Functors for unitary representations of classical real groups and affine Hecke algebras

We define exact functors from categories of Harish–Chandra modules for certain real classical groups to finite-dimensional modules over an associated graded affine Hecke algebra with parameters. We then study some of the basic properties of these functors. In particular, we show that they map irreducible spherical representations to irreducible spherical representations and, moreover, that they preserve unitarity. In the case of split classical groups, we thus obtain a functorial inclusion of the real spherical unitary dual (with “real infinitesimal character”) into the corresponding p-adic spherical unitary dual.     

Über projektive Moduln und Endlichkeitshindernisse bei Transformationsgruppen

We study projective modules in the category of functors from homogeneous spaces into abelian groups. Such functors have been considered by Bredon [1]. We show that protective functors are determined by a set of ordinary projective modules over suitable group rings. The general notions are applied to give a quick proof for the product formula of the finiteness obstruction for transformation groups. These finiteness obstructions are straightforward extensions of the Swan-Wall obstructions (see e. g. Quinn [7]). They are important in the study of homotopy representations (tom Dieck — Petrie [3], [4]). This work is also related to Rothenberg [8].     

Semisimple Types in GLn

This paper is concerned with the smooth representation theory of the general linear group G=GL(F) of a non-Archimedean local field F. The point is the (explicit) construction of a special series of irreducible representations of compact open subgroups, called semisimple types, and the computation of their Hecke algebras. A given semisimple type determines a Bernstein component of the category of smooth representations of G; that component is then the module category for a tensor product of affine Hecke algebras; every component arises this way. Moreover, all Jacquet functors and parabolic induction functors connecting G with its Levi subgroups are described in terms of standard maps between affine Hecke algebras. These properties of semisimple types depend on their special intertwining properties which in turn imply strong bounds on the support of coefficient functions.     

Parking functions and vertex operators

We introduce several associative algebras and families of vector spaces associated to these algebras. Using lattice vertex operators, we obtain dimension and character formulae for these spaces. In particular, we define a family of representations of symmetric groups which turn out to be isomorphic to parking function modules. We also construct families of vector spaces whose dimensions are Catalan numbers and Fuss–Catalan numbers respectively. Conjecturally, these spaces are related to spaces of global sections of vector bundles on (zero fibres of) Hilbert schemes and representations of rational Cherednik algebras.      

On representations of real Jacobi groups

We consider a category of continuous Hilbert space representations and a category of smooth Fr’echet representations,of a real Jacobi group G.By Mackey’s theory,they are respectively equivalent to certain categories of representations of a real reductive group L.Within these categories,we show that the two functors that take smooth vectors for G and for L are consistent with each other.By using Casselman-Wallach’s theory of smooth representations of real reductive groups,we define matrix coefficients for distributional vectors of certain representations of G.We also formulate Gelfand-Kazhdan criteria for real Jacobi groups which could be used to prove multiplicity one theorems for Fourier-Jacobi models.     

Interpolation sets and the Bohr topology of locally compact groups

Rosenthal's theorem describing those Banach spaces containing no copy of ?¹ is extended to topological groups replacing ?¹-basis by interpolation sets in the sense of Hartman and Ryll-Nardzewsky (Colloq. Math. 12 (1964) 23-39). This extension provides a characterization of those locally compact groups containing no interpolation sets and of those locally compact groups which respect compactness, i.e, such that every Bohr compact subset is compact. The approach followed in this paper sheds some light on other questions related to the duality theory of non-Abelian locally compact groups.     

Some remarks on the global structure of proper Lie groupoids in low codimensions

We observe that any connected proper Lie groupoid whose orbits have codimension at most two admits a globally effective representation, i.e. one whose kernel consists only of ineffective arrows, on a smooth vector bundle. As an application, we deduce that any such groupoid can up to Morita equivalence be presented as an extension, by some bundle of compact Lie groups, of some action groupoid G?X with G compact.     

Green fields

We introduce Green fields, as commutative Green biset functors with no non-trivial ideals. We state some of their properties and give examples of known Green biset functors which are Green fields. Among the properties, we prove some criterions ensuring that a Green field is semisimple. Finally, we describe a type of Green field for which its category of modules is equivalent to a category of vector spaces over a field.     

Irreducible decompositions of unitary representations of infinite- dimensional groups

This paper concerns the problem of irreducible decompositions of unitary representations of topological groups G, including the group Diff₀(M) of diffeomorphisms with compact support on smooth manifolds M. It is well known that the problem is affirmative, when G is a locally compact, separable group (cf. [3, 4]). We extend this result to infinite-dimensional groups with appropriate quasi-invariant measures, and, in particular, we show that every continuous unitary representation of Diff₀(M) has an irreducible decomposition under a fairly mild condition. This research was partially supported by a Grant-in-Aid for Scientific Research (No.14540167), Japan Socieity of the Promotion of Science.     

On simple modules over conformal Galilei algebras

We study irreducible representations of two classes of conformal Galilei algebras in 1-spatial dimension. We construct a functor which transforms simple modules with nonzero central charge over the Heisenberg subalgebra into simple modules over the conformal Galilei algebras. This can be viewed as an analogue of oscillator representations. We use oscillator representations to describe the structure of simple highest weight modules over conformal Galilei algebras. We classify simple weight modules with finite dimensional weight spaces over finite dimensional Heisenberg algebras and use this classification and properties of oscillator representations to classify simple weight modules with finite dimensional weight spaces over conformal Galilei algebras.     

Auslander-Reiten Triangles, Ziegler Spectra and Gorenstein Rings

We investigate (existence of) Auslander—Reiten triangles in a triangulated category in connection with torsion pairs, existence of Serre functors, representability of homological functors and realizability of injective modules. We also develop an Auslander—Reiten theory in a compactly generated triangulated category and we study the connections with the naturally associated Ziegler spectrum. Our analysis is based on the relative homological theory of purity and Brown's Representability Theorem. Our main interest lies in the structure of Auslander—Reiten triangles in the full subcategory of compact objects. We also study the connections and the interplay between Auslander—Reiten theory, pure-semisimplicity and the finite type property, Grothendieck groups, and we give applications to derived categories of Gorenstein rings.     

Unitary representations of the groups of measurable and continuous functions with values in the circle

We give a classification of unitary representations of certain Polish, not necessarily locally compact, groups: the groups of all measurable functions with values in the circle and the groups of all continuous functions on compact, second countable, zero-dimensional spaces with values in the circle. In the proofs of our classification results, certain structure theorems and factorization theorems for linear operators are used.     

Measurable Categories

We develop the theory of categories of measurable fields of Hilbert spaces and bounded fields of operators. We examine classes of functors and natural transformations with good measure theoretic properties, providing in the end a rigorous construction for the bicategory used in [3] and [4] as the basis for a representation theory of (Lie) 2-groups. Two important technical results are established along the way: first it is shown that all invertible additive bounded functors (and thus a fortiori all invertible *-functors) between categories of measurable fields of Hilbert spaces are induced by invertible measurable transformations between the underlying Borel spaces and second we establish the distributivity of Hilbert space tensor product over direct integrals over Lusin spaces with respect to σ-finite measures. The paper concludes with a general definition of measurable bicategories.     

Tilting Theoretical Approach to Moduli Spaces Over Preprojective Algebras

We apply tilting theory over preprojective algebras Λ to the study of moduli spaces of Λ-modules. We define the categories of semistable modules and give equivalences, so-called reflection functors, between them by using tilting modules over Λ. Moreover we prove that the equivalence induces an isomorphism of K-schemes between moduli spaces. In particular, we study the case when the moduli spaces are related to Kleinian singularities, and generalize some results of Crawley-Boevey (Am J Math 122:1027–1037, 2000).     

Geometric representation theory for unitary groups of operator algebras

Geometric realizations for the restrictions of GNS representations to unitary groups of C^∗-algebras are constructed. These geometric realizations use an appropriate concept of reproducing kernels on vector bundles. To build such realizations in spaces of holomorphic sections, a class of complex coadjoint orbits of the corresponding real Banach-Lie groups is described and some homogeneous holomorphic Hermitian vector bundles that are naturally associated with the coadjoint orbits are constructed.     

Duality between compactness and discreteness beyond pontryagin duality

One of the most striking results of Pontryagin’s duality theory is the duality between compact and discrete locally compact abelian groups. This duality also persists in part for objects associated with noncommutative topological groups. In particular, it is well known that the dual space of a compact topological group is discrete, while the dual space of a discrete group is quasicompact (i.e., it satisfies the finite covering theorem but is not necessarily Hausdorff). The converse of the former assertion is also true, whereas the converse of the latter is not (there are simple examples of nondiscrete locally compact solvable groups of height 2 whose dual spaces are quasicompact and non-Hausdorff (they are T ₁ spaces)). However, in the class of locally compact groups all of whose irreducible unitary representations are finite-dimensional, a group is discrete if and only if its dual space is quasicompact (and is automatically a T ₁ space). The proof is based on the structural theorem for locally compact groups all of whose irreducible unitary representations are finite-dimensional. Certain duality between compactness and discreteness can also be revealed in groups that are not necessarily locally compact but are unitarily, or at least reflexively, representable, provided that (in the simplest case) the irreducible representations of a group form a sufficiently large family and have jointly bounded dimensions. The corresponding analogs of compactness and discreteness cannot always be easily identified, but they are still duals of each other to some extent.     

Unbounded pseudodifferential calculus on Lie groupoids

We develop an abstract theory of unbounded longitudinal pseudodifferential calculus on smooth groupoids (also called Lie groupoids) with compact basis. We analyze these operators as unbounded operators acting on Hilbert modules over C^∗(G), and we show in particular that elliptic operators are regular. We construct a scale of Sobolev modules which are the abstract analogues of the ordinary Sobolev spaces, and analyze their properties. Furthermore, we show that complex powers of positive elliptic pseudodifferential operators are still pseudodifferential operators in a generalized sense.     

Gevrey functions and ultradistributions on compact Lie groups and homogeneous spaces

In this paper we give global characterisations of Gevrey–Roumieu and Gevrey–Beurling spaces of ultradifferentiable functions on compact Lie groups in terms of the representation theory of the group and the spectrum of the Laplace–Beltrami operator. Furthermore, we characterise their duals, the spaces of corresponding ultradistributions. For the latter, the proof is based on first obtaining the characterisation of their α-duals in the sense of Köthe and the theory of sequence spaces. We also give the corresponding characterisations on compact homogeneous spaces.     

A new approach to induction and imprimitivity results

Given a closed quantum subgroup of a locally compact quantum group, we study induction of unitary corepresentations of the quantum subgroup to the ambient quantum group. More generally, we study induction given a coaction of the quantum subgroup on a C^*-algebra. We prove imprimitivity theorems that unify the existing theorems for actions and coactions of groups. This means that we define quantum homogeneous spaces as C^*-algebras and that we prove Morita equivalence of crossed products and homogeneous spaces. We essentially use von Neumann algebraic techniques to prove these Morita equivalences between C^*-algebras.     

Equivariant embedding theorems and topological index maps

The construction of topological index maps for equivariant families of Dirac operators requires factoring a general smooth map through maps of a very simple type: zero sections of vector bundles, open embeddings, and vector bundle projections. Roughly speaking, a normally non-singular map is a map together with such a factorisation. These factorisations are models for the topological index map. Under some assumptions concerning the existence of equivariant vector bundles, any smooth map admits a normal factorisation, and two such factorisations are unique up to a certain notion of equivalence. To prove this, we generalise the Mostow Embedding Theorem to spaces equipped with proper groupoid actions. We also discuss orientations of normally non-singular maps with respect to a cohomology theory and show that oriented normally non-singular maps induce wrong-way maps on the chosen cohomology theory. For K-oriented normally non-singular maps, we also get a functor to Kasparov's equivariant KK-theory. We interpret this functor as a topological index map.