Hardy Spaces on Two{sheeted Covering Semigroups Koufany and Rsted

  • K Koufany, B Rsted, K Odd, Det
  • Published 1997

Abstract

In this paper we study the minimal complex Lie semigroups associated with three classical series of groups by using a holomorphic continuation of a certain Cayley transform for the group. In particular we show, that for the symplectic group the odd part of the Hardy space on the double cover is isomorphic to the classical Hardy space on the Siegel upper half space corresponding to the symplectic group of twice the rank of the given group. 0. Introduction In this paper we shall give a detailed description of the minimal complex Lie semigroups associated with three of the four classical series of groups with an Hermitian symmetric space. These were found by Ol'shanski ((21]) as were the associated Hardy spaces on these semigroups ((22]), and recently there has been much interest in analysis of this type of function space In particular one would like to calculate the Cauchy{Szegg o kernel explicitly, and to compare these new Hardy spaces with those for classical bounded domains (see 3], 18], 19] and 23]). By using a natural Cayley transform, which might be thought of as a holomorphic continuation of a causal compactiication of the Lie group, we show that for the symplectic group the odd part of the Hardy space on the double cover is indeed isomorphic to the classical Hardy space on the Siegel upper half space corresponding to the symplectic group of twice the rank of the given group. One of the main technical points in this work is the actual construction of the double cover semigroup, isomorphic to Howe's oscillator semigroup (see 12] and 16]), via a choice of square root of a certain Jacobian; this is close in spirit to the construction of the Riemann surface for p z. Thus for the metaplectic group we obtain an explicit formula for the Cauchy{Szegg o kernel as well as the branching law for the classical Hardy space with respect to the product of two metaplectic groups. For the two remaining series of groups the Cayley transform goes into the tube domains for the Jordan algebras of complex resp. quaternion Hermitian matrices; in the rst case the Ol'shanski Hardy space is never isomorphic to the classical one, and in the second case we show that even after a necessary passage to the double cover the classical Hardy space is strictly contained in the odd part of the Hardy space on the double …

Cite this paper

@inproceedings{Koufany1997HardySO, title={Hardy Spaces on Two\{sheeted Covering Semigroups Koufany and Rsted}, author={K Koufany and B Rsted and K Odd and Det}, year={1997} }