Yury A. Neretin
Multiplication of Conjugacy Classes, Colligations, and Characteristic Functions of Matrix Argument
Preprint series: ESI preprints

MSC:

22E65 Infinite-dimensional Lie groups and their Lie algebras, See also {17B65, 58B25, 58H05}

47A48 Operator colligations (= nodes)

15A72 Vector and tensor algebra, theory of invariants, See Also {13A50, 14D25}

16W20 Automorphisms and endomorphisms, actions of groups and semigroups and their fixed rings

Abstract: We extend the classical construction of operator
colligations and characteristic functions.
Consider the group $G$ of finite block unitary matrices
of size $\alpha+\infty+\dots+\infty$ ($k$ times). Consider
the subgroup $K=\U(\infty)$, which consists of block diagonal unitary matrices
(with blocks 1 of size $\alpha$ and a matrix $u\in K$ repeated $k$ times).
It appears that there is a natural multiplication
on the conjugacy classes $G//K$. We construct 'spectral data' of conjugacy classes,
which visualize the multiplication and are sufficient for a reconstruction of a conjugacy class.


Keywords: infinite-dimensional groups, colligations, conjugacy classes, invariants, categorical quotient, Grassmannians, characteristic functions, semigroups, unitary group