## 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.

### 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
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