Minimal Parabolic Subgroup Research Articles

Character formulas and Frobenius subgroups of algebraic groups

Let G be a connected semisimple algebraic group over a field of non-zero characteristic p, with a maximal torus T. On G we have the Frobenius endomorphism F induced by the pth power map on the underlying field. The (scheme-theoretic) kernel of F is the first Frobenius kernel G,, and the inverse image of T under F is the “Frobenius subgroup” G, T. The purpose of this paper is to study certain questions in the (rational) representation theory of G, T. Let B be a Bore1 subgroup of G which contains T. In [6], Cline, Parshall, and Scott obtained a result which describes the functor “induction from B to G” as the composite of functors of the form “induction from B to a minimal parabolic subgroup P” over a sequence of minimal parabolics coming from a given reduced expression of the long word in the Weyl group. In Section 5, we obtain the analogous result for G, T. In Section 6, we develop a theory of intertwines for a certain class of G, T-modules. This gives a direct proof of an extended strong linkage principle for G, T, independent of the strong linkage principle for G. (Originally, Jantzen [17] proved that strong linkage for G, T, and more generally for G, T, n > 1, follows from the strong linkage principle for G. 1 Recently we have applied this method to give a new proof of an extended strong linkage principle for G; see [ 121. From the results of Section 5 we can write down a formula for the image of one of the intertwines in our class of modules. (This also follows from results of Jantzen [16, 171.) Composing the intertwines in question, relative to the class of modules corresponding to a given character of T, gives a map whose image is the irreducible G, T-module of that highest weight. In case the character is a restricted dominant character, the image coincides with the irreducible G-module of that highest weight, by a result

A convexity theorem for semisimple symmetric spaces

In this paper we prove a generalization of the convexity theorem in [3] for a symmetric space G/H. Here G is a connected semisimple Lie group with finite center, σ an involution of G and H an open subgroup of the group G of fixed points for σ. The generalization involves Iwasawa decompositions related to minimal parabolic subgroups of G of arbitrary type instead of the particular type of parabolic subgroup considered in [3]. From now on we assume more generally that G is a real reductive group of the HarishChandra class; this will allow an inductive argument relative to the real rank of G. Let θ : G → G be a Cartan involution of G that commutes with σ; for its existence, see [20, Thm. 6.16]. The associated group K := G of fixed points is a maximal compact subgroup of G. For the infinitesimal involutions determined by σ and θ we use the same symbols: θ, σ : g → g; here g denotes the Lie algebra of G. With respect to the infinitesimal involutions, g decomposes as

Local coefficients as Artin factors for real groups

Introduction. The purpose of this paper is to prove the equality of certain local coefficients of arithmetic significance which were attached to representations of quasi-split real reductive algebraic groups in [27] with their corresponding Artin factors attached by local class field theory [21]. As a consequence, we establish an identity satisfied by certain normalized intertwining operators. It seems to be useful in applications of the trace formula [1, 29]. More precisely, let G be the group of real points of a quasi-split reductive algebraic group over R. Let A be the set of simple roots defined by a fixed minimal parabolic subgroup Po = M0A0U of G. Fix 6 A, and let P = P9 be the corresponding standard parabolic subgroup of G and write P = MAN for its Langlands decomposition. Fix a nondegenerate character x °f U i ^ (a, H{o)) be an irreducible admissible x-gic Banach (in particular x~generic unitary) representation of M (cf. Section 1). Given v G a£, the complex dual of the Lie algebra of A, let I(v,o,0) be the continuously (quasi-unitarily, if o is unitary) induced representation IndP^Ga® e v 9 and let V(v,o,0) be its space (Section 0). Then F ^ a , ^ = Vfoo^O). Now, let W be the Weyl group of Ao in G. Choose w G W such that w(0) A. Let N~ be the unipotent group opposite to N. Define N$ = U C wN~w~ where w is a representative of w in G. For/ G F(^,a,^)00, define

Multiplet classification of the reducible elementary representations of real semisimple Lie groups: the SO e (p, q) example

A multiplet classification of the reducible elementary representations (=generalized principal series representations) is proposed as a useful intermediate step in the important problem of determining the irreducible composition factors of the elementary representations and their multiplicities and, thus, of the distribution characters of all irreducible representations of real semisimple Lie groups. As an example, the results for the elementary representations of SOe(p, q) induced from its minimal parabolic subgroup are given. Further research is also indicated.

Knapp-Wallach Szegö integrals and generalized principal series representations: The parabolic rank one case

To each discrete series representation of a connected semisimple Lie group G with finite center, a G-equivariant embedding into a generalized principal series representation is given. This representation is induced from specified parameters on a maximal parabolic subgroup of G and the mapping is defined by an integral formula, analogous to the Szegö integral introduced by Knapp and Wallach for a minimal parabolic subgroup. In a limiting case, embeddings of limits of discrete series representations are obtained and used to exhibit a reducibility theorem.

Generalized Hua Operators and Parabolic Subgroups

Let G be a connected noncompact semisimple group with finite center. Fix G = KAN an Iwasawa decomposition of G. That is, K is a maximal compact subgroup of G, A a maximal vector subgroup of G with Ad A consisting of semisimple elements normalizing N, a maximal simply connected nilpotent subgroup of G. The space 52 = G/K is a Riemannian symmetric space, and if M is the centralizer of A in K the group B = MAN is a minimal parabolic subgroup of G, and the space G/B = K/M is called the maximal or Furstenberg boundary of U2. In general, a boundary of 52 will be a space of the form G/P where P is some parabolic subgroup of G which we may assume contains B. Fixing a parabolic subgroup P of G there is a Poisson kernel

Tensor products of principal series for the De Sitter group

The decomposition of the tensor product of two principal series representations is determined for the simply connected double covering, G = Spin(4, 1), of the DeSitter group. The main result is that this decomposition consists of two pieces, T, and Td, where T, is a continuous direct sum with respect to Plancherel measure on G of representations from the principal series only and Td is a discrete sum of representations from the discrete series of G. The multiplicities of representations occurring in T, and Td are all finite. Introduction. Let G = Spin(4, 1) be the simply connected double covering of the DeSitter group, G = KAN an Iwasawa decomposition of G, M the centralizer of A in K, and P = MAN the associated minimal parabolic subgroup of G. For a E M and T E A, a x T is a representation of P via a X T(man) = a(m)T(a) and a representation of the form 7(a, T) = Indp a x T is called a principal series representation of G. The main goal of this paper is to determine the decomposition of the tensor product of two principal series representations of G into irreducibles. It was shown in [7], by using Mackey's tensor product theorem and the Mackey-Anh reciprocity theorem, that this problem reduces to knowing how to decompose the restriction to MA of almost every principal series representation of G and each discrete series representation of G. For a representation v7 belonging to the principal series of G, the restriction of v7 to MA, (7)MAA was determined by using Mackey's subgroup theorem. However, in that paper, we were not able to determine explicitly (7)MA for a representation 'r belonging to the discrete series of G. This we do in ?3 of this paper by using Lie algebraic methods and the realizations of these representations given by Dixmier in [2]. This paper is organized as follows. In ?? 1 and 2 we summarize the main results concerning the structure and representation theory of G that we shall use. In ?3 we determine (7)MA when v7 is a discrete series representation of G. We also include the results of [7] concerning the decomposition of (@)MA when v7 is a principal series representation of G. In ?4 we show how to decompose the tensor product of two principal series representations of G. The main results are contained in Theorem 4. The basic methodology used in this paper to decompose principal series tensor products originates in the works of G. Mackey [6], N. Anh [1], and F. Williams [11]. Received by the editors April 15, 1980. 1980 Mathematics Subject Classification Primary 22E43, 81C40. 'This research was partially supported by a grant from the National Science Foundation. ? 1981 American Mathematical Society 0002-9947/81 /0000-0207/$04.75 121 This content downloaded from 157.55.39.118 on Fri, 22 Apr 2016 04:32:41 UTC All use subject to http://about.jstor.org/terms

Intertwining operators for semisimple groups, II

The purpose of the present paper is to expand the use of intertwining operators for semisimple Lie groups. In an earlier form (see [20]), these operators were meromorphic continuations of integral operators and exhibited equivalences among nonunitary principal series representations, those induced from a minimal parabolic subgroup. We demonstrated that there was an intimate connection between these operators and the irreducibility of the principal series, on the one hand, and the unitarity of the analytically continued representations (the complementary series), on the other hand. In the intervening years we have generalized the setting for intertwining operators significantly, and we have learned of new applications. For the most part, the expansion in the setting is that minimal parabolic subgroups have now been replaced by arbitrary parabolic subgroups M A N , and the representations that are studied are induced from a representation of M A N that is irreducible unitary on M, is one-dimensional on A, and is trivial on N. The expanded theory allows us to determine the degree of reducibility of all series of representations appearing in the Plancherel formula of the group, and to study complementary series attached to them. Such intertwining operators have also now found major applications in classifying representations (see Langlands [29] and Knapp-Zuckerman [-25]) and appear to have significance for some problems in number theory. Our objective in this paper is threefold: to develop the analytic properties of intertwining operators in what seems to be the appropriate degree of generality, to give a dimension formula for the commuting algebras of the unitary representations induced when the representation of M is in the discrete series and the character of A is unitary (and to give some further insights into these representations), and to illustrate a technique for dealing with complementary series. To make it possible to be more specific, we introduce some notation. Let G be a reductive group whose identity component has compact center; the precise axioms for G are given in w 1. Fix a Cartan involution 0 for G, and let P be a parabolic subgroup of G. Then P c~ OP decomposes into the product of commuting subgroups M and A, where A is a vector group and M satisfies the

Induced modules and extensions of representations

Let G be a connected affine algebraic group over an algebraically closed field k, and let B be a Borel subgroup of G. If V is a rational B-module we can ask for conditions under which V extends to G, that is, under which the action of B extends to a rational action of G on V. Since G/B is a complete variety, we have MtnlG_--__ M for any rational G-module M, cf. (1.3); so if any extension of V exists, it must be unique. The main result of this paper is that V is extendible to G if and only if it is extendible to every minimal parabolic subgroup P>B. Here "minimal" means that P properly contains B and that no other closed subgroup of P does so; equivalently, P has semisimple rank one. We also show that if P contains a Levi complement L, then V extends to P iff VILnn extends to L; so, for reductive G, the extendibility of V reduces to an SL 2 question. We obtain the main result as a corollary to a general theorem on induced modules. For any parabolic subgroup P >B, and any rational B-module V, let VIe denote the corresponding module induced from B to P. If P~ . . . . . P, is any sequence of such P's, we let VI e' ..... e. denote the P.-module vlv'lnlV2"-I r" obtained by successively inducing to P~_ 1, restricting to B, and inducing to P~. We prove the B-module structure of V[ e ...... e. depends only on the set-theoretic product P~ ... P., and that VI v' ..... e . _ VIG as P.-modules in case this product is G. At the same time we prove a "factorization" theorem for the affine coodinate ring R(G) of G. Let Px . . . . . P. be parabolic subgroups containing B such that G = P1... P.. Then we have R(G)~-R(P,)@n R(P,_ O@B"" @B R(PO, where in general for a right B-module M and a left B-module N, we denote by M@BN the set of fixed points of B in M @ N relative to the action b. (m@ n)= m b -1@ b n. The above results are in turn derived from simple considerations involving the Demazure desingularization of Schubert varieties [9], and are inspired by Kempf [12] and by Andersen's efforts [1] toward a reorganization of Kempfs work. A simplified version of Andersen's main reduction is given in the appendix.

Tensor products for 𝑆𝐿(2,𝑘)

Let G be SL ( 2 , k ) {\text {SL}}(2,k) where k is a locally compact, nondiscrete, totally disconnected topological field whose residual characteristic is not 2, π σ {\pi _\sigma } , be a principal series representation of G, and π ∈ G ^ \pi \in \hat G be arbitrary. We determine the decomposition of π σ ⊗ π {\pi _\sigma } \otimes \pi into irreducibles by reducing this problem to decomposing the restriction of each T ∈ G ^ T \in \hat G to a minimal parabolic subgroup B of G and decomposing certain tensor products of irreducibles of B.

𝑘-regular elements in semisimple algebraic groups

In this paper, Steinberg’s concept of a regular element in a semisimple algebraic group defined over an algebraically closed field is generalized to the concept of a k k -regular element in a semisimple algebraic group defined over an arbitrary field of characteristic zero. The existence of semisimple and unipotent k k -regular elements in a semisimple algebraic group defined over a field of characteristic zero is proved. The structure of all k k -regular unipotent elements is given. The number of minimal parabolic subgroups containing a k k -regular element is given. The number of conjugacy classes of R R -regular unipotent elements is given, where R R is the real field. The number of conjugacy classes of Q p {Q_p} -regular unipotent elements is shown to be finite, where Q p {Q_p} is the field of p p -adic numbers.

Singular Integrals and the Principal Series, III

The intertwining operators for nonunitary principal series representations of a real semisimple Lie group are used to construct intertwining operators for all the series of unitary representations appearing in the Plancherel formula. The resulting machinery reduces computations with Harish-Chandra's c-functions and the Plancherel measure to computations with those c-functions and measures obtained from a minimal parabolic subgroup.