### $\mathbb{Z}$-gradations of Lie algebras and infinitesimal generators.

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In this article, we prove a generalisation of Bochner-Godement theorem. Our result deals with Olshanski spherical pairs $(G,K)$ defined as inductive limits of increasing sequences of Gelfand pairs ${(G\left(n\right),K\left(n\right))}_{n\ge 1}$. By using the integral representation theory of G. Choquet on convex cones, we establish a Bochner type representation of any element $\varphi $ of the set ${\mathcal{P}}^{\u266e}\left(G\right)$ of $K$-biinvariant continuous functions of positive type on $G$.

In this note, we generalize the results in our previous paper on the Casimir operator and Berezin transform, by showing the $({L}^{2},{L}^{2})$-continuity of a generalized Berezin transform associated with a branching problem for a class of unitary representations defined by invariant elliptic operators; we also show, that under suitable general conditions, this generalized Berezin transform is $({L}^{p},{L}^{p})$-continuous for $1\le p\le \infty .$

We show that a Poisson Lie group (G,π) is coboundary if and only if the natural action of G×G on M=G is a Poisson action for an appropriate Poisson structure on M (the structure turns out to be the well known ${\pi}_{+}$). We analyze the same condition in the context of Hopf algebras. A quantum analogue of the ${\pi}_{+}$ structure on SU(N) is described in terms of generators and relations as an example.

In [8], we studied the problem of local solvability of complex coefficient second order left-invariant differential operators on the Heisenberg group ℍₙ, whose principal parts are "positive combinations of generalized and degenerate generalized sub-Laplacians", and which are homogeneous under the Heisenberg dilations. In this note, we shall consider the same class of operators, but in the presence of left invariant lower order terms, and shall discuss local solvability for these operators in a complete...

A result by G. H. Hardy ([11]) says that if f and its Fourier transform f̂ are $O\left(\right|x{|}^{m}{e}^{-\alpha x\xb2})$ and $O\left(\right|x\left|\u207f{e}^{-x\xb2/\left(4\alpha \right)}\right)$ respectively for some m,n ≥ 0 and α > 0, then f and f̂ are $P\left(x\right){e}^{-\alpha x\xb2}$ and ${P}^{\text{'}}\left(x\right){e}^{-x\xb2/\left(4\alpha \right)}$ respectively for some polynomials P and P’. If in particular f is as above, but f̂ is $o\left({e}^{-x\xb2/\left(4\alpha \right)}\right)$, then f = 0. In this article we will prove a complete analogue of this result for connected noncompact semisimple Lie groups with finite center. Our proof can be carried over to the real reductive groups of the Harish-Chandra class.