A Class of Parabolic k-Subgroups Associated with Symmetric k-Varieties (original) (raw)

On Orbit Decompositions for Symmetric k-Varieties

Orbit decompositions play a fundamental role in the study of symmetric k-varieties and their applications to representation theory and many other areas of mathematics, such as geometry, the study of automorphic forms and character sheaves. Symmetric k-varieties generalize symmetric varieties and are defined as the homogeneous spaces G k /H k , where G is a connected reductive algebraic group defined over a field k of characteristic not 2, H the fixed point group of an involution σ and G k (resp., H k ) the set of k-rational points of G (resp., H). In this contribution we give a survey of results on the various orbit decompositions which are of importance in the study of these symmetric k-varieties and their applications with an emphasis on orbits of parabolic k-subgroups acting on symmetric k-varieties. We will also discuss a number of open problems.

CONJUGACY CLASSES IN PARABOLIC SUBGROUPS OF SEMISIMPLE ALGEBRAIC GROUPS

Let G be a connected sernisimple algebraic group defined over an algebraically closed field K, let P be a parabolic subgroup of G and let V denote the unipotent radical of P. Assume that the set of unipotent conjugacy classes of G is finite. In this note we shall show that there exists veV such that C P (v), the conjugacy class of v in P, is an open subset of V.

On Rationality Properties of Involutions of Reductive Groups

Advances in Mathematics, 1993

Let k be a field of characteristic not two and G a connected linear reductive k-group. By a k-involution θ of G, we mean a k-automorphism θ 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 the representation theory of reductive groups are most rewarding. The application of discrete series of affine symmetric spaces to the cohomology of arithmetic subgroups [27] invites the study of Q-involutions. In the present paper, we give a treatment on rationality problems of general k-involutions. Here we generalize most of the earlier results [15, 16, 23, 29], sharpen some and add new ones. Let H be an open subgroup of the fixed point group G θ of an involution θ of G. In §1, we show that H 0 characterizes θ when G is semi-simple. It follows that (1.6) θ is defined over k if and only if H 0 is a k-subgroup of G. In §2, we deal with θ-stable k-split tori of G for a k-involution θ of G. The key, unlocking the door to rationality discussion, is the simple existence result (2.4) that every minimal parabolic k-subgroup P of G contains a θ-stable maximal k-split torus. In general, proper θ-stable parabolic k-subgroups of G do not exist. In §3, we present a simple criterion for their existence (3.4) and a structure theorem (3.5) for the minimal θ-stable parabolic k-subgroups of G. A parabolic subgroup Q of G is θ-split if Q and θ(Q) are opposite. In §4, we discuss θ-split parabolic k-subgroups of G following Vust [29]. It is known that G has proper θ-split parabolic k-subgroups if and only if the restriction of θ to the isotropic factor of G over k is nontrivial. The minimal θ-split parabolic k-subgroups of G are determined by the maximal (θ, k)-split tori of G (4.7) and are conjugate by elements of G k. However in general they are not H k-conjugate. To each k-involution θ of G, there correspond two root systems. The discussion is carried out in §5. As a consequence, we have the conjugacy theorem (5.8) for minimal θ-stable parabolic k-subgroups. Let P be a minimal parabolic k-subgroup and H a k-open subgroup of G θ. Consider the double coset space P k \ G k /H k. The geometry of these orbits is of importance in the study of Harish-Chandra modules [28] with k = C and of discrete series of affine symmetric spaces [8, 18] with k = R. The §6 deals with the reduction theory of the double coset space for general k. Our main result is Propositon 6.8 following Springer, and a slightly different characterization of Rossmann is given in 6.10. We also show that (6.16) P k \ G k /H k is finite when k is a local field. For k = R, this finiteness result is due to J. Wolf [30] (see also T. Matsuki [15]). Let be a root system in a finite dimensional real vector space V and θ an involution of V leaving invariant. Then θ induces an automorphism, also denoted by θ, of the Weyl group W of given by θ(w) = θ • w • θ, w ∈ W. An element w ∈ W is called a twisted involution if θ(w) = w −1. T. A. Springer initiated the study of the twisted involutions. The most elegant result is his decomposition theorem [23, Prop. 3.3]. Here we contribute certain uniqueness conditions of the decomposition ((iii) of 7.9). Inspired by the work of Matsuki [16], we present a new proof of constructive nature which yields also the classification of such decompositions (7.24). In §8, we establish some dimension formulas needed for our study on orbit closures. The double coset space P \ G/H has a unique open element, called the big cell. We characterize the big cell in 9.2. For g ∈ G k , let cl(PgH) denote the Zariski closure of PgH in G. The structure theorem of orbit closure is given in 9.5 in terms of the

On Orbit Closures of Symmetric Subgroups in Flag Varieties

Canadian Journal of Mathematics, 2000

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 θ. Generalizing work of Springer, we parametrize the (finite) orbit set K \ G/P and we determine the isotropy groups. As a consequence, we describe the closed (resp. affine) orbits in terms of θ-stable (resp. θ-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 on G/B having nonzero global sections, we show that the restriction map resX : H 0(G/B, ) → H 0(X, ) is surjective and that Hi (X, ) = 0 for i ≥ 1. Moreover, we describe the K-module H 0(X, ). This gives information on the restriction to K of the simple G-module H 0(G/B, ). Our construction is a geometric analogue of Vogan and Sepanski’s approach to extremal K-types.

The restriction of minuscule representations to parabolic subgroups

Mathematical Proceedings of the Cambridge Philosophical Society, 2003

Let G be a universal Chevalley group defined over an algebraically closed field F of arbitrary characteristic. In this paper we investigate the restrictions to parabolic subgroups of G of the irreducible FG-modules corresponding to minuscule highest weights via a combinatorial scheme of "chamber systems" indexed over certain cosets in the Weyl group.

Appendix: On parahoric subgroups

Advances in Mathematics, 2008

We give the proofs of some simple facts on parahoric subgroups and on Iwahori Weyl groups used in [H], [PR] and in [R]. 2000 Mathematics Subject Classification: Primary 11E95, 20G25; Secondary 22E20.

The projective characters of the symmetric groups that remain irreducible on subgroups

Journal of Algebra, 1991

An important problem in studying embeddings of one finite group in another is to determine all subgroups H of a group G for which an irreducible character of G remains irreducible upon restriction to H. Such questions arise, for example, in the study of maximal subgroups of the finite classical groups, where one seeks to classify triples H < G < G.L( V), where both H and G are quasisimple and act absolutely irreducibly in GL( V) (see Problem 4 in [1] and Problems 4 and 5 in 1211, for example). This problem has been solved when G is the symmetric group S,, or the alternating group A,, by Sax1 . In this paper we attack the same problem for faithful characters of a non-splitting double cover s,, of S,,. For each n 3 4, there are two such groups [IS]. Their irreducible representations, however, are very similar. Indeed. given a set of matrices in an irreducible representation of one double cover, we obtain a representation of the other by multiplying certain matrices by the scalars +i, where i = v' !-1. Consequently, subgroups for which a faithful character remains irreducible correspond in the two double covers. So with no loss of generality, we choose a particular s,,, following , which applies for all n. Here each transposition in S,, lifts to an element of order 4 in s,,, and disjoint transpositions in S,, generate a quaternion group Qx in 3,,.

On cuspidal representations of ppp-adic reductive groups

Bulletin of The American Mathematical Society, 1975

Let k be a p-adic field, and G a reductive connected algebraic group over k. Fix a maximal torus T of G which splits in an unramified extension of k, and which has the same split rank as the center of G. For each character 6 of T(k), satisfying some conditions, there is a cuspidal representation y$ of G{k) which is a sum of a finite number of irreducible representations; the correspondence 0 |-*-JQ is one-to-one on the orbits of such characters by the little Weyl group of T; furthermore, the formulas for the formal degree of JQ and its character for sufficiently regular elements of T(k) are given: they are formally the same as is the discrete series for real reductive groups.