On the irreducibility of locally analytic principal series representations

Let G be a p-adic connected reductive group with Lie algebra g. For a parabolic subgroup P in G and a finite-dimensional locally analytic representation V of P, we study the induced locally analytic G-representation W = Ind^G_P(V). Our result is the following criterion concerning the topological irreducibility of W: if the Verma module U(g) \otimes_{U(p)} V' associated to the dual representation V' is irreducible then W is topologically irreducible as well.Comment: 44 pages; final version. An appendix has been added in which it is shown that the canonical maps between certain completions of distribution algebras are injective. This fills a gap in a previous version; it was pointed out to us by a refere

Expressive Power in First Order Topology

A first order representation (fo.r.) in topology is an assignment of finitary relational structures of the same type to topological spaces in such a way that homeomorphic spaces get sent to isomorphic structures. We first define the notions one f.o.r. is at least as expressive as another relative to a class of spaces and one class of spaces is definable in another relative to an f.o.r. , and prove some general statements. Following this we compare some well-known classes of spaces and first order representations. A principal result is that if X and Y are two Tichonov spaces whose posets of zero-sets are elementarily equivalent then their respective rings of bounded continuous real-valued functions satisfy the same positiveuniversal sentences. The proof of this uses the technique of constructing ultraproducts as direct limits of products in a category theoretic setting

Excision for deformation K-theory of free products

Associated to a discrete group $G$, one has the topological category of finite dimensional (unitary) $G$-representations and (unitary) isomorphisms. Block sums provide this category with a permutative structure, and the associated $K$-theory spectrum is Carlsson's deformation $K$-theory of G. The goal of this paper is to examine the behavior of this functor on free products. Our main theorem shows the square of spectra associated to $G*H$ (considered as an amalgamated product over the trivial group) is homotopy cartesian. The proof uses a general result regarding group completions of homotopy commutative topological monoids, which may be of some independent interest.Comment: 32 pages, 1 figure. Final version: The title has changed, and the paper has been substantially revised to improve clarit

Gauge theory and mirror symmetry

Outlined in this paper is a description of \emph{equivariance} in the world of 2-dimensional extended topological quantum field theories, under a topological action of compactLie groups. In physics language, I am gauging the theories --- coupling them to a principal bundle on the surface world-sheet. I describe the data needed to gauge the theory, as well as the computation of the gauged theory, the result of integrating over all bundles. The relevant theories are A-models, such as arise from the Gromov-Witten theory of a symplectic manifold with Hamiltonian group action, and the mathematical description starts with a group action on the generating category (the Fukaya category, in this example) which is factored through the topology of the group. Their mirror description involves holomorphic symplectic manifolds and Lagrangians related to the Langlands dual group. An application recovers the complex mirrors of flag varieties proposed by Rietsch

A rich hierarchy of functionals of finite types

We are considering typed hierarchies of total, continuous functionals using complete, separable metric spaces at the base types. We pay special attention to the so called Urysohn space constructed by P. Urysohn. One of the properties of the Urysohn space is that every other separable metric space can be isometrically embedded into it. We discuss why the Urysohn space may be considered as the universal model of possibly infinitary outputs of algorithms. The main result is that all our typed hierarchies may be topologically embedded, type by type, into the corresponding hierarchy over the Urysohn space. As a preparation for this, we prove an effective density theorem that is also of independent interest.Comment: 21 page

Some remarks on the GNS representations of topological ∗^*-algebras

After an appropriate restatement of the GNS construction for topological $^*$-algebras we prove that there exists an isomorphism among the set \cycl(A) of weakly continuous strongly cyclic $^*$-representations of a barreled dual-separable $^*$-algebra with unit $A$, the space \hilb_A(A^*) of the Hilbert spaces that are continuously embedded in $A^*$ and are $^*$-invariant under the dual left regular action of $A$ and the set of the corresponding reproducing kernels. We show that these isomorphisms are cone morphisms and we prove many interesting results that follow from this fact. We discuss how these results can be used to describe cyclic representations on more general inner product spaces.Comment: 34 pages. Minor changes. To appear in J. Math. Phys. 49 (4) Apr-0

Higgs bundles and higher Teichm\"uller spaces

This paper is a survey on the role of Higgs bundle theory in the study of higher Teichm\"uller spaces. Recall that the Teichm\"uller space of a compact surface can be identified with a certain connected component of the moduli space of representations of the fundamental group of the surface into $\mathrm{PSL}(2,{\mathbb{R}})$. Higher Teichm\"uller spaces correspond to special components of the moduli space of representations when one replaces $\mathrm{PSL}(2,{\mathbb{R}})$ by a real non-compact semisimple Lie group of higher rank. Examples of these spaces are provided by the Hitchin components for split real groups, and maximal Toledo invariant components for groups of Hermitian type. More recently, the existence of such components has been proved for $\mathrm{SO}(p,q)$, in agreement with the conjecture of Guichard and Wienhard relating the existence of higher Teichm\"uller spaces to a certain notion of positivity on a Lie group that they have introduced. We review these three different situations, and end up explaining briefly the conjectural general picture from the point of view of Higgs bundle theory.Comment: arXiv admin note: substantial text overlap with arXiv:1511.0775