Subelliptic estimates on compact semisimple Lie groups

In this paper, we consider a natural subelliptic structure in semisimple, compact and connected Lie groups, and estimate the constant in the so-called subelliptic Friedrichs-Knapp-Stein inequality, which has implications in the regularity theory of p-energy minimizers.     

The spherical Bochner theorem on semisimple Lie groups

Let G be a connected semisimple Lie group with finite center and K a maximal compact subgroup. Denote (i) Harish-Chandra's Schwartz spaces by $\text{C}$^p(G)(0<p?2), (ii) the K-biinvariant elements in $\text{C}$^p(G) by $\text{I}$^p(G), (iii) the positive definite (zonal) spherical functions by $\text{P}$, and (iv) the spherical transform on $\text{C}$^p(G) by ? → $\text{}\text{?}$gj. For T a positive definite distribution on G it is established that (i) T extends uniquely onto $\text{C}$^l(G), (ii) there exists a unique measure μ of polynomial growth on $\text{P}$ such that T[ψ]=∫pψdμ for all ψ?I¹(G) (iii) all measures μ of polynomial growth on $\text{P}$ are obtained in this way, and (iv) T may be extended to a particular $\text{I}$^p(G) space (1 ? p ? 2) if and only if the support of μ lies in a certain easily defined subset of $\text{P}$. These results generalize a well-known theorem of Godement, and the proofs rely heavily on the recent harmonic analysis results of Trombi and Varadarajan.     

Weakly almost periodic functions on semisimple Lie groups

IfG is a semisimple analytic group with finite center, it is proved thatG admits only the “obvious” weakly almost periodic functions. The analysis yields also an intrinsic proof of Moore's ergodicity theorem [7].     

Generating sets for compact semisimple Lie groups

Let be a compact connected semisimple Lie group. We prove that the subset of consisting of pairs which topologically generate is Zariski open.

     

Stratonovich-Weyl correspondence for compact semisimple Lie groups

Let G be a compact Lie group, M a G-homogeneous space and π a unitary representation of G realized on a Hilbert space of functions on M. We give a general presentation of the Stratonovich-Weyl correspondence associated with π. In the case when G is a compact semisimple Lie group and π _λ an irreducible representation of G with highest weight λ, we study the Stratonovich-Weyl symbol of the derived operator d π _λ (X) for X in the Lie algebra of G and its behavior as λ goes to infinity.     

On subsemigroups of semisimple Lie groups

In this paper we classify the subsemigroups of any connected semisimple Lie groupG which areK-bi-invariant, whereG=KAN is an Iwasawa decomposition ofG.     

Automatic continuity of pseudocharacters on semisimple Lie groups

It is proved that an arbitrary pseudocharacter on a semisimple Lie group is continuous.     

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.