scholarly journals A CRITERION FOR HOMOGENEOUS PRINCIPAL BUNDLES

2010 ◽  
Vol 21 (12) ◽  
pp. 1633-1638
Author(s):  
INDRANIL BISWAS ◽  
GÜNTHER TRAUTMANN
Keyword(s):  
Vector Bundle ◽  
Algebraic Group ◽  
Parabolic Subgroup ◽  
Reductive Group ◽  
Linear Algebraic Group ◽  
Principal Bundles ◽  
Simply Connected

We consider principal bundles over G/P, where P is a parabolic subgroup of a semi-simple and simply connected linear algebraic group G defined over ℂ. We prove that a holomorphic principal H-bundle EH → G/P, where H is a complex reductive group, and is homogeneous if the adjoint vector bundle ad (EH) is homogeneous. Fix a faithful H-module V. We also show that EH is homogeneous if the vector bundle EH ×H V associated to it for the H-module V is homogeneous.

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.

scholarly journals On the Steinberg Character of a Finite Simple Group of Lie Type

1971 ◽  
Vol 12 (1) ◽  
pp. 1-14 ◽  
Author(s):  
Bhama Srinivasan
Keyword(s):  
Algebraic Group ◽  
Linear Algebraic Group ◽  
Simple Groups ◽  
Closed Field ◽  
Finite Simple Groups ◽  
Reductive Algebraic Group ◽  
Simple Algebraic Group ◽  
Simply Connected ◽  
Groups Of Lie Type ◽  
Steinberg Character

Let K be an algebraically closed field of characteristic ρ >0. If G is a connected, simple connected, semisimple linear algebraic group defined over K and σ an endomorphism of G onto G such that the subgroup Gσ of fixed points of σ is finite, Steinberg ([6] [7]) has shown that there is a complex irreducible character χ of Gσ with the following properties. χ vanishes at all elements of Gσ which are not semi- simple, and, if x ∈ G is semisimple, χ(x) = ±n(x) where n(x)is the order of a Sylow p-subgroup of (ZG(x))σ (ZG(x) is the centraliser of x in G). If G is simple he has, in [6], identified the possible groups Gσ they are the Chevalley groups and their twisted analogues over finite fields, that is, the ‘simply connected’ versions of finite simple groups of Lie type. In this paper we show, under certain restrictions on the type of the simple algebraic group G an on the characteristic of K, that χ can be expressed as a linear combination with integral coefficients of characters induced from linear characters of certain naturally defined subgroups of Gσ. This expression for χ gives an explanation for the occurence of n(x) in the formula for χ (x), and also gives an interpretation for the ± 1 occuring in the formula in terms of invariants of the reductive algebraic group ZG(x).

scholarly journals Sur une conjecture de Breuil–Herzig

2019 ◽  
Vol 2019 (751) ◽  
pp. 91-119 ◽  
Author(s):  
Julien Hauseux
Keyword(s):  
Parabolic Subgroup ◽  
Reductive Group ◽  
Principal Series ◽  
Nous Obtenons ◽  
Derived Subgroup ◽  
Nous Montrons ◽  
Simply Connected ◽  
Simplement Connexe ◽  
Do So

AbstractSoit G un groupe réductif p-adique de centre connexe et de groupe dérivé simplement connexe. Nous montrons que certaines “chaînes ” de séries principales de G n’existent pas et nous établissons plusieurs propriétés de la construction \Pi(\rho)^{\mathrm{ord}} de Breuil–Herzig. En particulier, nous obtenons une caractérisation naturelle de cette dernière et nous démontrons une conjecture de Breuil–Herzig. Pour cela, nous calculons le δ-foncteur \mathrm{H^{\bullet}Ord}_{P} des parties ordinaires dérivées d’Emerton relatif à un sous-groupe parabolique P de G sur une série principale. Nous énonçons une nouvelle conjecture sur les extensions entre représentations lisses modulo p de G obtenues par induction parabolique à partir de représentations supersingulières de sous-groupes de Levi de G et nous la démontrons pour les extensions par une série principale. Let G be a split p-adic reductive group with connected centre and simply connected derived subgroup. We show that certain “chains” of principal series of G do not exist and we establish several properties of the Breuil–Herzig construction \Pi(\rho)^{\mathrm{ord}}. In particular, we obtain a natural characterization of the latter and we prove a conjecture of Breuil–Herzig. In order to do so, we partially compute Emerton’s δ-functor \operatorname{H^{\bullet}Ord}_{P} of derived ordinary parts with respect to a parabolic subgroup on a principal series. We formulate a new conjecture on the extensions between smooth mod p representations of G parabolically induced from supersingular representations of Levi subgroups of G and we prove it in the case of extensions by a principal series.

scholarly journals Permutation characters of finite groups of Lie type

1979 ◽  
Vol 27 (3) ◽  
pp. 378-384 ◽  
Author(s):  
David B. Surowski
Keyword(s):  
Fixed Points ◽  
Algebraic Group ◽  
Weyl Group ◽  
Parabolic Subgroup ◽  
Finite Groups ◽  
Linear Algebraic Group ◽  
Finite Subgroup ◽  
Semisimple Element ◽  
Frobenius Endomorphism ◽  
Groups Of Lie Type

AbstractLet g be a connected reductive linear algebraic group, and let G = gσ be the finite subgroup of fixed points, where σ is the generalized Frobenius endomorphism of g. Let x be a regular semisimple element of G and let w be a corresponding element of the Weyl group W. In this paper we give a formula for the number of right cosets of a parabolic subgroup of G left fixed by x, in terms of the corresponding action of w in W. In case G is untwisted, it turns out thta x fixes exactly as many cosets as does W in the corresponding permutation representation.

scholarly journals A classification of equivariant principal bundles over nonsingular toric varieties

2016 ◽  
Vol 27 (14) ◽  
pp. 1650115 ◽  
Author(s):  
Indranil Biswas ◽  
Arijit Dey ◽  
Mainak Poddar
Keyword(s):  
Algebraic Group ◽  
Toric Variety ◽  
Toric Varieties ◽  
Linear Algebraic Group ◽  
Principal Bundles ◽  
Algebraic Torus ◽  
Complex Linear

We classify holomorphic as well as algebraic torus equivariant principal [Formula: see text]-bundles over a nonsingular toric variety [Formula: see text], where [Formula: see text] is a complex linear algebraic group. It is shown that any such bundle over an affine, nonsingular toric variety admits a trivialization in equivariant sense. We also obtain some splitting results.

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 A decomposition formula for representations

1987 ◽  
Vol 107 ◽  
pp. 63-68 ◽  
Author(s):  
George Kempf
Keyword(s):  
Irreducible Representation ◽  
General Formula ◽  
Parabolic Subgroup ◽  
Characteristic Zero ◽  
Maximal Torus ◽  
Reductive Group ◽  
Irreducible Representations ◽  
Levi Subgroup ◽  
Know How ◽  
Decomposition Formula

Let H be the Levi subgroup of a parabolic subgroup of a split reductive group G. In characteristic zero, an irreducible representation V of G decomposes when restricted to H into a sum V = ⊕mαWα where the Wα’s are distinct irreducible representations of H. We will give a formula for the multiplicities mα. When H is the maximal torus, this formula is Weyl’s character formula. In theory one may deduce the general formula from Weyl’s result but I do not know how to do this.

scholarly journals MULTIPLE FLAG IND-VARIETIES WITH FINITELY MANY ORBITS

2021 ◽  
Author(s):  
LUCAS FRESSE ◽  
IVAN PENKOV
Keyword(s):  
Algebraic Group ◽  
Linear Algebraic Group ◽  
Interesting Case ◽  
Analogous Problem ◽  
Restrictive Condition ◽  
Dimensional Case ◽  
Essential Difference ◽  
Parabolic Subgroups ◽  
Finite Dimensional ◽  
Finite Dimensional Case

AbstractLet G be one of the ind-groups GL(∞), O(∞), Sp(∞), and let P1, ..., Pℓ be an arbitrary set of ℓ splitting parabolic subgroups of G. We determine all such sets with the property that G acts with finitely many orbits on the ind-variety X1 × × Xℓ where Xi = G/Pi. In the case of a finite-dimensional classical linear algebraic group G, the analogous problem has been solved in a sequence of papers of Littelmann, Magyar–Weyman–Zelevinsky and Matsuki. An essential difference from the finite-dimensional case is that already for ℓ = 2, the condition that G acts on X1 × X2 with finitely many orbits is a rather restrictive condition on the pair P1, P2. We describe this condition explicitly. Using the description we tackle the most interesting case where ℓ = 3, and present the answer in the form of a table. For ℓ ≥ 4 there always are infinitely many G-orbits on X1 × × Xℓ.

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