Deformation of irreducible unitary representations of discrete series of Hermitian symmetric simple Lie groups in the class of pure pseudorepresentations
The deformation of irreducible unitary representations (IUR) of a simple Hermitian symmetric Lie group \(G\) is studied. It is shown that for each IUR \(\sigma \) of \(G\) in some Hilbert space \(H\) there is a one parameter family \(t\to \pi _{t}^{\sigma }\), \(t\in R\), of pairwise nonequivalent pure irreducible unitary \(\varepsilon _{t}\)-pseudorepresentations of \(G\) in the same space \(H\) such that the following two conditions hold: (1) \(\varepsilon _{t}\to 0\) as \(t\to 0\); (2) the norm \(\| \pi _{t}^{\sigma }( g) -\sigma ( g) \| \to 0\) as \(t\to 0\) for each \(g\in G\). This result is applied to construct explicitly the deformation of irreducible unitary principal series representations of \(G=SU(1,1)\). The universal covering group \(\widetilde{G}\) is defined and its nontrivial pseudocharacter \(\chi \) is evaluated. Then, a pure pseudorepresentation of the quotient group of \(\widetilde{G}\) by its center which is, in particular, a pure pseudorepresentation of the group \(SU(1,1)\), is constructed. It is proven that such pseudorepresentations are pairwise nonequivalent. It is also suggested that the deformation of the discrete series representations of \(SU(1,1)\) can be constructed in a similar way.
scientific article; zbMATH DE number 2115589