scholarly journals Translation-invariant line bundles on linear algebraic groups

2020 ◽  
pp. 1
Author(s):  
Zev Rosengarten
Keyword(s):  
Algebraic Groups ◽  
Line Bundles ◽  
Invariant Line ◽  
Translation Invariant ◽  
Linear Algebraic Groups

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.

scholarly journals Closedness of some subgroups in linear algebraic groups

1962 ◽  
Vol 14 (3) ◽  
pp. 272-275
Author(s):  
T. MIYATA ◽  
T. ODA ◽  
K. OTSUKA
Keyword(s):  
Algebraic Groups ◽  
Linear Algebraic Groups

scholarly journals Arithmetic purity of strong approximation for semi-simple simply connected groups

2020 ◽  
Vol 156 (12) ◽  
pp. 2628-2649
Author(s):  
Yang Cao ◽  
Zhizhong Huang
Keyword(s):  
Quadratic Form ◽  
Strong Approximation ◽  
Homogeneous Spaces ◽  
Algebraic Groups ◽  
Integral Points ◽  
Spin Group ◽  
Linear Algebraic Groups ◽  
Simply Connected ◽  
Quadratic Hypersurfaces ◽  
Almost Prime

In this article we establish the arithmetic purity of strong approximation for certain semisimple simply connected linear algebraic groups and their homogeneous spaces over a number field $k$. For instance, for any such group $G$ and for any open subset $U$ of $G$ with ${\mathrm {codim}}(G\setminus U, G)\geqslant 2$, we prove that (i) if $G$ is $k$-simple and $k$-isotropic, then $U$ satisfies strong approximation off any finite number of places; and (ii) if $G$ is the spin group of a non-degenerate quadratic form which is not compact over archimedean places, then $U$ satisfies strong approximation off all archimedean places. As a consequence, we prove that the same property holds for affine quadratic hypersurfaces. Our approach combines a fibration method with subgroup actions developed for induction on the codimension of $G\setminus U$, and an affine linear sieve which allows us to produce integral points with almost-prime polynomial values.

scholarly journals Morse inequalities for covering manifolds

2001 ◽  
Vol 163 ◽  
pp. 145-165 ◽  
Author(s):  
Radu Todor ◽  
Ionuţ Chiose ◽  
George Marinescu
Keyword(s):  
Line Bundles ◽  
Invariant Line ◽  
Complex Structures ◽  
Von Neumann ◽  
Holomorphic Sections ◽  
Galois Coverings ◽  
Von Neumann Dimension ◽  
The Stability ◽  
Bounded Below ◽  
Lefschetz Theorems

We study the existence of L2 holomorphic sections of invariant line bundles over Galois coverings. We show that the von Neumann dimension of the space of L2 holomorphic sections is bounded below under weak curvature conditions. We also give criteria for a compact complex space with isolated singularities and some related strongly pseudoconcave manifolds to be Moishezon. As applications we prove the stability of the previous Moishezon pseudoconcave manifolds under perturbation of complex structures as well as weak Lefschetz theorems.

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