%0 Journal Article
%T Canonical and Boundary Representations on Rank One Para-Hermitian Spaces
%A Anatoli A. Artemov
%J Applied Mathematics
%P 35-40
%@ 2152-7393
%D 2013
%I Scientific Research Publishing
%R 10.4236/am.2013.411A3006
%X This work studies the canonical representations (Berezin representations) for para-Hermitian symmetric spaces of rank one. These spaces are exhausted up to the covering by spaces&nbsp;<em>G/H</em>&nbsp;with <em>G</em> = SL(<em>n</em>,R),<em>H</em> = GL(<em>n</em>-1,R)&nbsp;. For Hermitian symmetric spaces <em>G/K</em>, canonical representations were introduced by Berezin and Vershik-Gelfand-Graev. They are unitary with respect to some invariant non-local inner product (the Berezin form). We consider canonical representations in a wider sense: we give up the condition of unitarity and let these representations act on spaces of distributions. For our spaces <em>G/H</em>, the canonical representations turn out to be tensor products of representations of maximal degenerate series and contragredient representations. We decompose the canonical representations into irreducible constituents and decompose boundary representations.
%K Para-Hermitian Symmetric Spaces
%K Overgroups
%K Canonical Representations
%K Boundary Representations
%K Poisson and Fourier Transforms
%U http://www.scirp.org/journal/PaperInformation.aspx?PaperID=39815