Principal series representations and harmonic spinors

Let G be a real reductive Lie group and G/H a reductive homogeneous space. We consider Kostant's cubic Dirac operator D on G/H twisted with a finite-dimensional representation of H. Under the assumption that G and H have the same complex rank, we construct a nonzero intertwining operator from principal series representations of G into the kernel of D. The Langlands parameters of these principal series are described explicitly. In particular, we obtain an explicit integral formula for certain solutions of the cubic Dirac equation D=0 on G/H.     

Harmonic spinors on semisimple symmetric spaces

Let G/H be a semisimple symmetric space. We consider a Dirac operator D on G/H twisted by a finite dimensional H-representation. We give an explicit integral formula for certain solutions of the equation D=0. In particular, some quotients of standard principal series representations are seen to occur in the kernel of D.     

Covariant representations of Hecke algebras and imprimitivity for crossed products by homogeneous spaces

For discrete Hecke pairs (G,H), we introduce a notion of covariant representation which reduces in the case where H is normal to the usual definition of covariance for the action of G/H on c₀(G/H) by right translation; in many cases where G is a semidirect product, it can also be expressed in terms of covariance for a semigroup action. We use this covariance to characterise the representations of c₀(G/H) which are multiples of the multiplication representation on ?²(G/H), and more generally, we prove an imprimitivity theorem for regular representations of certain crossed products by coactions of homogeneous spaces. We thus obtain new criteria for extending unitary representations from H to G.     

Orbifold construction for topological field theories

An equivariant topological field theory is defined on a cobordism category of manifolds with principal fiber bundles for a fixed (finite) structure group. We provide a geometric construction which for any given morphism $G?H$ of finite groups assigns in a functorial way to a G-equivariant topological field theory an H-equivariant topological field theory, the pushforward theory. When H is the trivial group, this yields an orbifold construction for G-equivariant topological field theories which unifies and generalizes several known algebraic notions of orbifoldization.     

Clifford Theory for Glider Representations

Classical Clifford theory studies the decomposition of simple G-modules into simple H-modules for some normal subgroup H ? G. In this paper we deal with chains of normal subgroups 1?G ₁?· · ·?G _d = G, which allow to consider fragments and in particular glider representations. These are given by a descending chain of vector spaces over some field K and relate different representations of the groups appearing in the chain. Picking some normal subgroup H ? G one obtains a normal subchain and one can construct an induced fragment structure. Moreover, a notion of irreducibility of fragments is introduced, which completes the list of ingredients to perform a Clifford theory.     

Branching problems for semisimple Lie groups and reproducing kernels

For a semisimple Lie group G satisfying the equal rank condition, the most basic family of unitary irreducible representations is the discrete series found by Harish-Chandra. In this paper, we study some of the branching laws for these when restricted to a subgroup H of the same type by combining the classical results with the recent work of T. Kobayashi. We analyze aspects of having differential operators being symmetry-breaking operators; in particular, we prove in the so-called admissible case that every symmetry breaking (H-map) operator is a differential operator. We prove discrete decomposability under Harish-Chandra's condition of cusp form on the reproducing kernels. Our techniques are based on realizing discrete series representations as kernels of elliptic invariant differential operators.     

Bimeasure algebras on locally compact groups

For locally compact groups G and H, let BM(G, H) denote the Banach space of bounded bilinear forms on C₀(G) × C₀(H). Using a consequence of the fundamental inequality of A. Grothendieck. a multiplication and an adjoint operation are introduced on BM(G, H) which generalize the convolution structure of M(G × H) and which make BM(G, H) into a K_G²-Banach ${}^1$-algebra, where K_G is Grothendieck's universal constant. Various topics relating to the ideal structure of BM(G, H) and the lifting of unitary representations of G × H to ${}^1$-representations of BM(G, H) are investigated.     

Moment sets and unitary dual for the diamond group

The diamond group G is a solvable group, semi-direct product of R with a (2n+1)-dimensional Heisenberg group Hn. We consider this group as a first example of a semi-direct product with the form R?N where N is nilpotent, connected and simply connected.Computing the moment sets for G, we prove that they separate the coadjoint orbits and its generic unitary irreducible representations.Then we look for the separation of all irreducible representations. First, moment sets separate representations for a quotient group G⁻ of G by a discrete subgroup, then we can extend G to an overgroup G⁺, extend simultaneously each unitary irreducible representation of G to G⁺ and separate the representations of G by moment sets for G⁺.     

Simple infinite-dimensional quotients of the group C-algebras ofcertain discrete 6-dimensional nilpotent groups

We study here the simple infinite-dimensional quotients of the group C^*-algebras of two discrete6-dimensional nilpotent groups H_6,1 and H_6,2 as the higher-dimensional analogues of the irrational rotation algebras. P Milnes and S. Walters, jointly and individually, have studied the lower-dimensional cases in a series of papers, and also have started the study of some other 6-dimensional groups. For G = H_6,1 or H_6,2, we can determine the crossed product presentations for the simple quotients of C^* (G), and matrix representations for those arising from non-faithful representations of the groups. The isomorphism classifications of these quotients are obtained using K-theoretic tools, namely, the K-groups and the range of trace on K₀. This marks the first use of K-theory in the classification of quotients for 6-dimensional groups.     

Knapp-Wallach Szegö integrals and generalized principal series representations: The parabolic rank one case

To each discrete series representation of a connected semisimple Lie group G with finite center, a G-equivariant embedding into a generalized principal series representation is given. This representation is induced from specified parameters on a maximal parabolic subgroup of G and the mapping is defined by an integral formula, analogous to the Szegö integral introduced by Knapp and Wallach for a minimal parabolic subgroup. In a limiting case, embeddings of limits of discrete series representations are obtained and used to exhibit a reducibility theorem.     

On the role of effective representations of Lie groupoids

In this paper, we undertake the study of the Tannaka duality construction for the ordinary representations of a proper Lie groupoid on vector bundles. We show that for each proper Lie groupoid G, the canonical homomorphism of G into the reconstructed groupoid T(G) is surjective, although — contrary to what happens in the case of groups — it may fail to be an isomorphism. We obtain necessary and sufficient conditions in order that G may be isomorphic to T(G) and, more generally, in order that T(G) may be a Lie groupoid. We show that if T(G) is a Lie groupoid, the canonical homomorphism G→T(G) is a submersion and the two groupoids have isomorphic categories of representations.     

Representation theory of 2-groups on Kapranov and Voevodsky's 2-vector spaces

In this paper the representation theory of 2-groups in 2-categories is considered, focusing the attention on the 2-category Rep2MatK(G) of representations of a 2-group G in (a version of) Kapranov and Voevodsky's 2-category of 2-vector spaces over a field K. The set of equivalence classes of such representations is computed in terms of the invariants π₀(G), π₁(G) and [α]∈H³(π₀(G),π₁(G)) classifying G, and the categories of intertwiners are described in terms of categories of vector bundles endowed with a projective action. In particular, it is shown that the monoidal category of finite dimensional linear representations (more generally, the category of [z]-projective representations, for any given cohomology class [z]∈H²(π₀(G),K^∗)) of the first homotopy group π₀(G) as well as its category of representations on finite sets both live in Rep2MatK(G), the first as the monoidal category of endomorphisms of the trivial representation (more generally, as the category of intertwiners between suitable 1-dimensional representations) and the second as a non-full subcategory of the homotopy category of Rep2MatK(G).     

Minimal degrees of faithful quasi-permutation representations for direct products of p-groups

In [2], the algorithms of c(G), q(G) and p(G), the minimal degrees of faithful quasi-permutation and permutation representations of a finite group G are given. The main purpose of this paper is to consider the relationship between these minimal degrees of non-trivial p-groups H and K with the group H×K.     

Geometric structure in smooth dual and local Langlands conjecture

This expository paper first reviews some basic facts about p-adic fields, reductive p-adic groups, and the local Langlands conjecture. If G is a reductive p-adic group, then the smooth dual of G is the set of equivalence classes of smooth irreducible representations of G. The representations are on vector spaces over the complex numbers. In a canonical way, the smooth dual is the disjoint union of subsets known as the Bernstein components. According to a conjecture due to ABPS (Aubert–Baum–Plymen–Solleveld), each Bernstein component has a geometric structure given by an appropriate extended quotient. The paper states this ABPS conjecture and then indicates evidence for the conjecture, and its connection to the local Langlands conjecture.     

Weighted zeta functions for quotients of regular coverings of graphs

Let G be a connected graph. We reformulate Stark and Terras' Galois Theory for a quotient H of a regular covering K of a graph G by using voltage assignments. As applications, we show that the weighted Bartholdi L-function of H associated to the representation of the covering transformation group of H is equal to that of G associated to its induced representation in the covering transformation group of K. Furthermore, we express the weighted Bartholdi zeta function of H as a product of weighted Bartholdi L-functions of G associated to irreducible representations of the covering transformation group of K. We generalize Stark and Terras' Galois Theory to digraphs, and apply to weighted Bartholdi L-functions of digraphs.     

The Continuous Zak Transform and Generalized Gabor Frames

Let G be a locally compact abelian group and H be a closed (not necessarily discrete) subgroup of G. In this article, we introduce the notion of Zak transform associated to H and obtain a necessary and sufficient condition to generate continuous Gabor frames for L ²(G). These results can be extended to non-abelian locally compact groups which are semidirect products. As an application, we obtain a characterization of admissible vectors for the regular and quasi regular representations.     

Supercuspidal Gelfand pairs

If G is a totally disconnected group and H is a closed subgroup then, according to the Gelfand-Kazhdan Lemma, if the double coset space H?G/H is preserved by an antiautomorphism of G of order two then (G,H) must be a Gelfand pair in the sense that Hom_H(π,1) has dimension at most one for each irreducible, admissible representation π of G. Under certain rather general restrictions, we show that if the symmetry property holds only for almost all double cosets, then (G,H) is a supercuspidal Gelfand pair in the sense that for all irreducible, supercuspidal representations π of G. There exist examples of supercuspidal Gelfand pairs which are not Gelfand pairs.     

On semi cover-avoiding subgroups of finite groups

A subgroup H of a finite group G is said to have the semi cover-avoiding property in G if there is a normal series of G such that H covers or avoids every normal factor of the series. In this paper, some new results are obtained based on the assumption that some subgroups have the semi cover-avoiding property in the group.     

Homological algebra for affine Hecke algebras

In this paper we study homological properties of modules over an affine Hecke algebra H. In particular we prove a comparison result for higher extensions of tempered modules when passing to the Schwartz algebra S, a certain topological completion of the affine Hecke algebra. The proof is self-contained and based on a direct construction of a bounded contraction of certain standard resolutions of H-modules.This construction applies for all positive parameters of the affine Hecke algebra. This is an important feature, since it is an ingredient to analyse how the irreducible discrete series representations of H arise in generic families over the parameter space of H. For irreducible non-simply laced affine Hecke algebras this will enable us to give a complete classification of the discrete series characters, for all positive parameters (we will report on this application in a separate article).