Harmonic analysis on the infinite-dimensional unitary group
The author considers the group \(U(\infty)\) whose elements are infinite unitary matrices \(U=\left[U_{ij}\right]\), \(i,j\in{\mathbb N}\), such that \(U_{ij}=\delta_{ij}\) for sufficiently large \(i+j\). The main goal of harmonic analysis on this group, as well as on another `large' group \(S(\infty)\), is to find a decomposition of the most `natural' unitary representations of this group into irreducible components. The article [\textit{G. Olshanski}, J. Funct. Anal. 205, No. 2, 464--524 (2003; Zbl 1036.43002)] contains a construction of a family \(T_{zw}\) of representations of \(U(\infty)\times U(\infty)\). The family is indexed by two complex parameters \(z\) and \(w\) satisfying the condition \(\Re(z+w)>-1/2\). As the author notes, the spectral type of \(T_{zw}\) is essentially dependent of whether or not \(z\) and \(w\) are integers. The case when both \(z\) and \(w\) are non-integer is already investigated in [\textit{A. Borodin} and \textit{G. Olshanski}, Ann. Math. (2) 161, No. 3, 1319--1422 (2005; Zbl 1082.43003)]. So, the present article is devoted to the case when both parameters are integer. As the author noticed, in this case, the spectrum of the decomposition changes dramatically. The representation \(T_{zw}\) is decomposed into a direct sum of subrepresentations \(T_{pqrs}\) which are called blocks. Then, the spectral type of \(T_{zw}\) is determined by a collection of measures on some finite dimensional faces \(\Omega(p,q;r,s)\). The result is similar to the one obtained in [\textit{S. Kerov, G. Olshanski} and \textit{A. Vershik}, Invent. Math. 158, No. 3, 551--642 (2004; Zbl 1057.43005)] for the group \(S(\infty)\) and corresponding one-parameter family of representations \(T_z\), for integer \(z\). The article is organized as follows. Chapter~1 contains the main definitions like characters of a topological group \(K\), and a summary of several useful results obtained in the area. Chapter~2 is devoted to the construction of \(T_{zw}\). Chapter~3 deals with the commutant \({\mathcal A}\) of the representation \(T_{zw}\) and with the block decomposition. In particular, it is shown that \({\mathcal A}\) is a commutative algebra, and an expression for the transition function of \(T_{zw}\) is obtained. In Section~4, it is proven that, for arbitrary \(z\) and \(w\), the space \(V_{zw}\) of \(K\)-invariant vectors is isomorphic to the space \({\mathcal F}_{zw}\) of functions on \({\mathbb GT}\) satisfying a pseudoharmonicity condition and a Hardy type condition. When \(z\) and \(w\) are integer, this isomorphism agrees with the block decomposition of \(T_{zw}\) obtained in the previous section. This isomorphism is used to construct a \(K\)-invariant vector \(v_{pqrs}\) in each of the blocks \(T_{pqrs}\). In Section~5, it is shown that the vector \(v_{pqrs}\) is a cyclic vector of the representation \(T_{pqrs}\). Section~6 is devoted to proving some lemmas from Section~5. Finally, in Section~7, the author shows that, if \(\Im z\geq 0\), \(\Im w\geq 0\), \(\Re(z+w)>-1/2\), and \(z\) and \(w\) are both integer (for noninteger indices, it was already known), the representations \(T_{zw}\) are pairwise disjoint, i.e., do not have equivalent nonzero subrepresentations. The results obtained, the methods developed, and a good review of other results obtained in this area allow us to say that the article might be very interesting for specialists in abstract harmonic analysis and representation theory.
scientific article; zbMATH DE number 6110566
scientific article