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Journal articles on the topic 'Semisimple algebraic groups'

Author: Grafiati

Published: 21 September 2024

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1

Nahlus, Nazih. "Homomorphisms of Lie Algebras of Algebraic Groups and Analytic Groups." Canadian Mathematical Bulletin 38, no. 3 (September 1, 1995): 352–59. http://dx.doi.org/10.4153/cmb-1995-051-7.

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AbstractLet be a Lie algebra homomorphism from the Lie algebra of G to the Lie algebra of H in the following cases: (i) G and H are irreducible algebraic groups over an algebraically closed field of characteristic 0, or (ii) G and H are linear complex analytic groups. In this paper, we present some equivalent conditions for ϕ to be a differential in the above two cases. That is, ϕ is the differential of a morphism of algebraic groups or analytic groups as appropriate.In the algebraic case, for example, it is shown that ϕ is a differential if and only if ϕ preserves nilpotency, semisimplicity, and integrality of elements. In the analytic case, ϕ is a differential if and only if ϕ maps every integral semisimple element of into an integral semisimple element of , where G0 and H0 are the universal algebraic subgroups of G and H. Via rational elements, we also present some equivalent conditions for ϕ to be a differential up to coverings of G in the algebraic case, and for ϕ to be a differential up to finite coverings of G in the analytic case.

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2

De Clercq, Charles. "Équivalence motivique des groupes algébriques semisimples." Compositio Mathematica 153, no. 10 (July 27, 2017): 2195–213. http://dx.doi.org/10.1112/s0010437x17007369.

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We prove that the standard motives of a semisimple algebraic group$G$with coefficients in a field of order$p$are determined by the upper motives of the group $G$. As a consequence of this result, we obtain a partial version of the motivic rigidity conjecture of special linear groups. The result is then used to construct the higher indexes which characterize the motivic equivalence of semisimple algebraic groups. The criteria of motivic equivalence derived from the expressions of these indexes produce a dictionary between motives, algebraic structures and the birational geometry of twisted flag varieties. This correspondence is then described for special linear groups and orthogonal groups (the criteria associated with other groups being obtained in De Clercq and Garibaldi [Tits$p$-indexes of semisimple algebraic groups, J. Lond. Math. Soc. (2)95(2017) 567–585]). The proofs rely on the Levi-type motivic decompositions of isotropic twisted flag varieties due to Chernousov, Gille and Merkurjev, and on the notion of pondered field extensions.

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3

De Clercq, Charles, and Skip Garibaldi. "Tits p-indexes of semisimple algebraic groups." Journal of the London Mathematical Society 95, no. 2 (January 16, 2017): 567–85. http://dx.doi.org/10.1112/jlms.12025.

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4

Gordeev, Nikolai, Boris Kunyavskiĭ, and Eugene Plotkin. "Word maps on perfect algebraic groups." International Journal of Algebra and Computation 28, no. 08 (December 2018): 1487–515. http://dx.doi.org/10.1142/s0218196718400052.

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We extend Borel’s theorem on the dominance of word maps from semisimple algebraic groups to some perfect groups. In another direction, we generalize Borel’s theorem to some words with constants. We also consider the surjectivity problem for particular words and groups, give a brief survey of recent results, present some generalizations and variations and discuss various approaches, with emphasis on new ideas, constructions and connections.

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5

Cassidy, Phyllis Joan. "The classification of the semisimple differential algebraic groups and the linear semisimple differential algebraic Lie algebras." Journal of Algebra 121, no. 1 (February 1989): 169–238. http://dx.doi.org/10.1016/0021-8693(89)90092-6.

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6

Avdeev, R. S. "On solvable spherical subgroups of semisimple algebraic groups." Transactions of the Moscow Mathematical Society 72 (2011): 1–44. http://dx.doi.org/10.1090/s0077-1554-2012-00192-7.

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7

Procesi, Claudio. "Book Review: Conjugacy classes in semisimple algebraic groups." Bulletin of the American Mathematical Society 34, no. 01 (January 1, 1997): 55–57. http://dx.doi.org/10.1090/s0273-0979-97-00689-7.

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8

Voskresenskii, V. E. "Maximal tori without effect in semisimple algebraic groups." Mathematical Notes of the Academy of Sciences of the USSR 44, no. 3 (September 1988): 651–55. http://dx.doi.org/10.1007/bf01159125.

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9

Mohrdieck, S. "Conjugacy classes of non-connected semisimple algebraic groups." Transformation Groups 8, no. 4 (December 2003): 377–95. http://dx.doi.org/10.1007/s00031-003-0429-3.

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10

Breuillard, Emmanuel, Ben Green, Robert Guralnick, and Terence Tao. "Strongly dense free subgroups of semisimple algebraic groups." Israel Journal of Mathematics 192, no. 1 (March 15, 2012): 347–79. http://dx.doi.org/10.1007/s11856-012-0030-3.

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11

Gupta, Shalini, and Sugandha Maheshwary. "Finite semisimple group algebra of a normally monomial group." International Journal of Algebra and Computation 29, no. 01 (February 2019): 159–77. http://dx.doi.org/10.1142/s0218196718500674.

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In this paper, the complete algebraic structure of the finite semisimple group algebra of a normally monomial group is described. The main result is illustrated by computing the explicit Wedderburn decomposition of finite semisimple group algebras of various normally monomial groups. The automorphism groups of these group algebras are also determined.

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12

Avdeev, R. S. "Excellent affine spherical homogeneous spaces of semisimple algebraic groups." Transactions of the Moscow Mathematical Society 71 (2010): 209. http://dx.doi.org/10.1090/s0077-1554-2010-00183-5.

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13

Barnea, Y., and M. Larsen. "Random generation in semisimple algebraic groups over local fields." Journal of Algebra 271, no. 1 (January 2004): 1–10. http://dx.doi.org/10.1016/j.jalgebra.2002.12.001.

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14

Kaneda, Masaharu. "On the alcove identification operator of semisimple algebraic groups." Communications in Algebra 15, no. 6 (January 1987): 1157–71. http://dx.doi.org/10.1080/00927878708823461.

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15

Can, Mahir Bilen. "Irreducible representations of semisimple algebraic groups and supersolvable lattices." Journal of Algebra 351, no. 1 (February 2012): 235–50. http://dx.doi.org/10.1016/j.jalgebra.2011.11.011.

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16

Kannan, S. Senthamarai. "Remarks on the wonderful compactification of semisimple algebraic groups." Proceedings Mathematical Sciences 109, no. 3 (August 1999): 241–56. http://dx.doi.org/10.1007/bf02843529.

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17

Garibaldi, Skip, and Daniel K. Nakano. "Bilinear and Quadratic Forms on Rational Modules of Split Reductive Groups." Canadian Journal of Mathematics 68, no. 2 (April 1, 2016): 395–421. http://dx.doi.org/10.4153/cjm-2015-042-5.

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AbstractThe representation theory of semisimple algebraic groups over the complex numbers (equivalently, semisimple complex Lie algebras or Lie groups, or real compact Lie groups) and the questions of whether a given complex representation is symplectic or orthogonal have been solved since at least the 1950s. Similar results for Weyl modules of split reductive groups over fields of characteristic different from z hold by using similar proofs. This paper considers analogues of these results for simple, induced, and tilting modules of split reductive groups over fields of prime characteristic as well as a complete answer for Weyl modules over fields of characteristic 2.

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18

Chen, Yu. "On rational subgroups of reductive algebraic groups over integral domains." Mathematical Proceedings of the Cambridge Philosophical Society 117, no. 2 (March 1995): 203–12. http://dx.doi.org/10.1017/s0305004100073059.

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Let G and G′ be reductive algebraic groups defined over infinite fields k and k′ respectively. The purpose of this paper is to show that G and G′ have isomorphic root systems if their rational subgroups G(R) and G′(R′), where R and R′ are integral domains with R ⊇ k and R′ ⊇ k′, are isomorphic to each other, except in one particular case (see Theorem 3·4). This has been proved by R. Steinberg in [6, theorem 31] for simple Chevalley groups over perfect fields. In particular, when G and G′ are semisimple and adjoint, every isomorphism between G(R) and G′(R′) induces an isomorphism between their irreducible components (see Proposition 3·3). These results imply that, when G and G′ are semisimple k-groups and when both are either simply connected or adjoint, then they are isomorphic to each other as algebraic groups if and only if their rational subgroups over an integral domain that contains k are isomorphic to each other, except in one particular case (see Corollary 3·5).

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19

CREUTZ, DARREN. "Stabilizers of actions of lattices in products of groups." Ergodic Theory and Dynamical Systems 37, no. 4 (March 24, 2016): 1133–86. http://dx.doi.org/10.1017/etds.2015.109.

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We prove that any ergodic non-atomic probability-preserving action of an irreducible lattice in a semisimple group, with at least one factor being connected and of higher-rank, is essentially free. This generalizes the result of Stuck and Zimmer [Stabilizers for ergodic actions of higher rank semisimple groups. Ann. of Math. (2)139(3) (1994), 723–747], who found that the same statement holds when the ambient group is a semisimple real Lie group and every simple factor is of higher-rank. We also prove a generalization of a result of Bader and Shalom [Factor and normal subgroup theorems for lattices in products of groups. Invent. Math.163(2) (2006), 415–454] by showing that any probability-preserving action of a product of simple groups, with at least one having property $(T)$, which is ergodic for each simple subgroup, is either essentially free or essentially transitive. Our method involves the study of relatively contractive maps and the Howe–Moore property, rather than relying on algebraic properties of semisimple groups and Poisson boundaries, and introduces a generalization of the ergodic decomposition to invariant random subgroups, which is of independent interest.

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20

Zeghib, A. "Ensembles invariants des flots géodésiques des variétés localement symétriques." Ergodic Theory and Dynamical Systems 15, no. 2 (April 1995): 379–412. http://dx.doi.org/10.1017/s0143385700008439.

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AbstractWe study the rectifiable invariant subsets of algebraic dynamical systems determined by ℝ-semisimple one parameter groups. We show that their ergodic components are algebraic. A more precise geometric description of these components is possible in some cases of geodesic flows of locally symmetric spaces with non-positive curvature.

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21

Popov, Vladimir L. "Cross-sections, quotients, and representation rings of semisimple algebraic groups." Transformation Groups 16, no. 3 (April 7, 2011): 827–56. http://dx.doi.org/10.1007/s00031-011-9137-6.

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22

Moy, Allen, and Goran Muić. "On existence of generic cusp forms on semisimple algebraic groups." Transactions of the American Mathematical Society 370, no. 7 (January 18, 2018): 4731–57. http://dx.doi.org/10.1090/tran/7081.

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23

Kordonskii, Vsevolod E., and E. A. Tevelev. "Non-stable linear actions of connected semisimple complex algebraic groups." Sbornik: Mathematics 186, no. 1 (February 28, 1995): 107–19. http://dx.doi.org/10.1070/sm1995v186n01abeh000006.

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24

Podkopaev, O. B. "On the Grothendieck group of simply connected semisimple algebraic groups." Journal of Mathematical Sciences 140, no. 5 (February 2007): 729–36. http://dx.doi.org/10.1007/s10958-007-0012-x.

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25

Gorodnik, Alexander, and Amos Nevo. "Discrepancy of rational points in simple algebraic groups." Compositio Mathematica 160, no. 4 (March 13, 2024): 836–77. http://dx.doi.org/10.1112/s0010437x23007716.

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The aim of the present paper is to derive effective discrepancy estimates for the distribution of rational points on general semisimple algebraic group varieties, in general families of subsets and at arbitrarily small scales. We establish mean-square, almost sure and uniform estimates for the discrepancy with explicit error bounds. We also prove an analogue of W. Schmidt's theorem, which establishes effective almost sure asymptotic counting of rational solutions to Diophantine inequalities in the Euclidean space. We formulate and prove a version of it for rational points on the group variety, with an effective bound which in some instances can be expected to be the best possible.

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26

Popov, V. L. "Embeddings of Automorphism Groups of Free Groups into Automorphism Groups of Affine Algebraic Varieties." Proceedings of the Steklov Institute of Mathematics 320, no. 1 (March 2023): 267–77. http://dx.doi.org/10.1134/s0081543823010121.

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Abstract For every integer $$n>0$$, we construct a new infinite series of rational affine algebraic varieties such that their automorphism groups contain the automorphism group $$\mathrm{Aut}(F_n)$$ of the free group $$F_n$$ of rank $$n$$ and the braid group $$B_n$$ on $$n$$ strands. The automorphism groups of such varieties are nonlinear for $$n\geq 3$$ and are nonamenable for $$n\geq 2$$. As an application, we prove that every Cremona group of rank $${\geq}\,3n-1$$ contains the groups $$\mathrm{Aut}(F_n)$$ and $$B_n$$. This bound is $$1$$ better than the bound published earlier by the author; with respect to $$B_n$$, the order of its growth rate is one less than that of the bound following from a paper by D. Krammer. The construction is based on triples $$(G,R,n)$$, where $$G$$ is a connected semisimple algebraic group and $$R$$ is a closed subgroup of its maximal torus.

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27

Mozaffarikhah, A., E. Momtahan, A. R. Olfati, and S. Safaeeyan. "p-semisimple modules and type submodules." Journal of Algebra and Its Applications 19, no. 04 (April 17, 2019): 2050078. http://dx.doi.org/10.1142/s0219498820500784.

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In this paper, we introduce the concept of [Formula: see text]-semisimple modules. We prove that a multiplication reduced module is [Formula: see text]-semisimple if and only if it is a Baer module. We show that a large family of abelian groups are [Formula: see text]-semisimple. Furthermore, we give a topological characterizations of type submodules (ideals) of multiplication reduced modules ([Formula: see text]-semisimple rings). Moreover, we observe that there is a one-to-one correspondence between type ideals of some algebraic structures on one hand and regular closed subsets of some related topological spaces on the other hand. This also characterizes the form of closed ideals in [Formula: see text].

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28

Ayala, Víctor, and María Torreblanca Todco. "Boundedness control sets for linear systems on Lie groups." Open Mathematics 16, no. 1 (April 18, 2018): 370–79. http://dx.doi.org/10.1515/math-2018-0035.

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AbstractLet Σ be a linear system on a connected Lie group G and assume that the reachable set 𝓐 from the identity element e ∈ G is open. In this paper, we give an algebraic condition to warrant the boundedness of the existent control set with a nonempty interior containing e. We concentrate the search for the class of decomposable groups which includes any solvable group and also every compact semisimple group.

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29

Moser-Jauslin, Lucy, and Ronan Terpereau. "Real structures on symmetric spaces." Proceedings of the American Mathematical Society 149, no. 8 (May 11, 2021): 3159–72. http://dx.doi.org/10.1090/proc/15520.

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We obtain a necessary and sufficient condition for the existence of equivariant real structures on complex symmetric spaces for semisimple algebraic groups and discuss how to determine the number of equivalence classes for such structures.

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30

LIEDÓ, M. A. "DEFORMATION QUANTIZATION OF COADJOINT ORBITS." International Journal of Modern Physics B 14, no. 22n23 (September 20, 2000): 2397–400. http://dx.doi.org/10.1142/s0217979200001916.

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A method for the deformation quantization of coadjoint orbits of semisimple Lie groups is proposed. It is based on the algebraic structure of the orbit. Its relation to geometric quantization and differentiable deformations is explored.

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31

Nesterenko, Maryna, Jiří Patera, and Agnieszka Tereszkiewicz. "Orthogonal Polynomials of Compact Simple Lie Groups." International Journal of Mathematics and Mathematical Sciences 2011 (2011): 1–23. http://dx.doi.org/10.1155/2011/969424.

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Recursive algebraic construction of two infinite families of polynomials innvariables is proposed as a uniform method applicable to every semisimple Lie group of rankn. Its result recognizes Chebyshev polynomials of the first and second kind as the special case of the simple group of typeA1. The obtained not Laurent-type polynomials are equivalent to the partial cases of the Macdonald symmetric polynomials. Recurrence relations are shown for the Lie groups of typesA1,A2,A3,C2,C3,G2, andB3together with lowest polynomials.

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32

Carter, R. W. "CONJUGACY CLASSES IN SEMISIMPLE ALGEBRAIC GROUPS (Mathematical Surveys and Monographs 43)." Bulletin of the London Mathematical Society 28, no. 6 (November 1996): 668–69. http://dx.doi.org/10.1112/blms/28.6.668.

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33

Andersen, Henning Haahr. "On the generic structure of cohomology modules for semisimple algebraic groups." Transactions of the American Mathematical Society 295, no. 1 (January 1, 1986): 397. http://dx.doi.org/10.1090/s0002-9947-1986-0831206-2.

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34

Merkurjev, A. S. "R-equivalence and rationality problem for semisimple adjoint classical algebraic groups." Publications mathématiques de l'IHÉS 84, no. 1 (December 1996): 189–213. http://dx.doi.org/10.1007/bf02698837.

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35

Chatterjee, Pralay. "On the surjectivity of the power maps of semisimple algebraic groups." Mathematical Research Letters 10, no. 5 (2003): 625–33. http://dx.doi.org/10.4310/mrl.2003.v10.n5.a6.

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36

Adams, Jeffrey. "The real Chevalley involution." Compositio Mathematica 150, no. 12 (August 27, 2014): 2127–42. http://dx.doi.org/10.1112/s0010437x14007374.

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AbstractThe Chevalley involution of a connected, reductive algebraic group over an algebraically closed field takes every semisimple element to a conjugate of its inverse, and this involution is unique up to conjugacy. In the case of the reals we prove the existence of a real Chevalley involution, which is defined over $\mathbb{R}$, takes every semisimple element of $G(\mathbb{R})$ to a $G(\mathbb{R})$-conjugate of its inverse, and is unique up to conjugacy by $G(\mathbb{R})$. We derive some consequences, including an analysis of groups for which every irreducible representation is self-dual, and a calculation of the Frobenius Schur indicator for such groups.

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37

Helminck, Aloysius G. "Algebraic groups with a commuting pair of involutions and semisimple symmetric spaces." Advances in Mathematics 71, no. 1 (September 1988): 21–91. http://dx.doi.org/10.1016/0001-8708(88)90066-7.

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38

De Visscher, Maud. "On the blocks of semisimple algebraic groups and associated generalized Schur algebras." Journal of Algebra 319, no. 3 (February 2008): 952–65. http://dx.doi.org/10.1016/j.jalgebra.2007.11.015.

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39

Labesse, Jean-Pierre, and Joachim Schwermer. "Corrigendum: On the cuspidal cohomology of S-arithmetic subgroups of reductive groups over number fields." Compositio Mathematica 157, no. 6 (May 26, 2021): 1207–10. http://dx.doi.org/10.1112/s0010437x21007181.

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The aim of this corrigendum is to correct an error in Corollary 10.7 to Theorem 10.6, one of the main results in the paper ‘On the cuspidal cohomology of $S$-arithmetic subgroups of reductive groups over number fields’. This makes necessary a thorough investigation of the conditions under which a Cartan-type automorphism exists on $G_1=\mathrm {Res}_{\mathbb {C}/\mathbb {R}}G_0$, where $G_0$ is a connected semisimple algebraic group defined over $\mathbb {R}$.

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40

Gordon, Carolyn S. "Naturally Reductive Homogeneous Riemannian Manifolds." Canadian Journal of Mathematics 37, no. 3 (June 1, 1985): 467–87. http://dx.doi.org/10.4153/cjm-1985-028-2.

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The simple algebraic and geometric properties of naturally reductive metrics make them useful as examples in the study of homogeneous Riemannian manifolds. (See for example [2], [3], [15]). The existence and abundance of naturally reductive left-invariant metrics on a Lie group G or homogeneous space G/L reflect the structure of G itself. Such metrics abound on compact groups, exist but are more restricted on noncompact semisimple groups, and are relatively rare on solvable groups. The goals of this paper are(i) to study all naturally reductive homogeneous spaces of G when G is either semisimple of noncompact type or nilpotent and(ii) to give necessary conditions on a Riemannian homogeneous space of an arbitrary Lie group G in order that the metric be naturally reductive with respect to some transitive subgroup of G.

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41

Gros, Michel, and Kaneda Masaharu. "UN SCINDAGE DU MORPHISME DE FROBENIUS SUR L’ALGÈBRE DES DISTRIBUTIONS D’UN GROUPE RÉDUCTIF." Quarterly Journal of Mathematics 71, no. 1 (December 20, 2019): 197–206. http://dx.doi.org/10.1093/qmathj/haz039.

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Abstract Pour un groupe algébrique semi-simple simplement connexe sur un corps algébriquement clos de caractéristique positive, nous avons précédemment construit un scindage de l’endomorphisme de Frobenius sur son algèbre des distributions. Nous généralisons la construction au cas de des groupes réductifs connexes et en dégageons les corollaires correspondants. For a simply connected semisimple algebraic group over an algebraically closed field of positive characteristic we have already constructed a splitting of the Frobenius endomorphism on its algebra of distributions. We generalize the construction to the case of general connected reductive groups and derive the corresponding corollaries.

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42

Ibraev, Sherali S., Larissa S. Kainbaeva, and Angisin Z. Seitmuratov. "On Restricted Cohomology of Modular Classical Lie Algebras and Their Applications." Mathematics 10, no. 10 (May 13, 2022): 1680. http://dx.doi.org/10.3390/math10101680.

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In this paper, we study the restricted cohomology of Lie algebras of semisimple and simply connected algebraic groups in positive characteristics with coefficients in simple restricted modules and their applications in studying the connections between these cohomology with the corresponding ordinary cohomology and cohomology of algebraic groups. Let G be a semisimple and simply connected algebraic group G over an algebraically closed field of characteristic p>h, where h is a Coxeter number. Denote the first Frobenius kernel and Lie algebra of G by G1 and g, respectively. First, we calculate the restricted cohomology of g with coefficients in simple modules for two families of restricted simple modules. Since in the restricted region the restricted cohomology of g is equivalent to the corresponding cohomology of G1, we describe them as the cohomology of G1 in terms of the cohomology for G1 with coefficients in dual Weyl modules. Then, we give a necessary and sufficient condition for the isomorphisms Hn(G1,V)≅Hn(G,V) and Hn(g,V)≅Hn(G,V), and a necessary condition for the isomorphism Hn(g,V)≅Hn(G1,V), where V is a simple module with highest restricted weight. Using these results, we obtain all non-trivial isomorphisms between the cohomology of G, G1, and g with coefficients in the considered simple modules.

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43

Stalder, Nicolas. "The semisimplicity conjecture for A-motives." Compositio Mathematica 146, no. 3 (March 18, 2010): 561–98. http://dx.doi.org/10.1112/s0010437x09004448.

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AbstractWe prove the semisimplicity conjecture for A-motives over finitely generated fields K. This conjecture states that the rational Tate modules V𝔭(M) of a semisimple A-motive M are semisimple as representations of the absolute Galois group of K. This theorem is in analogy with known results for abelian varieties and Drinfeld modules, and has been sketched previously by Tamagawa. We deduce two consequences of the theorem for the algebraic monodromy groups G𝔭(M) associated to an A-motive M by Tannakian duality. The first requires no semisimplicity condition on M and states that G𝔭(M) may be identified naturally with the Zariski closure of the image of the absolute Galois group of K in the automorphism group of V𝔭(M). The second states that the connected component of G𝔭(M) is reductive if M is semisimple and has a separable endomorphism algebra.

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44

Lu, Jiang-Hua. "On a dimension formula for spherical twisted conjugacy classes in semisimple algebraic groups." Mathematische Zeitschrift 269, no. 3-4 (September 30, 2010): 1181–88. http://dx.doi.org/10.1007/s00209-010-0776-4.

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45

Taylor, Jay. "The Structure of Root Data and Smooth Regular Embeddings of Reductive Groups." Proceedings of the Edinburgh Mathematical Society 62, no. 2 (November 29, 2018): 523–52. http://dx.doi.org/10.1017/s0013091518000597.

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AbstractWe investigate the structure of root data by considering their decomposition as a product of a semisimple root datum and a torus. Using this decomposition, we obtain a parametrization of the isomorphism classes of all root data. By working at the level of root data, we introduce the notion of a smooth regular embedding of a connected reductive algebraic group, which is a refinement of the commonly used regular embeddings introduced by Lusztig. In the absence of Steinberg endomorphisms, such embeddings were constructed by Benjamin Martin. In an unpublished manuscript, Asai proved three key reduction techniques that are used for reducing statements about arbitrary connected reductive algebraic groups, equipped with a Frobenius endomorphism, to those whose derived subgroup is simple and simply connected. Using our investigations into root data we give new proofs of Asai's results and generalize them so that they are compatible with Steinberg endomorphisms. As an illustration of these ideas, we answer a question posed to us by Olivier Dudas concerning unipotent supports.

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46

Ibraev, Sh Sh, A. Zh Seitmuratov, and L. S. Kainbayeva. "On simple modules with singular highest weights for so2l+1(K)." BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS 105, no. 1 (March 30, 2022): 52–65. http://dx.doi.org/10.31489/2022m1/52-65.

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In this paper, we study formal characters of simple modules with singular highest weights over classical Lie algebras of type B over an algebraically closed field of characteristic p ≥ h, where h is the Coxeter number. Assume that the highest weights of these simple modules are restricted. We have given a description of their formal characters. In particular, we have obtained some new examples of simple Weyl modules. In the restricted region, the representation theory of algebraic groups and its Lie algebras are equivalent. Therefore, we can use the tools of the representation theory of semisimple and simply-connected algebraic groups in positive characteristic. To describe the formal characters of simple modules, we construct Jantzen filtrations of Weyl modules of the corresponding highest weights.

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47

Gorodnik, Alexander, and Amos Nevo. "Lifting, restricting and sifting integral points on affine homogeneous varieties." Compositio Mathematica 148, no. 6 (October 11, 2012): 1695–716. http://dx.doi.org/10.1112/s0010437x12000516.

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AbstractIn [Gorodnik and Nevo,Counting lattice points, J. Reine Angew. Math.663(2012), 127–176] an effective solution of the lattice point counting problem in general domains in semisimpleS-algebraic groups and affine symmetric varieties was established. The method relies on the mean ergodic theorem for the action ofGonG/Γ, and implies uniformity in counting over families of lattice subgroups admitting a uniform spectral gap. In the present paper we extend some methods developed in [Nevo and Sarnak,Prime and almost prime integral points on principal homogeneous spaces, Acta Math.205(2010), 361–402] and use them to establish several useful consequences of this property, including:(1)effective upper bounds on lifting for solutions of congruences in affine homogeneous varieties;(2)effective upper bounds on the number of integral points on general subvarieties of semisimple group varieties;(3)effective lower bounds on the number of almost prime points on symmetric varieties;(4)effective upper bounds on almost prime solutions of congruences in homogeneous varieties.

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48

Testerman, D. M. "A1-Type Overgroups of Elements of Order p in Semisimple Algebraic Groups and the Associated Finite Groups." Journal of Algebra 177, no. 1 (October 1995): 34–76. http://dx.doi.org/10.1006/jabr.1995.1285.

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Avdeev, Roman S. "Extended weight semigroups of affine spherical homogeneous spaces of non-simple semisimple algebraic groups." Izvestiya: Mathematics 74, no. 6 (December 22, 2010): 1103–26. http://dx.doi.org/10.1070/im2010v074n06abeh002518.

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50

Ibraev, Sherali S., Larissa S. Kainbaeva, and Saulesh K. Menlikozhaeva. "On Cohomology of Simple Modules for Modular Classical Lie Algebras." Axioms 11, no. 2 (February 16, 2022): 78. http://dx.doi.org/10.3390/axioms11020078.

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In this article, we obtain some cohomology of classical Lie algebras over an algebraically closed field of characteristic p>h, where h is a Coxeter number, with coefficients in simple modules. We assume that these classical Lie algebras are Lie algebras of semisimple and simply connected algebraic groups. To describe the cohomology of simple modules, we will use the properties of the connections between ordinary and restricted cohomology of restricted Lie algebras.

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