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 .

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 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.

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.

One way of realizing representations of the Heisenberg group is by using Fock representations, whose representation spaces are Hilbert spaces of functions on complex vector space with inner products associated to points on a Siegel upper half space. We generalize such Fock representations using inner products associated to points on a Hermitian symmetric domain that is mapped into a Seigel upper half space by an equivariant holomorphic map. The representations of the Heisenberg group are then given by an automorphy factor associated to a Kuga fiber variety. We introduce theta functions associated to an equivariant holomorphic map and study connections between such generalized theta functions and Fock representations described above. Furthermore, we discuss Jacobi forms on Hermitian symmetric domains in connection with twisted torus bundles over symmetric spaces.

Westwick's convexity theorem on the numerical range is generalized in the context of compact connected Lie groups.

Pointwise bounds for characters of representations of the classical, compact, connected, simple Lie groups are obtained with which allow us to study the singularity of central measures. For example, we find the minimal integer k such that any continuous orbital measure convolved with itself k times belongs to L2. We also prove that if k = rank G then μ 2k ∈ L1 for all central, continuous measures μ. This improves upon the known classical result which required the exponent to be dimension of the group G.

For a real vector space V acted on by a group K and fixed x and y in V, we consider the problem of finding the minimum (respectively, maximum) distance, relative to a K-invariant convex function on V, between x and elements of the convex hull of the K-orbit of y. We solve this problem in the case where V is a Euclidean space and K is a finite reflection group acting on V. Then we use this result to obtain an analogous result in the case where K is a maximal compact subgroup of a reductive group G with adjoint action on the vector component ρ of a Cartan decomposition of Lie G. Our results generalize results of Li and Tsing and of Cheng concerning distances to the convex hulls of matrix orbits.

We extend an uncertainty principle due to Cowling and Price to Euclidean spaces, Heisenberg groups and the Euclidean motion group of the plane. This uncertainty principle is a generalisation of a classical result due to Hardy. We also show that on the real line this uncertainty principle is almost equivalent to Hardy's theorem.

The Plancherel formula for various semisimple homogeneous spaces with non-reductive stability group is derived within the framework of the Bonnet Plancherel formula for the direct integral decomposition of a quasi-regular representation. These formulas represent a continuation of the author's program to establish a new paradigm for concrete Plancherel analysis on homogeneous spaces wherein the distinction between finite and infinite multiplicity is de-emphasized. One interesting feature of the paper is the computation of the Bonnet nuclear operators corresponding to certain exponential representations (roughly those induced from infinite-dimensional representations of a subgroup). Another feature is a natural realization of the direct integral decomposition over a canonical set of concrete irreducible representations, rather than over the unitary dual.

Let K be a nonarchimedean local field, let L be a separable quadratic extension of K, and let h denote a nondegenerate sesquilinear formk on L3. The Bruhat-Tits building associated with SU3(h) is a tree. This is applied to the study of certain groups acting simply transitively on vertices of the building associated with SL(3, F), F = Q3 or F3((X)).