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Automatic continuity of pseudocharacters on semisimple Lie groups. (English. Russian original) Zbl 1116.22002

Math. Notes 80, No. 3, 435-441 (2006); translation from Mat. Zametki 80, No. 3, 456-464 (2006).
A real valued function \(f\) on a group \(G\) is a real quasicharacter if the set \(\{f(gh)-f(g)- f(h)| \;g,h\in G\}\) is bounded. The quasicharacter \(f\) is a pseudocharacter on \(G\) if \(f(g^n)=nf(g)\) for all \(g\in G\) and \(n \in {\mathbb{Z}}\). This notion is also known as homogeneous quasimorphism and it has been used very much in the theory of bounded cohomology, the theory of diffeomorphism groups, symplectic geometry, combinatorial group theory, and the theory of group representations. The author obtains the following results. Theorem 1. Every pseudocharacter on a simple Lie group is continuous. Theorem 2. Let \(G\) be a simple Lie group. If \(G\) is not a Hermitian symmetric group or if the center of the group \(G\) is finite, then every pseudocharacter on \(G\) is identically zero. If \(G\) is a Hermitian symmetric group with infinite center, then every pseudocharacter on \(G\) is a multiple of the Guichardet-Wigner pseudocharacter on \(G\). In particular, every pseudocharacter on a simple Lie group is continuous.

MSC:

22E30 Analysis on real and complex Lie groups
43A40 Character groups and dual objects
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