Maximal Parabolic Subgroup Research Articles

CONSTRUCTIONS OF GLOBAL INTEGRALS IN THE EXCEPTIONAL GROUP F4

Motivated by known examples of global integrals which represent automorphic L-functions, this paper initiates the study of a certain two-dimensional array of global integrals attached to any reductive algebraic group, indexed by maximal parabolic subgroups in one direction and by unipotent conjugacy classes in the other. Fourier coefficients attached to unipotent classes, the Gelfand-Kirillov dimension of automorphic representations, and an identity which, empirically, appears to constrain the unfolding process are presented in detail with examples selected from the exceptional groups. Two new Eulerian integrals are included among these examples.

Complex structures on product of circle bundles over complex manifolds

Soient L ¯ i →X i des fibrés en droites holomorphes sur des variétés complexes compactes, pour i=1,2. Soit S i le fibré en cercles associé par rapport à un produit scalaire hermitienne sur L ¯ i . On construit des structures complexes sur S=S 1 ×S 2 dites de type scalaire, diagonal, ou linéaire. Bien que des structures de type scalaire existent toujours, on construit des structures plus générales de type diagonal mais non-scalaire dans le cas où les L ¯ i sont des (ℂ * ) n i fibrés équivariants qui vérifient certaines hypothèses supplémentaires. Les structures complexes de type linéaire sont des variétés des drapeaux (généralisées) et les L i sont des fibrés en droites amples négatifs. Lorsque H 1 (X 1 ;ℝ)=0 et c 1 (L ¯ 1 ) est non-nulle la variété compacte S n’admet pas de structure symplectique et donc elle est non-Kählerienne par rapport à toute structure complexe.

A rank 3 geometry for the O’Nan group connected with the Livingstone graph

We construct a rank 3 geometry ( O 0 N) over the diagram a a a 1 10 1 c 8 5 8 whose automorphism group is the O’Nan sporadic simple group. The maximal parabolic subgroups are the Janko group J1, 2 × S5 and the Mathieu group M11. Our construction is based on a convenient amalgam of known geometries of rank 2 for J1 and M11 extracted from the subgroup lattice of O 0 N.

The Franke filtration of the spaces of automorphic forms supported in a maximal proper parabolic subgroup

The Franke filtration is a finite filtration of certain spaces of automorphic forms on the adelic points of a reductive linear algebraic group defined over a number field whose quotients can be described in terms of parabolically induced representations. Decomposing the space of automorphic forms according to their cuspidal support, the Franke filtration can be made more explicit. This paper describes explicitly the Franke filtration of the spaces of automorphic forms supported in a maximal proper parabolic subgroup, that is, in a cuspidal automorphic representation of its Levi factor. Such explicit description is important for applications to computation of automorphic cohomology, and thus the cohomology of congruence subgroups. As examples, the general linear group and split symplectic and special orthogonal groups are treated.

On the degrees of matrix coefficients of intertwining operators

Let G be a reductive algebraic group defined over a p-adic field F with residue field Fq and G = G(F ). Fix a special maximal compact subgroup K0 of G. For a maximal parabolic subgroup P = MU of G and a smooth irreducible representation π of M = M(F ) we consider the family of induced representations IP (π, s), s ∈ C, which extend the fixed K0-representation I K0 P∩K0(π|M∩K0), and the associated intertwining operators M(s) = MP |P (π, s) : IP (π, s)→ IP (π, s). For any open subgroup K of K0 the restriction M(s) : I0 P∩K0(π|M∩K0) K = IP (π, s) K → IP (π, s) = I K0 P∩K0 (π|M∩K0) of M(s) to the space of K-fixed vectors is a family of linear maps between finite-dimensional vector spaces which do not depend on s. It is well known that the matrix coefficients of the linear operators M(s) are rational functions of q−s, whose denominators can be controlled explicitly (e.g. [23, IV.1.1, IV.1.2]). In particular, their degrees are bounded independently of K and π. What can be said about the degrees of the numerators? In this note, we study the following conjecture, which should provide a bound of the correct order of magnitude. Let G′ be the derived group of G and set G′ = G′(F ). Note that K ′ 0 = K0 ∩ G′ is a special maximal compact subgroup of G′.

An example of orthogonal triple flag variety of finite type

Let G be the split special orthogonal group of degree 2n+1 over a field F of charF≠2. Then we describe G-orbits on the triple flag varieties G/P×G/P×G/P and G/P×G/P×G/B with respect to the diagonal action of G where P is a maximal parabolic subgroup of G of the shape (n,1,n) and B is a Borel subgroup. As by-products, we also describe GLn-orbits on G/B, Q2n-orbits on the full flag variety of GL2n where Q2n is the fixed-point subgroup in Sp2n of a nonzero vector in F2n and 1×Sp2n-orbits on the full flag variety of GL2n+1. In the same way, we can also solve the same problem for SO2n where the maximal parabolic subgroup P is of the shape (n,n).

Elements with finite Coxeter part in an affine Weyl group

Let Wa be an affine Weyl group and η:Wa→W0 be the natural projection to the corresponding finite Weyl group. We say that w∈Wa has finite Coxeter part if η(w) is conjugate to a Coxeter element of W0. The elements with finite Coxeter part are a union of conjugacy classes of Wa. We show that for each conjugacy class O of Wa with finite Coxeter part there exists a unique maximal proper parabolic subgroup WJ of Wa, such that the set of minimal length elements in O is exactly the set of Coxeter elements in WJ. Similar results hold for twisted conjugacy classes.

VISIBLE ACTIONS ON FLAG VARIETIES OF TYPE B AND A GENERALISATION OF THE CARTAN DECOMPOSITION

AbstractWe give a generalisation of the Cartan decomposition for connected compact Lie groups of type B motivated by the work on visible actions of Kobayashi [‘A generalized Cartan decomposition for the double coset space $(U(n_{1})\times U(n_{2})\times U(n_{3})) \backslash U(n)/ (U(p)\times U(q))$’, J. Math. Soc. Japan59 (2007), 669–691] for type A groups. Suppose that $G$ is a connected compact Lie group of type B, $\sigma $ is a Chevalley–Weyl involution and $L$, $H$ are Levi subgroups. First, we prove that $G=LG^{\sigma }H$ holds if and only if either (I) both $H$ and $L$ are maximal and of type A, or (II) $(G,H)$ is symmetric and $L$ is the Levi subgroup of an arbitrary maximal parabolic subgroup up to switching $H$ and $L$. This classification gives a visible action of $L$ on the generalised flag variety $G/H$, as well as that of the $H$-action on $G/L$ and of the $G$-action on $(G\times G)/(L\times H)$. Second, we find an explicit ‘slice’ $B$ with $\dim B=\mathrm {rank}\, G$ in case I, and $\dim B=2$ or $3$ in case II, such that a generalised Cartan decomposition $G=LBH$holds. An application to multiplicity-free theorems of representations is also discussed.

COMPUTATION OF A TWISTED CHARACTER OF A SMALL REPRESENTATION OF GL(3, E)

Let E/F be a quadratic extension of p-adic fields, p ≠ 2. Let [Formula: see text] be the involution of E over F. The representation π of GL (3, E) normalizedly induced from the trivial representation of the maximal parabolic subgroup is invariant under the involution [Formula: see text]. We compute — by purely local means — the σ-twisted character [Formula: see text] of π. We show that it is σ-unstable, namely its value at one σ-regular-elliptic conjugacy class within a stable such class is equal to negative its value at the other such conjugacy class within the stable class, or zero when the σ-regular-elliptic stable conjugacy class consists of a single such conjugacy class. Further, we relate this twisted character to the twisted endoscopic lifting from the trivial representation of the "unstable" twisted endoscopic group U (2, E/F) of GL (3, E). In particular π is σ-elliptic, that is, [Formula: see text] is not identically zero on the σ-elliptic set.

The tempered spectrum of quasi-split classical groups III: The odd orthogonal groups

Abstract. We study the tempered spectrum of special orthogonal groups of odd degree over p-adic fields of characteristic zero. Residues of intertwining operators for representations parabolically induced from arbitrary maximal parabolic subgroups are determined in terms of non-vanishing of weighted sums of twisted orbital integrals. This is connected with the theory of twisted endoscopy, and poles of the Langlands L-functions, in particular, with the Rankin product L-function between irreducible (generic) supercuspidal representations of the classical group SO2m+1(F) and those of GLk(F). These poles are described for, any m and k, explicitly in terms of twisted endoscopy. Moreover, we show these poles describe precisely when a supercuspidal representation of GLk(F) is the local component of the transfer from a special odd orthogonal group.

The P-radical classes in simple algebraic groups and finite groups of Lie type

Abstract.Given either a simple algebraic group or a finite group of Lie type, of rank at least 2, and a maximal parabolic subgroup, we determine which non-trivial unipotent classes have the property that their intersection with the parabolic subgroup is contained within its unipotent radical. Such classes are rare; listing them provides a basis for inductive arguments.

Locally parabolic subgroups in Coxeter groups of arbitrary ranks

Despite the significance of the notion of parabolic closures in Coxeter groups of finite ranks, the parabolic closure is not guaranteed to exist as a parabolic subgroup in a general case. In this paper, first we give a concrete example to clarify that the parabolic closure of even an irreducible reflection subgroup of countable rank does not necessarily exist as a parabolic subgroup. Then we propose a generalized notion of “locally parabolic closure” by introducing a notion of “locally parabolic subgroups”, which involves parabolic ones as a special case, and prove that the locally parabolic closure always exists as a locally parabolic subgroup. It is a subgroup of parabolic closure, and we give another example to show that the inclusion may be strict in general. Our result suggests that locally parabolic closure has more natural properties and provides more information than parabolic closure. We also give a result on maximal locally finite, locally parabolic subgroups in Coxeter groups, which generalizes a similar well-known fact on maximal finite parabolic subgroups.

Lattices in complete rank 2 Kac–Moody groups

Let Λ be a minimal Kac–Moody group of rank 2 defined over the finite field Fq, where q=pa with p prime. Let G be the topological Kac–Moody group obtained by completing Λ. An example is G=SL2(K), where K is the field of formal Laurent series over Fq. The group G acts on its Bruhat–Tits building X, a tree, with quotient a single edge. We construct new examples of cocompact lattices in G, many of them edge-transitive. We then show that if cocompact lattices in G do not contain p-elements, the lattices we construct are the only edge-transitive lattices in G, and that our constructions include the cocompact lattice of minimal covolume in G. We also observe that, with an additional assumption on p-elements in G, the arguments of Lubotzky (1990) [21] for the case G=SL2(K) may be generalised to show that there is a positive lower bound on the covolumes of all lattices in G, and that this minimum is realised by a non-cocompact lattice, a maximal parabolic subgroup of Λ.

Generalized Whittaker functions on GSp(2,R) associated with indefinite quadratic forms

We study the generalized Whittaker models for G = GSp(2;R) associated with indefinite binary quadratic forms when they arise from two standard representations of G: (i) a generalized principal series rep- resentation induced from the non-Siegel maximal parabolic subgroup and (ii) a (limit of) large discrete series representation. We prove the uniqueness of such models with moderate growth property. Moreover we express the values of the corresponding generalized Whittaker functions on a one-parameter subgroup of G in terms of the Meijer G-functions.

Degenerate principal series representations for quaternionic unitary groups

We give a complete description of all points of reducibility and the composition series of the degenerate principal series representations for quaternionic unitary groups which are induced from a character of the maximal parabolic subgroup with abelian unipotent radical. The case of even orthogonal groups is also included.

The [formula omitted] and [formula omitted] transforms as intertwining operators between generalized principal series representations of [formula omitted

In this article we connect topics from convex and integral geometry with well-known topics in representation theory of semisimple Lie groups by showing that the Cos λ and Sin λ transforms on the Grassmann manifolds Gr p ( K ) = SU ( n + 1 , K ) / S ( U ( p , K ) × U ( n + 1 − p , K ) ) are standard intertwining operators between certain generalized principal series representations induced from a maximal parabolic subgroup P p of SL ( n + 1 , K ) . The index p indicates the dependence of the parabolic on p. The general results of Knapp and Stein (1971, 1980) [20,21] and Vogan and Wallach (1990) [44] then show that both transforms have meromorphic extension to C and are invertible for generic λ ∈ C . Furthermore, known methods from representation theory combined with a Selberg type integral allow us to determine the K-spectrum of those operators.

Quasi-permutation Representations For The Borel And Maximal Parabolic Subgroups Of Sp(4,2n)

A square matrix over the complex field with non-negative integral trace is called a quasi-permutation matrix.Thus every permutation matrix over C is a quasi-permutation matrix . The minimal degree of a faithful representation of G by quasi-permutation matrices over the complex numbers is denoted by c(G), and r(G) denotes the minimal degree of a faithful rational valued complex character of G . In this paper c(G) and r(G) are calculated for the Borel or maximal parabolic subgroups of \( SP(4,2^f)\) .

Restriction of Tangent Bundle and Semistability

Let G be a simple linear algebraic group defined over ℂ and P ⊂ G a maximal proper parabolic subgroup such that m: = dim ℂ G/P ≥ 5. Let ι: Z 1 ∩ Z 2↪G/P be a smooth complete intersection such that degree(Z i ) ≥ (m − 1)·index(G/P)/m, i = 1, 2. Then the vector bundle ι*T(G/P) → Z 1 ∩ Z 2 is semistable.

On the principal symbols of $K_C$- invariant differential operators on Hermitian symmetric spaces

Let $(G,K)$ be one of the following Hermitian symmetric pair: $SU(p,q),S(U(p) \: \times \: U(q)))$, $(Sp(n,$\mathbf{R}$ ,U(n))$, or $(SO^*(2n),U(n))$. Let $G_C$ and $K_C$ be the complexifications of $G$ and $K$, respectively, $Q$ the maximal parabolic subgroup of $G_C$ whose Levi part is $K_C$, and $V$ the holomorphic tangent space at the origin of $G/K$. It is known that the ring of $K_C$ -invariant differential operators on $V$ has a generating system $\{\mathit{\Gamma_k}\}$ given in terms of determinant or Pfaffian that plays an essential role in the Capelli identities. Our main result is that determinant or Pfaffian of a deformation of the twisted moment map on the holomorphic cotangent bundle of $G_C / Q$ provides a generating function for the principal symbols of $\mathit{\Gamma_k}$'s.

Rank One Reducibility for Metaplectic Groups via Theta Correspondence

Abstract We calculate reducibility for the representations of metaplectic groups induced from cuspidal representations of maximal parabolic subgroups via theta correspondence, in terms of the analogous representations of the odd orthogonal groups. We also describe the lifts of all relevant subquotients.