scholarly journals Ghost classes in $${\mathbb {Q}}$$-rank two orthogonal Shimura varieties

2020 ◽  
Vol 296 (3-4) ◽  
pp. 1209-1233 ◽  
Author(s):  
Jitendra Bajpai ◽  
Matias V. Moya Giusti
Keyword(s):  
Algebraic Group ◽  
Algebraic Groups ◽  
Irreducible Representations ◽  
Shimura Varieties ◽  
Hodge Structures ◽  
Highest Weight ◽  
Weight Property ◽  
Eisenstein Cohomology ◽  
Cohomology Spaces ◽  
Mixed Hodge Structures

Abstract In this article, the existence of ghost classes for the Shimura varieties associated to algebraic groups of orthogonal similitudes of signature (2, n) is investigated. We make use of the study of the weights in the mixed Hodge structures associated to the corresponding cohomology spaces and results on the Eisenstein cohomology of the algebraic group of orthogonal similitudes of signature $$(1, n-1)$$ ( 1 , n - 1 ) . For the values of $$n = 4, 5$$ n = 4 , 5 we prove the non-existence of ghost classes for most of the irreducible representations (including most of those with an irregular highest weight). For the rest of the cases, we prove strong restrictions on the possible weights in the space of ghost classes and, in particular, we show that they satisfy the weak middle weight property.

RESTRICTIONS OF SOME REPRESENTATIONS OF THE SYMPLECTIC GROUP TO SUBGROUPS OF TYPE A1

2007 ◽  
Vol 06 (04) ◽  
pp. 697-701
Author(s):  
ANNA A. OSINOVSKAYA
Keyword(s):  
Algebraic Group ◽  
Symplectic Group ◽  
Algebraic Groups ◽  
Type A ◽  
Irreducible Representations ◽  
Symplectic Groups ◽  
Inductive System ◽  
Highest Weight ◽  
Composition Factors

Restrictions of modular irreducible representations of the symplectic algebraic group to naturally embedded long subgroups of type A1 are studied. Let ω = m1ω1 + ⋯ + mnωn be the highest weight of such representation. The composition factors of such restrictions are determined in the case of m1 + ⋯ + mn + 3 ≤ p < mn-1 + 2mn + 3 that completes the description of restrictions of classical algebraic groups to naturally embedded A1-subgroups and gives an example of a new inductive system of representations of symplectic groups that has no analogues in characteristic 0.

scholarly journals Representations of Algebraic Groups

1963 ◽  
Vol 22 ◽  
pp. 33-56 ◽  
Author(s):  
Robert Steinberg
Keyword(s):  
Algebraic Group ◽  
Algebraic Groups ◽  
Vector Spaces ◽  
Irreducible Representations ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Prime Field ◽  
Infinite Degree ◽  
Semisimple Algebraic Groups

Our purpose here is to study the irreducible representations of semisimple algebraic groups of characteristic p 0, in particular the rational representations, and to determine all of the representations of corresponding finite simple groups. (Each algebraic group is assumed to be defined over a universal field which is algebraically closed and of infinite degree of transcendence over the prime field, and all of its representations are assumed to take place on vector spaces over this field.)

Mumford-Tate Groups

2012 ◽  
Author(s):  
Mark Green ◽  
Phillip Griffiths ◽  
Matt Kerr
Keyword(s):  
Hodge Structure ◽  
Algebraic Groups ◽  
Hodge Decomposition ◽  
Hodge Structures ◽  
The Third ◽  
Direct Sum ◽  
Hodge Filtration ◽  
Mixed Hodge Structures

This chapter provides an introduction to the basic definitions and properties of Mumford-Tate groups in both the case of Hodge structures and of mixed Hodge structures. Hodge structures of weight n are sometimes called pure Hodge structures, and the term “Hodge structure” then refers to a direct sum of pure Hodge structures. The chapter presents three definitions of a Hodge structure of weight n, given in historical order. In the first definition, a Hodge structure of weight n is given by a Hodge decomposition; in the second, it is given by a Hodge filtration; in the third, it is given by a homomorphism of ℝ-algebraic groups. In the first two definitions, n is assumed to be positive and the p,q's in the definitions are non-negative. In the third definition, n and p,q are arbitrary. For the third definition, the Deligne torus integers are used.

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 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.

scholarly journals Small Degree Representations of Finite Chevalley Groups in Defining Characteristic

2001 ◽  
Vol 4 ◽  
pp. 135-169 ◽  
Author(s):  
Frank Lübeck
Keyword(s):  
Algebraic Groups ◽  
Small Degree ◽  
Irreducible Representations ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Linear Algebraic Groups ◽  
Large Rank ◽  
Projective Representations ◽  
Computer Calculations ◽  
Simply Connected

AbstractThe author has determined, for all simple simply connected reductive linear algebraic groups defined over a finite field, all the irreducible representations in their defining characteristic of degree below some bound. These also give the small degree projective representations in defining characteristic for the corresponding finite simple groups. For large rank l, this bound is proportional to l3, and for rank less than or equal to 11 much higher. The small rank cases are based on extensive computer calculations.

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