We describe the supercuspidal representations of Sp4(F), for F a non-archimedean local field of residual characteristic different from two, and determine which are generic.

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.

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.

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 .

Introduction. Polyhedra in 3-dimensional hyperbolic space which give rise to discrete groups generated by reflections in their faces have been investigated in [14], [17], [29] and in the case of tetrahedra there are precisely nine compact non-congruent ones with dihedral angles integral submultiples of π [14]. These polyhedral groups give rise to hyperbolic 3-orbifolds and examples of these have been studied, for example, in [3], [15], [18], [24], [25].

Let ℝ∞ be the direct limit of the Euclidean spaces ℝn. Now the orthogonal group O(∞) acts on ℝn and the direct limit O(∞) of the groups O(∞) acts on ℝ∞. The infinite pin group Pin(∞) is an extension of ℤ2 by O(∞) and admits the following presentation: the generators are the unit vectors xf in ℝ∞ and the relations are

Let G and H be Hausdorff locally compact groups. By R(G, H) we denote the space of continuous homomorphisms of G into H equipped with the compact-open topology, namely that which is generated by subsets of the form

where K is any compact subset of G and U is any open subset of H. Further, let R0(G, H) be the subset of R(G, H) consisting of elements r satisfying the conditions that the quotient space H/r(G) is compact and r is proper, i.e., the action of G on H defined as the action by left translations of r(G) on H is proper. It has been shown by H. Abels [2] that if G and H are such that at least one of them has a compact defining subset then R0(G,H) is open in R(G,H). Moreover, for each r0 ∈ R0(G,H) there exists a neighbourhood M and a compact subset F of H such that r(G) F = H for all r ∈ M and for each compact subset K of H the union is a relatively compact subset of G. It is furthermore shown in [1] that if G contains no small subgroups and H is connected then the subset of R0(G, H) consisting of isomorphisms of G into H is open in R(G, H). These results, in the case in which H is a connected Lie Group and G is a discrete group, have been established by Weil in [5] and [6] appendix 1.

Every n-dimensional manifold admits an embedding in R2n by the result of H. Whitney [11]. Lie groups are parallelizable and so by the theorem of M. W. Hirsch [5] there is an immersion of any Lie group in codimension one. However no general theorem is known which asserts that a parallelizable manifold embeds in Euclidean space of dimension less than 2n. Here we give a method for constructing smooth embeddings of compact Lie groups in Euclidean space. The construction is a fairly direct one using the geometry of the Lie group, and works very well in some cases. It does not give reasonable results for the group Spin (n) except for low values of n. We also give a method for constructing some embeddings of Spin (n), this uses the embedding of SO(n) that was constructed by the general method and an embedding theorem of A. Haefliger [3]. Although this is a very ad hoc method, it has some interest as it seems to be the first application of Haefliger's theorem which gives embedding results appreciably below twice the dimension of the manifold. The motivation for this work was to throw some light on the problem of the existence of low codimensional embeddings of parallelizable manifolds.