The variety of topological groups generated by the class of all abelian kω-groups has been shown to equal the variety of topological groups generated by the free abelian topological group on [0, 1]. In this paper it is proved that the free abelian topological group on a compact Hausdorff space X generates the same variety if and only if X is not scattered.

Let F be an arbitrary local field. Consider the standard embedding and the two-sided action of GLn(F)×GLn(F) on GLn+1(F). In this paper we show that any GLn(F)×GLn(F)-invariant distribution on GLn+1(F) is invariant with respect to transposition. We show that this implies that the pair (GLn+1(F), GLn(F)) is a Gelfand pair. Namely, for any irreducible admissible representation (π,E) of GLn+1(F), . For the proof in the archimedean case, we develop several tools to study invariant distributions on smooth manifolds.

A nine-dimensional exponential Lie group G and a linear form ℓ on the Lie algebra of G are presented such that for all Pukanszky polarizations 𝔭 at ℓ the canonically associated unitary representation ρ=ρ(ℓ,𝔭) of G has the property that ρ(ℒ1(G)) does not contain any nonzero operator given by a compactly supported kernel function. This example shows that one of Leptin’s results is wrong, and it cannot be repaired.

The eta invariant of the Dirac operator over a non-compact cofinite quotient of PSL(2,ℝ) is defined through a regularized trace following Melrose. It reduces to the standard definition in terms of eigenvalues in the case of a totally non-trivial spin structure. When the S1-fibers are rescaled, the metric becomes of non-exact fibered-cusp type near the ends. We completely describe the continuous spectrum of the Dirac operator with respect to the rescaled metric and its dependence on the spin structure, and show that the adiabatic limit of the eta invariant is essentially the volume of the base hyperbolic Riemann surface with cusps, extending some of the results of Seade and Steer.

Suppose that X is a smooth quasiprojective variety over ℂ and ρ:π1(X,x)→SL(2,ℂ) is a Zariski-dense representation with quasiunipotent monodromy at infinity. Then ρ factors through a map X→Y with Y either a Deligne–Mumford (DM) curve or a Shimura modular stack.

We give some general results concerning continuity of measurable homomorphisms of topological groups. As a consequence we show that a Christensen measurable homomorphism of a Polish abelian group into a locally compact topological group is continuous. We also obtain similar results for the universally measurable homomorphisms and the homomorphisms that have the Baire property.

The description of irreducible representations of a group G can be seen as a problem in harmonic analysis; namely, decomposing a suitable space of functions on G into irreducibles for the action of G×G by left and right multiplication. For a split p-adic reductive group G over a local non-archimedean field, unramified irreducible smooth representations are in bijection with semisimple conjugacy classes in the ‘Langlands dual’ group. We generalize this description to an arbitrary spherical variety X of G as follows. Irreducible unramified quotients of the space are in natural ‘almost bijection’ with a number of copies of AX*/WX, the quotient of a complex torus by the ‘little Weyl group’ of X. This leads to a description of the Hecke module of unramified vectors (a weak analog of geometric results of Gaitsgory and Nadler), and an understanding of the phenomenon that representations ‘distinguished’ by certain subgroups are functorial lifts. In the course of the proof, rationality properties of spherical varieties are examined and a new interpretation is given for the action, defined by Knop, of the Weyl group on the set of Borel orbits.

The equivalence between contact and Pansu differentiable maps on Carnot groups is established within the class of maps that are C1 with respect to the ambient Euclidean structure.

We establish various properties of the definition of cohomology of topological groups given by Grothendieck, Artin and Verdier in SGA4, including a Hochschild–Serre spectral sequence and a continuity theorem for compact groups. We use these properties to compute the cohomology of the Weil group of a totally imaginary field, and of the Weil-étale topology of a number ring recently introduced by Lichtenbaum (both with integer coefficients).

Let F be a non-Archimedean local field and let p be the residual characteristic of F. Let G=GL2(F) and let P be a Borel subgroup of G. In this paper we study the restriction of irreducible smooth representations of G on -vector spaces to P. We show that in a certain sense P controls the representation theory of G. We then extend our results to smooth -modules of finite length and unitary K-Banach space representations of G, where is the ring of integers of a complete discretely valued field K with residue field .

A pro-Lie group is a projective limit of finite dimensional Lie groups. It is proved that a surjective continuous group homomorphism between connected pro-Lie groups is open. In fact this remains true for almost connected pro-Lie groups where a topological group is called almost connected if the factor group modulo the identity component is compact. As consequences we get a Closed Graph Theorem and the validity of the Second Isomorphism Theorem for pro-Lie groups in the almost connected context.

A mapping f : G → s from a left topological group G into a semigroup S is a local homomorphism if for every x є G \ {e}, there is a neighborhood Ux of e such that f (xy) = f (x)f (y) for all y є Ux \ {e}. A local homomorphism f : G → S is onto if for every neighborhood U of e, f(U \ {e}) = S. We show that

(1) every countable regular left topological group containing a discrete subset with exactly one accumulation point admits a local homomorphism onto N,

(2) it is consistent that every countable topological group containing a discrete subset with exactly one accumulation point admits a local homomorphism onto any countable semigroup,

(3) it is consistent that every countable nondiscrete maximally almost periodic topological group admits a local homomorphism onto the countably infinite right zero semigroup.

A distribution on a Heisenberg type group of homogeneous dimension Q is a biradial kernel of type α if it coincides with a biradial function, homogeneous of degree α — Q, and smooth away from the identity. We prove that a distribution is a biradial kernel of type α, 0 < α < Q, if and only if its Gelfand transform, defined on the Heisenberg fan, extends to a smooth even function on the upper half plane, homogeneous of degree −α/2. A similar result holds for radial kernels on the Heisenberg group.

To each filter ℱ on ω, a certain linear subalgebra A(ℱ) of Rω, the countable product of lines, is assigned. This algebra is shown to have many interesting topological properties, depending on the properties of the filter ℱ. For example, if ℱ is a free ultrafilter, then A(ℱ) is a Baire subalgebra of ℱω for which the game OF introduced by Tkachenko is undetermined (this resolves a problem of Hernández, Robbie and Tkachenko); and if ℱ1 and ℱ2 are two free filters on ω that are not near coherent (such filters exist under Martin's Axiom), then A (ℱ1) and A(ℱ2) are two o-bounded and OF-undetermined subalgebras of ℱω whose product A(ℱ1) × A(ℱ2) is OF-determined and not o-bounded (this resolves a problem of Tkachenko). It is also shown that the statement that the product of two o-bounded subrings of ℱω is o-bounded is equivalent to the set-theoretic principle NCF (Near Coherence of Filters); this suggests that Tkachenko's question on the productivity of the class of o-bounded topological groups may be undecidable in ZFC.

It is shown that for the computation of the Kazhdan constant for a compact group only the regular representation restricted to the orthogonal complement of the constant functions needs to be taken into account.

In this paper we study the kernels and the Lp–Lq boundedness properties of some intertwining operators associated to representations of SL(n, R).

Let a1… ad be a basis of the Lie algebra g of a connected Lie group G and let M be a Lie subgroup of,G. If dx is a non-zero positive quasi-invariant regular Borel measure on the homogeneous space X = G/M and S: X × G → C is a continuous cocycle, then under a rather weak condition on dx and S there exists in a natural way a (weakly*) continuous representation U of G in Lp (X;dx) for all p ε [1,].

Let Ai be the infinitesimal generator with respect to U and the direction ai, for all i ∈ { 1… d}. We consider n–th order strongly elliptic operators H = ΣcαAα with complex coefficients cα. We show that the semigroup S generated by the closure of H has a reduced heat kernel K and we derive upper bounds for k and all its derivatives.

Pointwise bounds for characters of representations of the compact, connected, simple, exceptional Life groups are obtained. It is a classical result that if μ is a central, continuous measure on such a group, then μdimG is absolutely continuous. Our estimates on the size of characters allow us to prove that the exponent, dimension of G, can be replaced by approximately the rank of G. Similar results were obtained earlier for the classical, compact Lie groups.

We study a convolution semigroup satisfying Gaussian estimates on a group G of polynomial volume growth. If Q is a subgroup satisfying a certain geometric condition, we obtain high order regularity estimates for the semigroup in the direction of Q. Applications to heat kernels and convolution powers are given.

We extend an uncertainty principle due to Cowling and Price to threadlike nilpotent Lie groups. This uncertainty principle is a generalization of a classical result due to Hardy. We are thus extending earlier work on Rn and Heisenberg groups.