A note on the multipliers and projective representations of semi-simple Lie groups (Q2736868)

Let \(G\) be a connected semisimple real Lie group and \(T\) be the circle group. A projective representation of \(G\) in a Hilbert space \(H\) is a measurable function \(\pi\) from \(G\) into the group \(U(H)\) of all unitary operators on \(H\), along with a \(T\)-valued function \(m\) on \(G\times G\) satisfying the following conditions: \(\pi(1)=Id\) and \(\pi(gg')=m(g,g')\pi(g)\pi(g')\) for all \(g\) and \(g'\) in \(G\). In the special case where \(m\) is the constant function \(1\), \(\pi\) is an ordinary representation of \(G\) in \(H\). The function \(m\) is called the multiplier of \(\pi\). It is well known that the set \(M(G)\) of all multipliers on \(G\) has a natural structure of an abelian group. The second cohomology group of \(G\) is some quotient \(H^{2}(G,T)\) of \(M(G)\). In the paper under review, the authors prove that there is a natural isomorphism between \(H^{2}(G,T)\) and the Pontryagin dual of the fundamental group of \(G\). Moreover, after defining the notion of equivalence of projective representations, they prove that there is a natural bijection between the set of (equivalence classes of) projective representations of \(G\) and the set of (equivalence classes of) some ordinary representations of the universal cover of \(G\). As an example, they consider the case of the Möbius group.