Aloysius G. Helminck

Algebraic Groups with a Commuting Pair of Involutions and Semisimple Symmetric Spaces

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 describe all semisimple locally symmetric spaces together with their line structure. Involutorial automorphisms of g occur in several places in the literature. Cartan has already shown that for F= C, the isomorphism classes of involutorial automorphisms of g correspond bijectively to the isomorphism classes of real semisimple Lie algebras, which correspond in their turn to the isomorphism classes of Riemannian symmetric spaces (see Helgason [ 111). If one lifts this involution to the group G, then the present work gives a characteristic free description of these isomorphism classes. In a similar manner we can show that semisimple locally symmetric spaces correspond to pairs of commuting involutorial automorphisms of g. Namely let (go, a) be a semisimple locally symmetric pair; i.e., go is a real semisimple Lie algebra and rr E Aut(g,) an involution. Then by a result of Berger [2], there exists a Cartan involution 8 of go, such that 00 = ea. If we denote the complexilication of go by g, then o and 8 induce a pair of commuting involutions of g. Conversely, if c, 8 E Aut(g) are commuting involutions, then c and 8 determine two locally semisimple symmetric pairs. For if u is a O- and O-stable compact real form with conjugation r, then (ger, (~1 ge,) and (g,,, f3 I ger) are semisimple locally symmetric pairs where

On Orbit Closures of Symmetric Subgroups in Flag Varieties

We study K-orbits in G/P where G is a complex connected reductive group, P ⊆ G is a parabolic subgroup, and K ⊆ G is the fixed point subgroup of an involutive automorphism �. Generalizing work of Springer, we parametrize the (finite) orbit set K\ G/P and we determine the isotropy groups. As a conse- quence, we describe the closed (resp. affine) orbits in terms of �-stable (resp. �-split) parabolic subgroups. We also describe the decomposition of any (K, P)-double coset in G into (K, B)-double cosets, where B⊆ P is a Borel subgroup. Finally, for certain K-orbit closures X ⊆ G/B, and for any homogeneous line bundle L on G/B having nonzero global sections, we show that the restriction map resX : H 0 (G/B,L)→ H 0 (X,L) is surjective and that H i (X,L) = 0 for i ≥ 1. Moreover, we describe the K-module H 0 (X,L). This gives information on the restriction to K of the simple G-module H 0 (G/B,L). Our construction is a geometric analogue of Vogan and Sepanskis approach to extremal K-types.

Tori Invariant under an Involutorial Automorphism II

Abstract The geometry of the orbits of a minimal parabolick-subgroup acting on a symmetrick-variety is essential in several areas, but its main importance is in the study of the representations associated with these symmetrick-varieties (see for example [5, 6, 20, and 31]). Up to an action of the restricted Weyl group ofG, these orbits can be characterized by theHk-conjugacy classes of maximalk-split tori, which are stable underk-involutionθassociated with the symmetrick-variety. HereHis a openk-subgroup of the fixed point group ofθ. This is the second in a series of papers in which we characterize and classify theHk-conjugacy classes of maximalk-split tori. The first paper in this series dealt with the case of algebraically closed fields. In this paper we lay the foundation for a characterization and classification for the case of nonalgebraically closed fields. This includes a partial classification in the cases, where the base field is the real numbers, p -adic numbers, finite fields, and number fields.

CLASSIFICATION OF INVOLUTIONS OF SL(2, k)

ABSTRACT In this paper we give a simple characterization of the isomorphy classes of involutions of 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 is k -anisotropic. In those cases the corresponding symmetric k -variety consists of semisimple elements.

Orbits and invariants associated with a pair of commuting involutions

Let σ, θ be commuting involutions of the connected reductive algebraic group G where σ, θ and G are defined over a (usually algebraically closed) field k, char k = 2. We have fixed point groups H := G and K := G and an action (H × K ) × G → G, where ((h, k), g) → hgk−1, h ∈ H, k ∈ K, g ∈ G. Let G//(H × K ) denote Spec O(G)H×K (the categorical quotient). Let A be maximal among subtori S of G such that θ(s) = σ(s) = s−1 for all s ∈ S. There is the associated Weyl group W := WH×K (A). We show: • The inclusion A → G induces an isomorphism A/W ∼ → G//(H × K ). In particular, the closed (H × K )-orbits are precisely those which intersect A. • The fibers of G → G//(H × K ) are the same as those occurring in certain associated symmetric varieties. In particular, the fibers consist of finitely many orbits. We investigate: • The structure of W and its relation to other naturally occurring Weyl groups and to the action of σθ on the A-weight spaces of . • The relation of the orbit type stratifications of A/W and G//(H × K ). Along the way we simplify some of Richardson’s proofs for the symmetric case σ = θ, and at the end we quickly recover results of Berger, Flensted-Jensen, Hoogenboom and Matsuki [Ber57, FJ78, Hoo84, Mat97] for the case k = .

A class of parabolic -subgroups associated with symmetric -varieties

Let G be a connected reductive algebraic group defined over a field k of characteristic not 2, σ an involution of G defined over k, H a k-open subgroup of the fixed point group of σ, 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 generically contributes to the Plancherel decomposition of L2(Gk/Hk) if and only if the parabolic k-subgroup is σ-split. So for a study of these induced representations a detailed description of the Hk-conjucagy classes of these σ-split parabolic k-subgroups is needed. In this paper we give a description of these conjugacy classes for general symmetric kvarieties. This description can be refined to give a more detailed description in a number of cases. These results are of importance for studying representations for real and p�-adic symmetric k-varieties.

Smoothness of Quotients Associated With a Pair of Commuting Involutions

Let �, � be commuting involutions of the connected semisimple algebraic group G where �, � and G are defined over an algebraically closed field k, char k = 0. Let H := Gand K := Gbe the fixed point groups. We have an action (H × K) × G → G, where ((h, k), g) 7→ hgk 1 , h ∈ H, k ∈ K, g ∈ G. Let G//(H × K) denote the categorical quotient Spec O(G) H×K. We determine when this quotient is smooth. Our results are a generalization of those of Steinberg (Ste75), Pittie (Pit72) and Richardson (Ric82) in the symmetric case where � = � and H = K.

Computing Orbits of Minimal Parabolic k -subgroups Acting on Symmetric k -varieties

In this paper we present an algorithm to compute the orbits of a minimal parabolic k -subgroup acting on a symmetric k -variety and most of the combinatorial structure of the orbit decomposition. This algorithm can be implemented in LiE, GAP4, Magma, Maple or in a separate program. These orbits are essential in the study of symmetric k -varieties and their representations. In a similar way to the special case of a Borel subgroup acting on the symmetric variety, (see A. G. Helminck. Computing B -orbits on G/H. J. Symb. Comput.,21 , 169?209, 1996.) one can use the associated twisted involutions in the restricted Weyl group to describe these orbits (see A. G. Helminck and S. P. Wang. On rationality properties of involutions of reductive groups. Adv. Math., 99, 26?96, 1993). However, the orbit structure in this case is much more complicated than the special case of orbits of a Borel subgroup. We will first modify the characterization of the orbits of minimal parabolic k -subgroups acting on the symmetric k -varieties given in Helminck and Wang (1993), to illuminate the similarity to the one for orbits of a Borel subgroup acting on a symmetric variety in Helminck (1996). Using this characterization we show how the algorithm in Helminck (1996) can be adjusted and extended to compute these twisted involutions as well.

Computing B -orbits on G/H

Abstract 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 HarishChandra modules (see Vogan, 1983). There are several descriptions of these orbits, but in practice it is actually very difficult and cumbersome to compute the orbits and their closures. Since the characterizations of theseorbits are very combinatorial in nature, this work could conceivably be done by a computer. In this paper we prove a number of additional properties of these orbits and combine these with properties of the various descriptions of these orbits to obtain an efficient algorithm. This algorithm can be implemented on a computer by using existing symbolic manipulation programs or by writing an independent program.

Symmetry and Spaces

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