A generalized Cartan decomposition for the double coset space \((U(n_1)\times U(n_2)\times U(n_3))\setminus U(n)/(U(p)\times U(q))\). (English) Zbl 1124.22003
Double coset spaces \(L\backslash G/H\), where \(L\subset G\supset H\) is a triple of reductive Lie groups, are studied. In the non-symmetric case, i.e. when one of the pairs \((G,L)\) and \((G,H)\) is non-symmetric, no structure theory on the double coset space is known.
The author, motivated by recent works on ‘visible actions’ on complex manifolds and multiplicity-free representations, takes \(\left(U(n_1)\times U(n_2)\times U(n_3)\right)\backslash U(n)/\left(U(p)\times U(q)\right)\) as a test ‘non-symmetric and visible’ case and develops new techniques in finding an explicit decomposition of the double coset space. In particular, an analog of the Cartan decomposition \(G=LBH\) is proved for an explicit subset \(B\) of \(O(n)\). There are also applications to representation theory.
The author, motivated by recent works on ‘visible actions’ on complex manifolds and multiplicity-free representations, takes \(\left(U(n_1)\times U(n_2)\times U(n_3)\right)\backslash U(n)/\left(U(p)\times U(q)\right)\) as a test ‘non-symmetric and visible’ case and develops new techniques in finding an explicit decomposition of the double coset space. In particular, an analog of the Cartan decomposition \(G=LBH\) is proved for an explicit subset \(B\) of \(O(n)\). There are also applications to representation theory.
Reviewer: Janusz Grabowski (Warszawa)
MSC:
22E46 | Semisimple Lie groups and their representations |
32A37 | Other spaces of holomorphic functions of several complex variables (e.g., bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA)) |
43A85 | Harmonic analysis on homogeneous spaces |
11F67 | Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols |
53C50 | Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics |
53D20 | Momentum maps; symplectic reduction |