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A weighted Plancherel formula II. The case of the ball

Genkai Zhang (1992)

Studia Mathematica

The group SU(1,d) acts naturally on the Hilbert space L ² ( B d μ α ) ( α > - 1 ) , where B is the unit ball of d and d μ α the weighted measure ( 1 - | z | ² ) α d m ( z ) . It is proved that the irreducible decomposition of the space has finitely many discrete parts and a continuous part. Each discrete part corresponds to a zero of the generalized Harish-Chandra c-function in the lower half plane. The discrete parts are studied via invariant Cauchy-Riemann operators. The representations on the discrete parts are equivalent to actions on some holomorphic...

Generalizations of Milne’s U ( n + 1 ) q -Chu-Vandermonde summation

Jian-Ping Fang (2016)

Czechoslovak Mathematical Journal

We derive two identities for multiple basic hyper-geometric series associated with the unitary U ( n + 1 ) group. In order to get the two identities, we first present two known q -exponential operator identities which were established in our earlier paper. From the two identities and combining them with the two U ( n + 1 ) q -Chu-Vandermonde summations established by Milne, we arrive at our...

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