scholarly journals A Coincidence Theorem for Holomorphic Maps toG/P

2003 ◽  
Vol 46 (2) ◽  
pp. 291-298 ◽  
Author(s):  
Parameswaran Sankaran
Keyword(s):  
Algebraic Group ◽  
Parabolic Subgroup ◽  
Holomorphic Maps ◽  
Projective Varieties ◽  
Maximal Parabolic Subgroup ◽  
Simple Algebraic Group ◽  
Coincidence Theorem ◽  
Complex Projective

AbstractThe purpose of this note is to extend to an arbitrary generalized Hopf and Calabi-Eckmann manifold the following result of Kalyan Mukherjea: Letdenote a Calabi-Eckmann manifold. If f, g : Vn→are any two holomorphic maps, at least one of them being non-constant, then there exists a coincidence: f(x) = g(x) for some x ∈ Vn. Our proof involves a coincidence theorem for holomorphic maps to complex projective varieties of the formG/PwhereGis complex simple algebraic group andP⊂Gis a maximal parabolic subgroup, where one of the maps is dominant.

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

2012 ◽  
Vol 15 (5) ◽  
Author(s):  
R. Lawther
Keyword(s):  
Finite Group ◽  
Algebraic Group ◽  
Parabolic Subgroup ◽  
Finite Groups ◽  
Algebraic Groups ◽  
Unipotent Radical ◽  
Maximal Parabolic Subgroup ◽  
Simple Algebraic Group ◽  
A Finite Group ◽  
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.

scholarly journals Unipotent elements forcing irreducibility in linear algebraic groups

2018 ◽  
Vol 21 (3) ◽  
pp. 365-396 ◽  
Author(s):  
Mikko Korhonen
Keyword(s):  
Algebraic Group ◽  
Parabolic Subgroup ◽  
Algebraic Groups ◽  
Closed Field ◽  
Unipotent Element ◽  
Tilting Modules ◽  
Simple Algebraic Group ◽  
Algebraically Closed Field ◽  
Linear Algebraic Groups ◽  
Highest Weight

Abstract Let G be a simple algebraic group over an algebraically closed field K of characteristic {p>0} . We consider connected reductive subgroups X of G that contain a given distinguished unipotent element u of G. A result of Testerman and Zalesski [D. Testerman and A. Zalesski, Irreducibility in algebraic groups and regular unipotent elements, Proc. Amer. Math. Soc. 141 2013, 1, 13–28] shows that if u is a regular unipotent element, then X cannot be contained in a proper parabolic subgroup of G. We generalize their result and show that if u has order p, then except for two known examples which occur in the case {(G,p)=(C_{2},2)} , the subgroup X cannot be contained in a proper parabolic subgroup of G. In the case where u has order {>p} , we also present further examples arising from indecomposable tilting modules with quasi-minuscule highest weight.

scholarly journals Zero Cycles on a Twisted Cayley Plane

2008 ◽  
Vol 51 (1) ◽  
pp. 114-124 ◽  
Author(s):  
V. Petrov ◽  
N. Semenov ◽  
K. Zainoulline
Keyword(s):  
Algebraic Group ◽  
Parabolic Subgroup ◽  
Simple Algebraic Group ◽  
Homogeneous Variety

AbstractLet k be a field of characteristic not 2, 3. Let G be an exceptional simple algebraic group over k of type F4, 1E6 or E7 with trivial Tits algebras. Let X be a projective G-homogeneous variety. If G is of type E7, we assume in addition that the respective parabolic subgroup is of type P7. The main result of the paper says that the degree map on the group of zero cycles of X is injective.

scholarly journals Torus quotient of Richardson varieties in orthogonal and symplectic grassmannians

2019 ◽  
Vol 19 (10) ◽  
pp. 2050186
Author(s):  
Arpita Nayek ◽  
S. K. Pattanayak
Keyword(s):  
Line Bundle ◽  
Algebraic Group ◽  
Parabolic Subgroup ◽  
Maximal Torus ◽  
Ample Line Bundle ◽  
Maximal Parabolic Subgroup ◽  
Type Formula ◽  
Richardson Variety ◽  
Simply Connected ◽  
Symplectic Grassmannians

For any simple, simply connected algebraic group [Formula: see text] of type [Formula: see text] and [Formula: see text] and for any maximal parabolic subgroup [Formula: see text] of [Formula: see text], we provide a criterion for a Richardson variety in [Formula: see text] to admit semistable points for the action of a maximal torus [Formula: see text] with respect to an ample line bundle on [Formula: see text].

scholarly journals Regular and residual Eisenstein series and the automorphic cohomology of Sp(2,2)

2009 ◽  
Vol 146 (1) ◽  
pp. 21-57 ◽  
Author(s):  
Harald Grobner
Keyword(s):  
Algebraic Group ◽  
Eisenstein Series ◽  
Symplectic Group ◽  
Automorphic Forms ◽  
General Theorem ◽  
Simple Algebraic Group ◽  
Ramanujan Conjecture ◽  
Eisenstein Cohomology ◽  
Derivatives Of

AbstractLetGbe the simple algebraic group Sp(2,2), to be defined over ℚ. It is a non-quasi-split, ℚ-rank-two inner form of the split symplectic group Sp8of rank four. The cohomology of the space of automorphic forms onGhas a natural subspace, which is spanned by classes represented by residues and derivatives of cuspidal Eisenstein series. It is called Eisenstein cohomology. In this paper we give a detailed description of the Eisenstein cohomologyHqEis(G,E) ofGin the case of regular coefficientsE. It is spanned only by holomorphic Eisenstein series. For non-regular coefficientsEwe really have to detect the poles of our Eisenstein series. SinceGis not quasi-split, we are out of the scope of the so-called ‘Langlands–Shahidi method’ (cf. F. Shahidi,On certainL-functions, Amer. J. Math.103(1981), 297–355; F. Shahidi,On the Ramanujan conjecture and finiteness of poles for certainL-functions, Ann. of Math. (2)127(1988), 547–584). We apply recent results of Grbac in order to find the double poles of Eisenstein series attached to the minimal parabolicP0ofG. Having collected this information, we determine the square-integrable Eisenstein cohomology supported byP0with respect to arbitrary coefficients and prove a vanishing result. This will exemplify a general theorem we prove in this paper on the distribution of maximally residual Eisenstein cohomology classes.

Tangent bundle of hypersurfaces in G/P

2008 ◽  
Vol 4 (1) ◽  
pp. 91-100
Author(s):  
Indranil Biswas ◽  
Georg Schumacher
Keyword(s):  
Line Bundle ◽  
Algebraic Group ◽  
Parabolic Subgroup ◽  
Tangent Bundle ◽  
Linear Algebraic Group ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Smooth Hypersurface ◽  
Canonical Line Bundle

AbstractLet G be a simple linear algebraic group defined over an algebraically closed field k of characteristic p ≥ 0, and let P be a maximal proper parabolic subgroup of G. If p > 0, then we will assume that dimG/P ≤ p. Let ι : H ↪ G/P be a reduced smooth hypersurface in G/P of degree d. We will assume that the pullback homomorphism is an isomorphism (this assumption is automatically satisfied when dimH ≥ 3). We prove that the tangent bundle of H is stable if the two conditions τ(G/P) ≠ d and hold; here n = dimH, and τ(G/P) ∈ is the index of G/P which is defined by the identity = where L is the ample generator of Pic(G/P) and is the anti–canonical line bundle of G/P. If d = τ(G/P), then the tangent bundle TH is proved to be semistable. If p > 0, and then TH is strongly stable. If p > 0, and d = τ(G/P), then TH is strongly semistable.

Restriction of the Tangent Bundle of G/P to a Hypersurface

2010 ◽  
Vol 53 (2) ◽  
pp. 218-222
Author(s):  
Indranil Biswas
Keyword(s):  
Algebraic Group ◽  
Parabolic Subgroup ◽  
Tangent Bundle ◽  
Linear Algebraic Group ◽  
Smooth Hypersurface

AbstractLet P be a maximal proper parabolic subgroup of a connected simple linear algebraic group G, defined over ℂ, such that n := dimℂG/P ≥ 4. Let ι : Z ↪ G/P be a reduced smooth hypersurface of degree at least (n – 1) · degree(T(G/P))/n. We prove that the restriction of the tangent bundle ι*TG/P is semistable.

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