scholarly journals Langlands parameters of derived functor modules and Vogan diagrams

2003 ◽  
Vol 92 (1) ◽  
pp. 31 ◽  
Author(s):  
Paul D. Friedman
Keyword(s):  
Lie Group ◽  
Simple Root ◽  
Maximal Compact Subgroup ◽  
Compact Subgroup ◽  
Classical Groups ◽  
Parabolic Subalgebra ◽  
Finite Center ◽  
Derived Functor ◽  
Langlands Parameters ◽  
Reductive Lie Group

Let $G$ be a linear reductive Lie group with finite center, let $K$ be a maximal compact subgroup, and assume that $\mathrm{rank } G = \mathrm {rank } K$. Let $\ g= l\oplus u$ be a $\theta$ stable parabolic subalgebra obtained by building $l$ from a subset of the compact simple roots and form $A_g(\lambda)$. Suppose $\Lambda=\lambda+2\delta( u\cap p)$ is $K$-dominant and the infinitesimal character, $\lambda+\delta$, of $A_{g}(\lambda)$ is nondominant due to a noncompact simple root. By interpreting these conditions on the level of Vogan diagrams, a conjecture by Knapp is (essentially) settled for the groups $G=SU(p,q),\, Sp(p,q)$, and $SO^*(2n)$, thereby determining the Langlands parameters of natural irreducible subquotient of $A_{ g}(\lambda)$. For the remaining classical groups, simple supplementary conditions are given under which the Langlands parameters may be determined.

scholarly journals Character formulas for discrete series on semisimple Lie groups

1976 ◽  
Vol 64 ◽  
pp. 47-61 ◽  
Author(s):  
Rebecca A. Herb
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Maximal Compact Subgroup ◽  
Discrete Series ◽  
Compact Subgroup ◽  
Semisimple Lie Groups ◽  
Analytic Group ◽  
Finite Center ◽  
Real Analytic ◽  
Analytic Subgroup

Let G be a connected, semisimple real Lie group with finite center, K a maximal compact subgroup of G. Assume rank G = rank K. Let be the Lie algebra of G, its complexification. If Gc is the simplyconnected complex analytic group corresponding to assume G is the real analytic subgroup of Gc corresponding to .

Homogeneous Complex Manifolds with more than One End

1989 ◽  
Vol 41 (1) ◽  
pp. 163-177 ◽  
Author(s):  
B. Gilligan ◽  
K. Oeljeklaus ◽  
W. Richthofer
Keyword(s):  
Lie Group ◽  
Homogeneous Spaces ◽  
Maximal Compact Subgroup ◽  
Compact Subgroup ◽  
Complex Manifolds ◽  
Connected Subgroup ◽  
Homogeneous Complex ◽  
Homogeneous Complex Manifold ◽  
Homogeneous Cone ◽  
Closed Connected Subgroup

For homogeneous spaces of a (real) Lie group one of the fundamental results concerning ends (in the sense of Freudenthal [8] ) is due to A. Borel [6]. He showed that if X = G/H is the homogeneous space of a connected Lie group G by a closed connected subgroup H, then X has at most two ends. And if X does have two ends, then it is diffeomorphic to the product of R with the orbit of a maximal compact subgroup of G.In the setting of homogeneous complex manifolds the basic idea should be to find conditions which imply that the space has at most two ends and then, when the space has exactly two ends, to display the ends via bundles involving C* and compact homogeneous complex manifolds. An analytic condition which ensures that a homogeneous complex manifold X has at most two ends is that X have non-constant holomorphic functions and the structure of such a space with exactly two ends is determined, namely, it fibers over an affine homogeneous cone with its vertex removed with the fiber being compact [9], [13].

Minimal actions of semisimple groups

1996 ◽  
Vol 16 (4) ◽  
pp. 821-831 ◽  
Author(s):  
Garrett Stuck
Keyword(s):  
Parabolic Subgroup ◽  
Lie Group ◽  
Maximal Compact Subgroup ◽  
Compact Subgroup ◽  
Semisimple Lie Group ◽  
Semisimple Groups ◽  
Open Problems ◽  
Minimal Action ◽  
Orbit Structure ◽  
Low Dimension

AbstractWe show that a C0 minimal action of a semisimple Lie group without compact factors is either locally free or induced from a minimal action of a proper parabolic subgroup. We describe the orbit structure of the action restricted to a maximal compact subgroup, and then apply this to minimal actions in low dimension. We give some examples, applications, and open problems.

scholarly journals ON THE EINSTEIN–KÄHLER METRIC AND THE HOLONOMY OF A LINE BUNDLE

2002 ◽  
Vol 45 (1) ◽  
pp. 83-90
Author(s):  
Kenji Tsuboi
Keyword(s):  
Line Bundle ◽  
Complex Manifold ◽  
Lie Group ◽  
Maximal Compact Subgroup ◽  
Compact Subgroup ◽  
Base Space ◽  
Subject Classification ◽  
Compact Complex Manifold ◽  
Identity Component ◽  
Determinant Line Bundle

AbstractIn this paper we give a relation between the Futaki invariant for a compact complex manifold $M$ and the holonomy of a determinant line bundle over a loop in the base space of any principal $G$-bundle, where $G$ is the identity component of the maximal compact subgroup of the complex Lie group consisting of all biholomorphic automorphisms of $M$. Using the property of the Futaki invariant, we show that the holonomy is an obstruction to the existence of the Einstein–Kähler metrics on $M$. Our main result is Theorem 2.1.AMS 2000 Mathematics subject classification: Primary 32Q20. Secondary 58J28; 58J52

scholarly journals AnLp−Lqversion of Hardy's theorem for spherical Fourier transform on semisimple Lie groups

2004 ◽  
Vol 2004 (33) ◽  
pp. 1757-1769 ◽  
Author(s):  
S. Ben Farah ◽  
K. Mokni ◽  
K. Trimèche
Keyword(s):  
Fourier Transform ◽  
Maximal Compact Subgroup ◽  
Compact Subgroup ◽  
Semisimple Lie Groups ◽  
Semisimple Lie Group ◽  
Positive Real ◽  
Finite Center ◽  
Hardy’S Theorem ◽  
Spherical Fourier Transform ◽  
Almost Everywhere

We consider a real semisimple Lie groupGwith finite center andKa maximal compact subgroup ofG. We prove anLp−Lqversion of Hardy's theorem for the spherical Fourier transform onG. More precisely, leta,bbe positive real numbers,1≤p,q≤∞, andfaK-bi-invariant measurable function onGsuch thatha−1f∈Lp(G)andeb‖λ‖2ℱ(f)∈Lq(𝔞+*)(hais the heat kernel onG). We establish that ifab≥1/4andporqis finite, thenf=0almost everywhere. Ifab<1/4, we prove that for allp,q, there are infinitely many nonzero functionsfand ifab=1/4withp=q=∞, we havef=const ha.

scholarly journals ON THE GEOMETRY OF THE SUPERMULTIPLET IN M-THEORY

2011 ◽  
Vol 08 (07) ◽  
pp. 1519-1551 ◽  
Author(s):  
HISHAM SATI
Keyword(s):  
Partition Function ◽  
Lie Group ◽  
Maximal Compact Subgroup ◽  
Compact Subgroup ◽  
Internal Space ◽  
Bilinear Forms ◽  
M Theory ◽  
Kaluza Klein ◽  
String Theories

The massless supermultiplet of 11-dimensional supergravity can be generated from the decomposition of certain representation of the exceptional Lie group F4 into those of its maximal compact subgroup Spin(9). In an earlier paper, a dynamical Kaluza–Klein origin of this observation is proposed with internal space the Cayley plane, 𝕆P2, and topological aspects are explored. In this paper we consider the geometric aspects and characterize the corresponding forms which contribute to the action as well as cohomology classes, including torsion, which contribute to the partition function. This involves constructions with bilinear forms. The compatibility with various string theories are discussed, including reduction to loop bundles in ten dimensions.

scholarly journals Differential equations and an analog of the Paley-Wiener theorem for linear semisimple Lie groups

1976 ◽  
Vol 64 ◽  
pp. 17-29 ◽  
Author(s):  
Kenneth D. Johnson
Keyword(s):  
Differential Equations ◽  
Lie Groups ◽  
Lie Group ◽  
Maximal Compact Subgroup ◽  
Compact Subgroup ◽  
Nilpotent Subgroup ◽  
Iwasawa Decomposition ◽  
Semisimple Lie Groups ◽  
Semisimple Lie Group ◽  
Simply Connected

Let G be a noncompact linear semisimple Lie group. Fix G = KAN an Iwasawa decomposition of G. That is, K is a maximal compact subgroup of G, A is a vector subgroup with AdA consisting of semisimple transformations and A normalizes N, a simply connected nilpotent subgroup of G.

scholarly journals Yang–Mills connections on compact complex tori

2015 ◽  
Vol 07 (02) ◽  
pp. 293-307
Author(s):  
Indranil Biswas
Keyword(s):  
Algebraic Group ◽  
Maximal Compact Subgroup ◽  
Compact Subgroup ◽  
Structure Group ◽  
Kähler Structure ◽  
Yang Mills ◽  
Complex Torus ◽  
Complex Tori

Let G be a connected reductive complex affine algebraic group and K ⊂ G a maximal compact subgroup. Let M be a compact complex torus equipped with a flat Kähler structure and (EG, θ) a polystable Higgs G-bundle on M. Take any C∞ reduction of structure group EK ⊂ EG to the subgroup K that solves the Yang–Mills equation for (EG, θ). We prove that the principal G-bundle EG is polystable and the above reduction EK solves the Einstein–Hermitian equation for EG. We also prove that for a semistable (respectively, polystable) Higgs G-bundle (EG, θ) on a compact connected Calabi–Yau manifold, the underlying principal G-bundle EG is semistable (respectively, polystable).

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