scholarly journals Decomposition of Splitting Invariants in Split Real Groups

2011 ◽  
Vol 63 (5) ◽  
pp. 1083-1106 ◽  
Author(s):  
Tasho Kaletha
Keyword(s):  
Local Field ◽  
Maximal Torus ◽  
Decomposition Theorem ◽  
Reduction Formula ◽  
Transfer Factors ◽  
Connected Group ◽  
Real Group ◽  
Cohomological Invariant ◽  
Simply Connected ◽  
The Given

Abstract For a maximal torus in a quasi-split semi-simple simply-connected group over a local field of characteristic 0, Langlands and Shelstad constructed a cohomological invariant called the splitting invariant, which is an important component of their endoscopic transfer factors. We study this invari- ant in the case of a split real group and prove a decomposition theorem which expresses this invariant for a general torus as a product of the corresponding invariants for simple tori. We also show how this reduction formula allows for the comparison of splitting invariants between different tori in the given real group.

scholarly journals Some Studies on Kaehlerian Homogeneous Spaces

1957 ◽  
Vol 11 ◽  
pp. 77-92 ◽  
Author(s):  
Jun-Ichi Hano ◽  
Yozô Matsushima
Keyword(s):  
Differential Geometry ◽  
Homogeneous Spaces ◽  
Decomposition Theorem ◽  
Canonical Decomposition ◽  
Connected Group ◽  
Group Of Automorphisms ◽  
De Rham ◽  
Real Analytic ◽  
Riemannian Case ◽  
Simply Connected

The present paper is devoted to the study of differential geometry of Kaehlerian homogeneous spaces. In section 1 we deal with the canonical decomposition of a simply connected complete Kaehlerian space and that of its largest connected group of automorphisms. We know that a simply connected complete Riemannian space V is the product of Riemannian spaces V0, V1, …, Vn, where V0 is a Euclidean space and V1, …,Vn are not locally flat and their homogeneous holonomy groups are irreducible [2]. Moreover, if V is homogeneous, so are all Vk [10]. We shall show that if V is Kaehlerian space with real analytic metric (resp. Kaehlerian homogeneous space), each factor Vk is also Kaehlerian (resp. Kaehlerian homogeneous) and that V is the product of V0, V1, …, Vn as Kaehlerian space. We call this decomposition the de Rham decomposition of the Kaehlerian space V. Although this result is supposedly known, there is no published proof as yet. Using this decomposition theorem we shall show that the largest connected group of automorphisms of a simply connected complete Kaehlerian space with real analytic metric is the direct product of those of the factors of the de Rham decomposition. In the Riemannian case this result has be been established in [3] by one of the authors of the present paper.

scholarly journals R-GROUP AND WHITTAKER SPACE OF SOME GENUINE REPRESENTATIONS

2021 ◽  
pp. 1-61
Author(s):  
Fan Gao
Keyword(s):  
Series Representation ◽  
Principal Series ◽  
Symplectic Groups ◽  
Principal Series Representation ◽  
Connected Group ◽  
Covering Group ◽  
Simply Connected

Abstract For a unitary unramified genuine principal series representation of a covering group, we study the associated R-group. We prove a formula relating the R-group to the dimension of the Whittaker space for the irreducible constituents of such a principal series representation. Moreover, for certain saturated covers of a semisimple simply connected group, we also propose a simpler conjectural formula for such dimensions. This latter conjectural formula is verified in several cases, including covers of the symplectic groups.

PRESCRIBED HORIZONTAL AND VERTICAL TREES PROBLEM OF QUADRATIC DIFFERENTIALS

2006 ◽  
Vol 08 (03) ◽  
pp. 381-399
Author(s):  
THOMAS KWOK-KEUNG AU ◽  
TOM YAU-HENG WAN
Keyword(s):  
Riemann Surface ◽  
Quadratic Differential ◽  
Quadratic Differentials ◽  
Sufficient Condition ◽  
Holomorphic Quadratic Differential ◽  
Simply Connected ◽  
The Given

A sufficient condition for the existence of holomorphic quadratic differential on a non-compact simply-connected Riemann surface with prescribed horizontal and vertical trees is obtained. In particular, for any pair of complete ℝ-trees of finite vertices with (n + 2) infinite edges, there exists a polynomial quadratic differential on ℂ of degree n such that the associated vertical and horizontal trees are isometric to the given pair.

A Decomposition Theorem for Complex Nilmanifolds

1987 ◽  
Vol 30 (3) ◽  
pp. 377-378
Author(s):  
Jean-Jacques Loeb ◽  
Karl Oeljeklaus ◽  
Wolfgang Richthofer
Keyword(s):  
Lie Group ◽  
Discrete Subgroup ◽  
Decomposition Theorem ◽  
Simply Connected

AbstractA complex nilmanifold X is isomorphic to a product X ⋍ ℂp x N/┌, where N is a simply connected nilpotent complex Lie group and ┌ is a discrete subgroup of N not contained in a proper connected complex subgroup of N. The pair (N, ┌) is uniquely determined up to holomorphic group isomorphisms.

A decomposition theorem forh-cobordant smooth simply-connected compact 4-manifolds

1996 ◽  
Vol 123 (1) ◽  
pp. 343-348
Author(s):  
C. L. Curtis ◽  
M. H. Freedman ◽  
W. C. Hsiang ◽  
R. Stong
Keyword(s):  
Decomposition Theorem ◽  
Simply Connected

scholarly journals Some Results on the Cohomology of Line Bundles on the Three Dimensional Flag Variety

Mathematics ◽  
2019 ◽  
Vol 7 (3) ◽  
pp. 295
Author(s):  
Muhammad Anwar
Keyword(s):  
Borel Subgroup ◽  
Three Dimensional ◽  
Linear Algebraic Group ◽  
Flag Variety ◽  
Line Bundles ◽  
Closed Field ◽  
Two Dimensional ◽  
Prime Characteristic ◽  
Simply Connected ◽  
The Given

Let k be an algebraically closed field of prime characteristic and let G be a semisimple, simply connected, linear algebraic group. It is an open problem to find the cohomology of line bundles on the flag variety G / B , where B is a Borel subgroup of G. In this paper we consider this problem in the case of G = S L 3 ( k ) and compute the cohomology for the case when ⟨ λ , α ∨ ⟩ = − p n a − 1 , ( 1 ≤ a ≤ p , n > 0 ) or ⟨ λ , α ∨ ⟩ = − p n − r , ( r ≥ 2 , n ≥ 0 ) . We also give the corresponding results for the two dimensional modules N α ( λ ) . These results will help us understand the representations of S L 3 ( k ) in the given cases.

scholarly journals On Splitting of the Normalizer of a Maximal Torus in E6(q)

2019 ◽  
Vol 26 (02) ◽  
pp. 329-350
Author(s):  
Alexey Galt ◽  
Alexey Staroletov
Keyword(s):  
Finite Group ◽  
Weyl Group ◽  
Maximal Torus ◽  
Group Of Lie Type ◽  
Maximal Tori ◽  
A Finite Group ◽  
Simply Connected

Let G be a finite group of Lie type E6 over 𝔽q (adjoint or simply connected) and W be the Weyl group of G. We describe maximal tori T such that T has a complement in its algebraic normalizer N(G, T). It is well known that for each maximal torus T of G there exists an element w ∊ W such that N(G, T )/T ≃ CW(w). When T does not have a complement isomorphic to CW(w), we show that w has a lift in N(G, T) of the same order.

Regular Germs For -Adic Sp(4)

1989 ◽  
Vol 41 (4) ◽  
pp. 626-641
Author(s):  
Kim Yangkon
Keyword(s):  
Algebraic Group ◽  
Local Field ◽  
Maximal Torus ◽  
Simple Algebraic Group

Shalika [6] proved the existence of germs (associated with a connected semi-simple algebraic group and a maximal torus over a non-archimedean local field), established many of their properties, and conjectured that the germ associated to the trivial unipotent class in GL(n) should be a constant. R. Howe and Harish-Chandra proved that it is a constant and J. Rogawski [5] proved that it had the value predicted previously by J. Shalika.

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