caa a22 4500 445862351 CHVBK 20180317145447.0 cr unu---uuuuu 170323e20110901xx s 000 0 eng 10.1007/s00031-011-9151-8 doi (NATIONALLICENCE)springer-10.1007/s00031-011-9151-8 On generalized Cartan subspaces [Elektronische Daten] [Aloysius Helminck, Gerald Schwarz] Let G be a connected reductive algebraic group defined over a field k of characteristic not 2, θ an involution of G defined over k, H a k-open subgroup of the fixed point group of θ and G k (resp. H k) the set of k-rational points of G (resp. H). The variety G k/H k is called a symmetric k-variety. For real and p-adic symmetric k-varieties the space L 2(G k/H k) of square integrable functions decomposes into several series, one for each H k-conjugacy class of Cartan subspaces of G k/H k. In this paper we give a characterization of the H k-conjugacy classes of these Cartan subspaces in the case that there exists a splitting extension of order 2 and (G, σ) is (σ, k)-split conjugate (see Subsection 3.8). This condition is satisfied for k the real numbers and several other fields for which the symmetric k-variety has a splitting extension of order 2. For $$ k = \mathbb{R} $$ we prove a number of additional results as well. Springer Science+Business Media, LLC, 2011 Helminck Aloysius Department of Mathematics, North Carolina State University, 27695, Raleigh, NC, USA aut Schwarz Gerald Department of Mathematics, Brandeis University, 02454-9110, Waltham, MA, USA aut Transformation Groups SP Birkhäuser Verlag Boston 16/3(2011-09-01), 783-805 1083-4362 16:3<783 2011 16 31 https://doi.org/10.1007/s00031-011-9151-8 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s00031-011-9151-8 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Helminck Aloysius Department of Mathematics, North Carolina State University, 27695, Raleigh, NC, USA aut NATIONALLICENCE 700 1- Schwarz Gerald Department of Mathematics, Brandeis University, 02454-9110, Waltham, MA, USA aut NATIONALLICENCE 773 0- Transformation Groups SP Birkhäuser Verlag Boston 16/3(2011-09-01), 783-805 1083-4362 16:3<783 2011 16 31 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer