Abstract
We extend the Paley–Wiener theorem for Riemannian symmetric spaces to an important class of infinite-dimensional symmetric spaces. For this we define a notion of propagation of symmetric spaces and examine the direct (injective) limit symmetric spaces defined by propagation. This relies on some of our earlier work on invariant differential operators and the action of Weyl group invariant polynomials under restriction.
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Notes
More detailed information is given by the Satake–Tits diagram for M; see [1] or [9, pp. 530–534]. In that classification the case \(\operatorname {SU}(p,1)\), p≧1, is denoted by AIV, but here it appears in AIII. The case \(\operatorname {SO}(p,q)\), p+q odd, p≧q>1, is denoted by BI as in this case the Lie algebra \(\mathfrak {g}_{\mathbb{C}}=\mathfrak{so}(p+q,\mathbb{C})\) is of type B. The case \(\operatorname {SO}(p,q)\), with p+q even, p≧q>1 is denoted by DI as in this case \(\mathfrak {g}_{\mathbb{C}}\) is of type D. Finally, the case \(\operatorname {SO}(p,1)\), p even, is denoted by BII and \(\operatorname {SO}(p,1)\), p odd, is denoted by DII.
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Communicated by Jiri Dadok.
The research of G. Ólafsson was supported by NSF grants DMS-0801010 and DMS-1101337.
The research of J.A. Wolf was partially supported by NSF grant DMS-0652840.
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Ólafsson, G., Wolf, J.A. The Paley–Wiener Theorem and Limits of Symmetric Spaces. J Geom Anal 24, 1–31 (2014). https://doi.org/10.1007/s12220-013-9467-9
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DOI: https://doi.org/10.1007/s12220-013-9467-9