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On generalized Cartan subspaces

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Abstract

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.

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Authors and Affiliations

  1. Department of Mathematics, North Carolina State University, Raleigh, NC, 27695, USA

    Aloysius G. Helminck

  2. Department of Mathematics, Brandeis University, Waltham, MA, 02454-9110, USA

    Gerald W. Schwarz

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  1. Aloysius G. Helminck
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  2. Gerald W. Schwarz
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Correspondence to Aloysius G. Helminck.

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Dedicated to T. A. Springer on his 85th birthday

First author is partially supported by NSF grant DMS-0532140 and NSA grant H98230-06-1-0098.

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Helminck, A.G., Schwarz, G.W. On generalized Cartan subspaces. Transformation Groups 16, 783–805 (2011). https://doi.org/10.1007/s00031-011-9151-8

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