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Introduction. Let k be a field of characteristic not two and G a connected linear reductive k-group. By a k-involution Î¸ of G, we mean a k-automorphism Î¸ of G of order two. For k = R, C or an algebraically closed field, such involutions have been extensively studied emerging from different interests. As manifested in [8, 18, 28], the interactions with theâ€¦ (More)

Let G be a connected reductive algebraic group defined over an algebraically closed field F of characteristic not 2. Denote the Lie algebra of G by 9. In this paper we shall classify the isomorphism classes of ordered pairs of commuting involutorial automorphisms of G. This is shown to be independent of the characteristic of F and can be applied to describeâ€¦ (More)

Let Ïƒ , Î¸ be commuting involutions of the connected reductive algebraic group G, where Ïƒ , Î¸ , andG are defined over a (usually algebraically closed) field k, char k = 2. We have fixed point groupsH := G andK := G and an action (HÃ—K)Ã—G â†’ G, where ((h, k), g) â†’ hgkâˆ’1, h âˆˆ H , k âˆˆ K , g âˆˆ G. Let G/(H Ã—K) denote SpecO(G)HÃ—K (the categorical quotient). Let A beâ€¦ (More)

Let G be a connected reductive algebraic group defined over a field k of characteristic not 2, Ïƒ an involution of G defined over k, H a k-open subgroup of the fixed point group of Ïƒ, Gk (resp. Hk) the set of k-rational points of G (resp. H) and Gk/Hk the corresponding symmetric k-variety. A representation induced from a parabolic k-subgroup of G genericallyâ€¦ (More)

Let G be a connected reductive algebraic group defined over a field k of characteristic not 2, Î¸ an involution of G defined over k, H a k-open subgroup of the fixed point group of Î¸ and Gk (resp. Hk) the set of k-rational points of G (resp. H). The variety Gk/Hk is called a symmetric k-variety. These varieties occur in many problems in representationâ€¦ (More)

- Aloysius G. Helminck
- J. Symb. Comput.
- 1996

The orbits of a Borel subgroup acting on a symmetric variety G/H occur in several areas of mathematics. For example, these orbits and their closures are essential in the study of Harish Chandra modules (see [Vog83]). There are several descriptions of these orbits, but in practice it is actually very difficult and cumbersome to compute the orbits and theirâ€¦ (More)

In this paper we give a simple characterization of the isomorphy classes of involutions of SL(2, k) with k any field of characteristic not 2. We also classify the isomorphy classes of involutions for k algebraically closed, the real numbers, the -adic numbers and finite fields. We determine in which cases the corresponding fixed point group H isâ€¦ (More)

- Aloysius G. Helminck, G. F. HELMINCK
- 1999

In this paper we present a geometric realization of infinite dimensional analogues of the finite dimensional representations of the general linear group. This requires a detailed analysis of the structure of the flag varieties involved and the line bundles over them. In general the action of the restricted linear group can not be lifted to the line bundlesâ€¦ (More)

- Aloysius G. Helminck, G. F. HELMINCK
- 2002

In this paper we introduce the infinite-dimensional flag varieties associated with integrable systems of the KdVand Toda-type and we discuss the structure of these manifolds. As an example we treat the Fubini-Study metric on the projective space associated with a separable complex Hilbert space and we conclude by showing that all flag varieties introducedâ€¦ (More)

The geometry of the orbits of a minimal parabolic k-subgroup acting on a symmetric k-variety is essential in several areaâ€™s, but its main importance is in the study of the representations associated with these symmetric k-varieties (see for example [6], [5], [20] and [31]). Up to an action of the restricted Weyl group of G, these orbits can be characterizedâ€¦ (More)