting those geodesics whose lifts lie in the split Cartan subgroup. This is the first arithmetic prime geodesic theorem of higher rank for a non-cocompact group.

In this paper, we discuss continuity properties of the Wick-type star product on the 2-sphere, interpreted as a coadjoint orbit. Star products on coadjoint orbits in general have been constructed by different techniques. We compare the constructions of Alekseev–Lachowska and Karabegov, and we prove that they agree in general. In the case of the 2-sphere, we establish the continuity of the star product, thereby allowing for a completion to a Fréchet algebra.

The cohomology of an arithmetic congruence subgroup of a connected reductive algebraic group defined over a number field is captured in the automorphic cohomology of that group. The residual Eisenstein cohomology is by definition the part of the automorphic cohomology represented by square-integrable residues of Eisenstein series. The existence of residual Eisenstein cohomology classes depends on a subtle combination of geometric conditions (coming from cohomological reasons) and arithmetic conditions in terms of analytic properties of automorphic L -functions (coming from the study of poles of Eisenstein series). Hence, there are almost no unconditional results in the literature regarding the very existence of non-trivial residual Eisenstein cohomology classes. In this paper, we show the existence of certain non-trivial residual cohomology classes in the case of the split symplectic, and odd and even special orthogonal groups of rank two, as well as the exceptional group of type G2{\mathrm{G}_{2}}, defined over a totally real number field. The construction of cuspidal automorphic representations of GL2{\mathrm{GL}_{2}} with prescribed local and global properties is decisive in this context.

We prove an invariance property of intersections of Kudla–Rapoport divisors on a unitary Rapoport–Zink space.

We study maximal associative subalgebras of an arbitrary finite-dimensional associative algebra B over a field 𝕂{\mathbb{K}} and obtain full classification/description results of such algebras. This is done by first obtaining a complete classification in the semisimple case and then lifting to non-semisimple algebras. The results are sharpest in the case of algebraically closed fields and take special forms for algebras presented by quivers with relations. We also relate representation theoretic properties of the algebra and its maximal and other subalgebras and provide a series of embeddings between quivers, incidence algebras and other structures which relate indecomposable representations of algebras and some subalgebras via induction/restriction functors. Some results in literature are also re-derived as a particular case, and other applications are given.

We construct a bialgebra object in the category of linear maps ℒ⁢ℳ{\mathcal{LM}} from a cocommutative rack bialgebra. The construction does extend to some non-cocommutative rack bialgberas, as is illustrated by a concrete example. As a separate result, we show that the Loday complex with adjoint coefficients embeds into the rack bialgebra deformation complex for the rack bialgebra defined by a Leibniz algebra.

We prove the following rank rigidity result for proper CAT⁡(0){\operatorname{CAT}(0)} spaces with one-dimensional Tits boundaries: Let Γ be a group acting properly discontinuously, cocompactly, and by isometries on such a space X . If the Tits diameter of ∂⁡X{\partial X} equals π and Γ does not act minimally on ∂⁡X{\partial X}, then ∂⁡X{\partial X} is a spherical building or a spherical join. If X is also geodesically complete, then X is a Euclidean building, higher rank symmetric space, or a nontrivial product. Much of the proof, which involves finding a Tits-closed convex building-like subset of ∂⁡X{\partial X}, does not require the Tits diameter to be π, and we give an alternate condition that guarantees rigidity when this hypothesis is removed, which is that a certain invariant of the group action be even.

For an interval finite quiver Q , we introduce a class of flat representations. We classify the indecomposable projective objects in the category rep⁢(Q){\mathrm{rep}(Q)} of pointwise finite dimensional representations. We show that an object in rep⁢(Q){\mathrm{rep}(Q)} is projective if and only if it is a direct sum of countably generated flat representations.