scholarly journals Asymptotics of the powers in finite reductive groups

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Amit Kulshrestha ◽  
Rijubrata Kundu ◽  
Anupam Singh
Keyword(s):  
Reductive Group ◽  
Reductive Groups ◽  
Regular Elements

Abstract Let 𝐺 be a connected reductive group defined over F q \mathbb{F}_{q} . Fix an integer M ≥ 2 M\geq 2 , and consider the power map x ↦ x M x\mapsto x^{M} on 𝐺. We denote the image of G ⁢ ( F q ) G(\mathbb{F}_{q}) under this map by G ⁢ ( F q ) M G(\mathbb{F}_{q})^{M} and estimate what proportion of regular semisimple, semisimple and regular elements of G ⁢ ( F q ) G(\mathbb{F}_{q}) it contains. We prove that, as q → ∞ q\to\infty , the set of limits for each of these proportions is the same and provide a formula. This generalizes the well-known results for M = 1 M=1 where all the limits take the same value 1. We also compute this more explicitly for the groups GL ⁢ ( n , q ) \mathrm{GL}(n,q) and U ⁢ ( n , q ) \mathrm{U}(n,q) and show that the set of limits are the same for these two group, in fact, in bijection under q ↦ - q q\mapsto-q for a fixed 𝑀.

Central extensions and groups locally of minimal type

2015 ◽  
Author(s):  
Brian Conrad ◽  
Gopal Prasad
Keyword(s):  
Root System ◽  
Reductive Group ◽  
Reductive Groups ◽  
Central Extensions ◽  
General Lemma ◽  
Quotient Maps ◽  
Central Quotient ◽  
Characteristic 2 ◽  
Minimal Type ◽  
Quadratic Case

This chapter deals with central extensions and groups locally of minimal type. It begins with a discussion of the general lemma on the behavior of the scheme-theoretic center with respect to the formation of central quotient maps between pseudo-reductive groups; this lemma generalizes a familiar fact in the connected reductive case. The chapter then considers four phenomena that go beyond the quadratic case, along with a pseudo-reductive group of minimal type that is locally of minimal type. It shows that the pseudo-split absolutely pseudo-simple k-groups of minimal type with a non-reduced root system are classified over any imperfect field of characteristic 2. In this classification there is no effect if the “minimal type” hypothesis is relaxed to “locally of minimal type.”

Bruhat Order and Transfer for Complex Reductive Groups

1992 ◽  
Vol 44 (5) ◽  
pp. 911-923
Author(s):  
Martin Andler
Keyword(s):  
Reductive Group ◽  
Bruhat Order ◽  
Reductive Groups ◽  
Natural Way

AbstractLet G be a complex reductive group, and G^ its set of irreducible admissible representations. The Bruhat order on G^ is defined in a natural way. We prove that this Bruhat order is preserved by transfer. This gives new proofs of some results by the author on L-functions.

scholarly journals Algorithms for representation theory of real reductive groups

2009 ◽  
Vol 8 (2) ◽  
pp. 209-259 ◽  
Author(s):  
Jeffrey Adams ◽  
Fokko du Cloux
Keyword(s):  
Representation Theory ◽  
Lie Groups ◽  
Maximal Compact Subgroup ◽  
Reductive Group ◽  
Compact Subgroup ◽  
Flag Variety ◽  
Irreducible Representations ◽  
Structure Theory ◽  
Effective Algorithm ◽  
Reductive Groups

AbstractThe admissible representations of a real reductive groupGare known by work of Langlands, Knapp, Zuckerman and Vogan. This paper describes an effective algorithm for computing the irreducible representations ofGwith regular integral infinitesimal character. The algorithm also describes structure theory ofG, including the orbits ofK(ℂ) (a complexified maximal compact subgroup) on the flag variety. This algorithm has been implemented on a computer by the second author, as part of the ‘Atlas of Lie Groups and Representations’ project.

scholarly journals ON THE EXISTENCE OF ADMISSIBLE SUPERSINGULAR REPRESENTATIONS OF -ADIC REDUCTIVE GROUPS

2020 ◽  
Vol 8 ◽  
Author(s):  
FLORIAN HERZIG ◽  
KAROL KOZIOŁ ◽  
MARIE-FRANCE VIGNÉRAS
Keyword(s):  
Reductive Group ◽  
Finite Extension ◽  
Reductive Groups

Suppose that $\mathbf{G}$ is a connected reductive group over a finite extension $F/\mathbb{Q}_{p}$ and that $C$ is a field of characteristic  $p$ . We prove that the group $\mathbf{G}(F)$ admits an irreducible admissible supercuspidal, or equivalently supersingular, representation over  $C$ .

scholarly journals Stratifications with respect to actions of real reductive groups

2008 ◽  
Vol 144 (1) ◽  
pp. 163-185 ◽  
Author(s):  
Peter Heinzner ◽  
Gerald W. Schwarz ◽  
Henrik Stötzel
Keyword(s):  
Critical Points ◽  
Maximal Compact Subgroup ◽  
Reductive Group ◽  
Compact Subgroup ◽  
Kähler Manifold ◽  
Reductive Groups ◽  
Cartan Decomposition ◽  
Kahler Manifold ◽  
Real Submanifold

AbstractWe study the action of a real reductive group G on a real submanifold X of a Kähler manifold Z. We suppose that the action of G extends holomorphically to an action of the complexified group $G^{\mathbb {C}}$ and that with respect to a compatible maximal compact subgroup U of $G^{\mathbb {C}}$ the action on Z is Hamiltonian. There is a corresponding gradient map $\mu _{\mathfrak {p}}\colon X\to \mathfrak {p}^*$ where $\mathfrak {g}=\mathfrak {k}\oplus \mathfrak {p}$ is a Cartan decomposition of $\mathfrak {g}$. We obtain a Morse-like function $\eta _{\mathfrak {p}}:=\Vert \mu _{\mathfrak {p}}\Vert ^2$ on X. Associated with critical points of $\eta _{\mathfrak {p}}$ are various sets of semistable points which we study in great detail. In particular, we have G-stable submanifolds Sβ of X which are called pre-strata. In cases where $\mu _{\mathfrak {p}}$ is proper, the pre-strata form a decomposition of X and in cases where X is compact they are the strata of a Morse-type stratification of X. Our results are generalizations of results of Kirwan obtained in the case where $G=U^{\mathbb {C}}$ and X=Z is compact.

scholarly journals Lifting Representations of Finite Reductive Groups I: Semisimple Conjugacy Classes

2014 ◽  
Vol 66 (6) ◽  
pp. 1201-1224 ◽  
Author(s):  
Jeffrey D. Adler ◽  
Joshua M. Lansky
Keyword(s):  
Fixed Point ◽  
Finite Group ◽  
Fixed Points ◽  
Point Group ◽  
Reductive Group ◽  
Conjugacy Classes ◽  
Irreducible Representations ◽  
Reductive Groups ◽  
A Finite Group ◽  
The Relationship

AbstractSuppose that is a connected reductive group defined over a field k, and ┌ is a finite group acting via k-automorphisms of satisfying a certain quasi-semisimplicity condition. Then the identity component of the group of -fixed points in is reductive. We axiomatize the main features of the relationship between this fixed-point group and the pair (,┌) and consider any group G satisfying the axioms. If both and G are k-quasisplit, then we can consider their duals *and G*. We show the existence of and give an explicit formula for a natural map from the set of semisimple stable conjugacy classes in G*(k) to the analogous set for *(k). If k is finite, then our groups are automatically quasisplit, and our result specializes to give a map of semisimple conjugacy classes. Since such classes parametrize packets of irreducible representations of G(k) and (k), one obtains a mapping of such packets.

Characters of Non-Connected, Reductive p-Abic Groups

1987 ◽  
Vol 39 (1) ◽  
pp. 149-167 ◽  
Author(s):  
Laurent Clozel
Keyword(s):  
Irreducible Representation ◽  
Characteristic Zero ◽  
Reductive Group ◽  
Base Change ◽  
Reductive Groups ◽  
Local Integrability ◽  
Precise Control ◽  
Distribution Character ◽  
Connected Case

In this paper, we extend to non-connected, reductive groups over p-adic field of characteristic zero Harish-Chandra's theorem on the local integrability of characters.Harish-Chandra's theorem states that the distribution character of an admissible, irreducible representation of a (connected) reductive p-adic group is locally integrable. We show that this extends to any reductive group; just as in the connected case, one even gets a very precise control over the singularities of the character along the singular elements.As will be seen, the proof in the non-connected case is an easy extension of Harish-Chandra's. The reader may wonder why we have bothered to write its generalization completely. The reason is that the original article [8] does not contain proofs for the crucial lemmas, and this makes it impossible to explain why the theorem extends. Because this result is needed for work of Arthur and the author on base change, it has been thought necessary to give complete arguments.

scholarly journals Compactifications of reductive groups as moduli stacks of bundles

2015 ◽  
Vol 152 (1) ◽  
pp. 62-98 ◽  
Author(s):  
Johan Martens ◽  
Michael Thaddeus
Keyword(s):  
Stability Condition ◽  
Reductive Group ◽  
Weyl Chamber ◽  
Reductive Groups ◽  
Moduli Stacks ◽  
Moduli Stack

Let $G$ be a split reductive group. We introduce the moduli problem of bundle chains parametrizing framed principal $G$-bundles on chains of lines. Any fan supported in a Weyl chamber determines a stability condition on bundle chains. Its moduli stack provides an equivariant toroidal compactification of $G$. All toric orbifolds may be thus obtained. Moreover, we get a canonical compactification of any semisimple $G$, which agrees with the wonderful compactification in the adjoint case, but which in other cases is an orbifold. Finally, we describe the connections with Losev–Manin’s spaces of weighted pointed curves and with Kausz’s compactification of $GL_{n}$.

scholarly journals The ℓ-modular representation of reductive groups over finite local rings of length two

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Nariel Monteiro
Keyword(s):  
Residue Field ◽  
Reductive Group ◽  
Group Scheme ◽  
Modular Representation ◽  
Group Algebras ◽  
Large Field ◽  
Local Rings ◽  
Reductive Groups ◽  
Good For ◽  
Prime 2

Abstract Let O 2 \mathcal{O}_{2} and O 2 ′ \mathcal{O}^{\prime}_{2} be two distinct finite local rings of length two with residue field of characteristic 𝑝. Let G ⁢ ( O 2 ) \mathbb{G}(\mathcal{O}_{2}) and G ⁢ ( O 2 ′ ) \mathbb{G}(\mathcal{O}^{\prime}_{2}) be the groups of points of any reductive group scheme 𝔾 over ℤ such that 𝑝 is very good for G × F q \mathbb{G}\times\mathbb{F}_{q} or G = GL n \mathbb{G}=\operatorname{GL}_{n} . We prove that there exists an isomorphism of group algebras K ⁢ G ⁢ ( O 2 ) ≅ K ⁢ G ⁢ ( O 2 ′ ) K\mathbb{G}(\mathcal{O}_{2})\cong K\mathbb{G}(\mathcal{O}^{\prime}_{2}) , where 𝐾 is a sufficiently large field of characteristic different from 𝑝.

scholarly journals LIMIT MULTIPLICITIES FOR PRINCIPAL CONGRUENCE SUBGROUPS OF AND

2014 ◽  
Vol 14 (3) ◽  
pp. 589-638 ◽  
Author(s):  
Tobias Finis ◽  
Erez Lapid ◽  
Werner Müller
Keyword(s):  
Substantial Reduction ◽  
Reductive Group ◽  
Main Tool ◽  
Principal Congruence ◽  
Plancherel Measure ◽  
Reductive Groups ◽  
Congruence Subgroups ◽  
Limiting Behavior ◽  
Discrete Spectra ◽  
Recent Refinement

We study the limiting behavior of the discrete spectra associated to the principal congruence subgroups of a reductive group over a number field. While this problem is well understood in the cocompact case (i.e., when the group is anisotropic modulo the center), we treat groups of unbounded rank. For the groups $\text{GL}(n)$ and $\text{SL}(n)$ we show that the suitably normalized spectra converge to the Plancherel measure (the limit multiplicity property). For general reductive groups we obtain a substantial reduction of the problem. Our main tool is the recent refinement of the spectral side of Arthur’s trace formula obtained in [Finis, Lapid, and Müller, Ann. of Math. (2) 174(1) (2011), 173–195; Finis and Lapid, Ann. of Math. (2) 174(1) (2011), 197–223], which allows us to show that for $\text{GL}(n)$ and $\text{SL}(n)$ the contribution of the continuous spectrum is negligible in the limit.

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