Vol 24, No 128 (2019)

This is the summary of a part of the talk delivered at the workshop held at the Tambov University in September 2012, reporting several results on Jacobi groups and its holomorphic representations published by the authors.

Earlier we described canonical (labelled by λ ∈C ) and accompanying boundary representations of the group G = SU (1,1) on the Lobachevsky plane D in sections of linear bundles and decomposed canonical representations into irreducible ones. Now we decompose representations acting on distributions concentrated at the boundary of D . In the generic case 2λ ∉N they are diagonalizable, in the exceptional case Jordan blocks appear.

The conditions of continuity of the implicit set-valued map and the inverse setvalued map acting in topological spaces are proposed. For given mappings f : T ×X → Y , y : T → Y , where T,X,Y are topological spaces, the space Y is Hausdorff, the equation f(t,x) = y(t) with the parameter t ∈ T relative to the unknown x ∈ X is considered. It is assumed that for some multi-valued map U : T ⇉ X for all t ∈ T the inclusion f(t,U(t)) ∋ y(t) is satisfied. An implicit mapping R U : T ⇉ X , which associates with each value of the parameter t ∈ T the set of solutions x(t) ∈ U(t) of this equation. It is proved that R U is upper semicontinuous at the point t 0 ∈ T , if the following conditions are satisfied: for any x ∈ X the map f is continuous at ( t 0 ,x) , the map y is continuous at t 0 , a multi-valued map U is upper semicontinuous at the point t 0 and the set U( t 0 ) ⊂ X is compact. If, in addition, with the value of the parameter t 0 , the solution to the equation is unique, then the map R U is continuous at t 0 and any section of this map is also continuous at t 0 . The listed results are applied to the study of a multi-valued inverse mapping. Namely, for a given map g : X → T we consider the equation g(x) = y with respect to the unknown x ∈ X . We obtain conditions for upper semicontinuity and continuity of the map V U : T ⇉ X , V U (t) = {x ∈ U(t) : g(x) = t} , t ∈ T .

We offer a variant of Radon transforms for a pair X and Y of hyperboloids in R3 defined by [x,x] = 1 and [y,y] = -1 , y 1 ≥ 1 , respectively, here [x,y] = - x 1 y 1 + x 2 y 2 + x 3 y 3 . For a kernel of these transforms we take δ([x,y]) , δ(t) being the Dirac delta function. We obtain two Radon transforms D(X) → C ∞ (Y) and D(Y) → C ∞ (X) . We describe kernels and images of these transforms. For that we decompose a sesqui-linear form with the kernel δ([x,y]) into inner products of Fourier components.