In these lectures we plan to present a survey of certain aspects of harmonic analysis on a Heisenberg nilmanifold Γ\Hn. Using Weil-Brezin-Zak transform we obtain an explicit decomposition of L(Γ\H) into irreducible subspaces invariant under the right regular representation of the Heisenberg group. We then study the Segal-Bargmann transform associated to the Laplacian on a nilmanifold and charac...

what we mean by alliteration of homogeneous words is that such alliteration is based on two or more lexical basis that is conspicuously coordinative and homogeneous with each other. they are homogeneous in that their form, structure, and tone, are similar. we can name two homogeneous words, two bases of homonym. it is evident that such words, though homogeneous from one hand, should necessarily...

Let G be a connected, simply connected nilpotent Lie group, identified with real algebraic subgroup of UT(n,R), and let Γ lattice in G, π:G→G∕Γ the quotient map. For semialgebraic X⊆G, more generally definable set an o-minimal structure on field, we consider topological closure π(X) compact nilmanifold G∕Γ. Our theorem describes cl(π(X)) terms finitely many families cosets subgroups G. The unde...

We consider the compactification of a Yang-Mills theory on three-dimensional nilmanifold. The generates in four space-time dimensions, coupled to specific scalar sector. geometry gives rise masses for zero-modes, proportional twist parameter study simple example an SU(3) model broken by non-trivial vacuum potential which three mass scales, two being at tree level, and third one loop level. poin...

We study equidistribution properties of translations on nilmanifolds along functions polynomial growth from a Hardy field. More precisely, if X = G/Γ is nilmanifold, a1, …, ak ∈ G are commuting nilrotations, and f1, fk field then we show that