We consider a class of fractal subsets of ${{\mathbb{R}}^{d}}$ formed in a manner analogous to the construction of the Sierpinski carpet. We prove a uniform Harnack inequality for positive harmonic functions; study the heat equation, and obtain upper and lower bounds on the heat kernel which are, up to constants, the best possible; construct a locally isotropic diffusion $X$ and determine its basic properties; and extend some classical Sobolev and Poincaré inequalities to this setting.

Using the close relationship between coactions of discrete groups and Fell bundles, we introduce a procedure for inducing a ${{C}^{*}}$ -coaction $\delta :D\to D\otimes {{C}^{*}}\left( G/N \right)$ of a quotient group $G/N$ of a discrete group $G$ to a ${{C}^{*}}$-coaction $\text{Ind}\,\delta \text{:}\,\text{Ind}\,D\to \text{Ind}\,D\otimes {{C}^{*}}\left( G \right)\,\text{of }G$. We show that induced coactions behave in many respects similarly to induced actions. In particular, as an analogue of the well known imprimitivity theorem for induced actions we prove that the crossed products $\text{Ind}\,D{{\times }_{\text{Ind}\,\delta }}\,G\,\text{and }D{{\times }_{\delta }}\,G/N$ are always Morita equivalent. We also obtain nonabelian analogues of a theorem of Olesen and Pedersen which show that there is a duality between induced coactions and twisted actions in the sense of Green. We further investigate amenability of Fell bundles corresponding to induced coactions.

This paper starts by introducing a bi-periodic summation formula for automorphic forms on a group $G\left( E \right)$, with periods by a subgroup $G\left( F \right)$, where $E/F$ is a quadratic extension of number fields. The split case, where $E=F\oplus F$, is that of the standard trace formula. Then it introduces a notion of stable bi-conjugacy, and stabilizes the geometric side of the bi-period summation formula. Thus weighted sums in the stable bi-conjugacy class are expressed in terms of stable bi-orbital integrals. These stable integrals are on the same endoscopic groups $H$ which occur in the case of standard conjugacy.

The spectral side of the bi-period summation formula involves periods, namely integrals over the group of $F$-adele points of $G$, of cusp forms on the group of $E$-adele points on the group $G$. Our stabilization suggests that such cusp forms—with non vanishing periods—and the resulting bi-period distributions associated to “periodic” automorphic forms, are related to analogous bi-period distributions associated to “periodic” automorphic forms on the endoscopic symmetric spaces $H\left( E \right)/H\left( F \right)$. This offers a sharpening of the theory of liftings, where periods play a key role.

The stabilization depends on the “fundamental lemma”, which conjectures that the unit elements of the Hecke algebras on $G$ and $H$ have matching orbital integrals. Even in stating this conjecture, one needs to introduce a “transfer factor”. A generalization of the standard transfer factor to the bi-periodic case is introduced. The generalization depends on a new definition of the factors even in the standard case.

Finally, the fundamental lemma is verified for $\text{SL}\left( 2 \right)$.

In general, the tensor product, $A\otimes B$, of the lattices $A$ and $B$ with zero is not a lattice (it is only a join-semilattice with zero). If $A\otimes B$ is a capped tensor product, then $A\otimes B$ is a lattice (the converse is not known). In this paper, we investigate lattices $A$ with zero enjoying the property that $A\otimes B$ is a capped tensor product, for every lattice $B$ with zero; we shall call such lattices amenable.

The first author introduced in 1966 the concept of a sharply transferable lattice. In 1972, H. Gaskill defined, similarly, sharply transferable semilattices, and characterized them by a very effective condition $\left( \text{T} \right)$.

We prove that a finite lattice $A$ is amenable iff it is sharply transferable as a join-semilattice.

For a general lattice $A$ with zero, we obtain the result: $A$ is amenable iff $A$ is locally finite and every finite sublattice of $A$ is transferable as a join-semilattice.

This yields, for example, that a finite lattice $A$ is amenable iff $A\otimes \text{F}\left( 3 \right)$ is a lattice iff $A$ satisfies $\left( \text{T} \right)$, with respect to join. In particular, ${{M}_{3}}\,\otimes \,\text{F}\left( 3 \right)$ is not a lattice. This solves a problem raised by R. W. Quackenbush in 1985 whether the tensor product of lattices with zero is always a lattice.

I consider a two-parameter family ${{B}_{s,t}}$ of unitary transforms mapping an ${{L}^{2}}$-space over a Lie group of compact type onto a holomorphic ${{L}^{2}}$-space over the complexified group. These were studied using infinite-dimensional analysis in joint work with $\text{B}$. Driver, but are treated here by finite-dimensional means. These transforms interpolate between two previously known transforms, and all should be thought of as generalizations of the classical Segal-Bargmann transform. I consider also the limiting cases $s\,\to \,\infty \,\text{and}\,s\,\to \,t/2$.

In this paper we use Langlands-Shahidi method and the result of Langlands which says that non self-conjugate maximal parabolic subgroups do not contribute to the residual spectrum, to prove the holomorphy of several completed automorphic $L$-functions on the whole complex plane which appear in constant terms of the Eisenstein series. They include the exterior square $L$-functions of $\text{G}{{\text{L}}_{\text{n}}},\,n$ odd, the Rankin-Selberg $L$-functions of $\text{G}{{\text{L}}_{n}}\times \,\text{G}{{\text{L}}_{m}},\,n\,\ne \,m$, and $L$-functions $L\left( s,\,\sigma ,\,r \right)$, where $\sigma $ is a generic cuspidal representation of $\text{S}{{\text{O}}_{10}}$ and $r$ is the half-spin representation of GSpin $\left( 10,\,\mathbb{C} \right)$. The main part is proving the holomorphy and non-vanishing of the local normalized intertwining operators by reducing them to natural conjectures in harmonic analysis, such as standard module conjecture.

Our objective in this sequel to $[18]$ is to develop extensions, to representations of tensor algebras over ${{C}^{*}}$-correspondences, of two fundamental facts about isometries on Hilbert space: The Wold decomposition theorem and Beurling’s theorem, and to apply these to the analysis of the invariant subspace structure of certain subalgebras of Cuntz-Krieger algebras.

When $H$ is a finite dimensional, semisimple, almost cocommutative Hopf algebra, we examine a table of characters which extends the notion of the character table for a finite group. We obtain a formula for the structure constants of the representation ring in terms of values in the character table, and give the example of the quantum double of a finite group. We give a basis of the centre of $H$ which generalizes the conjugacy class sums of a finite group, and express the class equation of $H$ in terms of this basis. We show that the representation ring and the centre of $H$ are dual character algebras (or signed hypergroups).