scholarly journals ENDOSCOPY FOR HECKE CATEGORIES, CHARACTER SHEAVES AND REPRESENTATIONS

2020 ◽  
Vol 8 ◽  
Author(s):  
GEORGE LUSZTIG ◽  
ZHIWEI YUN
Keyword(s):  
Finite Field ◽  
Reductive Group ◽  
The Other ◽  
Torus Action ◽  
Similar Relationship ◽  
Unipotent Representations ◽  
Character Sheaves ◽  
Trivial Monodromy ◽  
Unipotent Character ◽  
Endoscopic Group

For a reductive group $G$ over a finite field, we show that the neutral block of its mixed Hecke category with a fixed monodromy under the torus action is monoidally equivalent to the mixed Hecke category of the corresponding endoscopic group $H$ with trivial monodromy. We also extend this equivalence to all blocks. We give two applications. One is a relationship between character sheaves on $G$ with a fixed semisimple parameter and unipotent character sheaves on the endoscopic group $H$ , after passing to asymptotic versions. The other is a similar relationship between representations of $G(\mathbb{F}_{q})$ with a fixed semisimple parameter and unipotent representations of $H(\mathbb{F}_{q})$ .

scholarly journals Le Groupe GLn Tordu, Sur un Corps Fini

2006 ◽  
Vol 182 ◽  
pp. 313-379 ◽  
Author(s):  
J.-L. Waldspurger
Keyword(s):  
Irreducible Representation ◽  
Finite Field ◽  
Complex Number ◽  
Outer Automorphism ◽  
The Other ◽  
Characteristic Functions ◽  
Connected Component ◽  
Connected Group ◽  
Unipotent Representations ◽  
Character Sheaves

AbstractLet q be a finite field, G = GLn(q), θ be the outer automorphism of G, suitably normalized. Consider the non-connected group G ⋊ {1, θ} and its connected component = Gθ. We know two ways to produce functions on , with complex values and invariant by conjugation by G: on one hand, let π be an irreducible representation of G we can and do extend to a representation π+ of G ⋊ {1, θ}, then the restriction trace to of the character of π+ is such a function; on the other hand, Lusztig define character-sheaves a, whose characteristic functions ϕ(a) are such functions too. We consider only “quadratic-unipotent” representations. For all such representation π, we define a suitable extension π+, a character-sheave f(π) and we prove an identity trace = γ(π)ϕ(f(π)) with an explicit complex number γ(π).

Effects of load and duration of tension on pain induced by muscular contraction

1962 ◽  
Vol 203 (4) ◽  
pp. 735-738 ◽  
Author(s):  
Syuk Ryun Park ◽  
Simon Rodbard
Keyword(s):  
Severe Pain ◽  
Total Duration ◽  
Muscular Contraction ◽  
The Other ◽  
Cube Root ◽  
Similar Relationship ◽  
Moderate Pain ◽  
Slight Pain ◽  
Mercury Column ◽  
Local Accumulation

Pain was induced in the ischemic forearm in more than 300 tests in seven subjects by an exercise which consisted of compression of an air bulb to raise or maintain a mercury column at 50, 100, or 200 mm for periods of 1, 2, 4, or 5.5 sec. The rate of pain development could be correlated with the product ( P) of the number of contractions, square root of the load (in mm Hg), and cube root of the duration of contraction (in sec). Slight pain appeared at a product of about 345 P, moderate pain at approximately 433 P, severe pain at 536 P, and intolerable pain at 626 P. A similar relationship could be shown in maintained contraction provided the total duration was treated as if each 5.5 sec of the maintained contraction constituted a separate contraction. Ischemia of the arm for periods up to 15 min had no effect on product. Recovery from the effect of exercise was complete in 10 min. Simultaneous exercise of the other arm had no effect on the rate of pain development. The results support the concept that muscle pain results from the local accumulation of a slowly diffusible material released during muscle contraction.

scholarly journals Shintani descent for algebraic groups and almost characters of unipotent groups

2016 ◽  
Vol 152 (8) ◽  
pp. 1697-1724 ◽  
Author(s):  
Tanmay Deshpande
Keyword(s):  
Finite Field ◽  
Algebraic Group ◽  
Algebraic Groups ◽  
Unipotent Group ◽  
Connected Component ◽  
Unipotent Groups ◽  
Character Sheaves ◽  
Treat All ◽  
Trace Of Frobenius

In this paper, we extend the notion of Shintani descent to general (possibly disconnected) algebraic groups defined over a finite field $\mathbb{F}_{q}$. For this, it is essential to treat all the pure inner $\mathbb{F}_{q}$-rational forms of the algebraic group at the same time. We prove that the notion of almost characters (introduced by Shoji using Shintani descent) is well defined for any neutrally unipotent algebraic group, i.e. an algebraic group whose neutral connected component is a unipotent group. We also prove that these almost characters coincide with the ‘trace of Frobenius’ functions associated with Frobenius-stable character sheaves on neutrally unipotent groups. In the course of the proof, we also prove that the modular categories that arise from Boyarchenko and Drinfeld’s theory of character sheaves on neutrally unipotent groups are in fact positive integral, confirming a conjecture due to Drinfeld.

scholarly journals Character sheaves and generalized Springer correspondence

2003 ◽  
Vol 170 ◽  
pp. 47-72 ◽  
Author(s):  
Anne-Marie Aubert
Keyword(s):  
Finite Field ◽  
Algebraic Group ◽  
Algebraic Closure ◽  
Reductive Algebraic Group ◽  
Good For ◽  
Character Sheaves ◽  
Springer Correspondence

AbstractLetGbe a connected reductive algebraic group over an algebraic closure of a finite field of characteristicp. Under the assumption thatpis good forG, we prove that for each character sheafAonGwhich has nonzero restriction to the unipotent variety ofG, there exists a unipotent classCAcanonically attached toA, such thatAhas non-zero restriction onCA, and any unipotent classCinGon whichAhas non-zero restriction has dimension strictly smaller than that ofCA.

scholarly journals Genus-2 curves and Jacobians with a given number of points

2015 ◽  
Vol 18 (1) ◽  
pp. 170-197 ◽  
Author(s):  
Reinier Bröker ◽  
Everett W. Howe ◽  
Kristin E. Lauter ◽  
Peter Stevenhagen
Keyword(s):  
Finite Field ◽  
Elliptic Curves ◽  
Polynomial Time ◽  
Rational Points ◽  
The Other ◽  
Base Field ◽  
Arbitrary Base ◽  
The Family ◽  
Genus 2 ◽  
Genus 2 Curves

AbstractWe study the problem of efficiently constructing a curve $C$ of genus $2$ over a finite field $\mathbb{F}$ for which either the curve $C$ itself or its Jacobian has a prescribed number $N$ of $\mathbb{F}$-rational points.In the case of the Jacobian, we show that any ‘CM-construction’ to produce the required genus-$2$ curves necessarily takes time exponential in the size of its input.On the other hand, we provide an algorithm for producing a genus-$2$ curve with a given number of points that, heuristically, takes polynomial time for most input values. We illustrate the practical applicability of this algorithm by constructing a genus-$2$ curve having exactly $10^{2014}+9703$ (prime) points, and two genus-$2$ curves each having exactly $10^{2013}$ points.In an appendix we provide a complete parametrization, over an arbitrary base field $k$ of characteristic neither two nor three, of the family of genus-$2$ curves over $k$ that have $k$-rational degree-$3$ maps to elliptic curves, including formulas for the genus-$2$ curves, the associated elliptic curves, and the degree-$3$ maps.Supplementary materials are available with this article.

Irreducible Automorphisms of Certainp-Groups

1977 ◽  
Vol 29 (2) ◽  
pp. 333-348 ◽  
Author(s):  
D. Ž. Djoković ◽  
J. Malzan
Keyword(s):  
Finite Field ◽  
Maximal Subgroup ◽  
Finite Groups ◽  
General Problem ◽  
The Other ◽  
Frattini Subgroup ◽  
Other Hand

The chief purpose of this paper is to find all pairs (G, θ) whereGis a finite specialp-group, andθis an automorphism ofGacting trivially on the Frattini subgroup and irreducibly on the Frattini quotient. This problem arises in the context of describing finite groups having an abelian maximal subgroup. In fact, we solve a more general problem for a wider class ofp-groups, which we callspecial F-groups,whereFis a finite field of characteristicp.We point out that ifpis odd, then anF-group has exponentp.On the other hand, every special 2-group is also a specialGF(2)-group.

scholarly journals Explicit isogenies in quadratic time in any characteristic

2016 ◽  
Vol 19 (A) ◽  
pp. 267-282 ◽  
Author(s):  
Luca De Feo ◽  
Cyril Hugounenq ◽  
Jérôme Plût ◽  
Éric Schost
Keyword(s):  
Finite Field ◽  
Elliptic Curves ◽  
Large Class ◽  
The Other ◽  
Base Field ◽  
Other Hand ◽  
Previous Algorithm ◽  
The Cost ◽  
Interesting Alternative ◽  
Refined Version

Consider two ordinary elliptic curves$E,E^{\prime }$defined over a finite field$\mathbb{F}_{q}$, and suppose that there exists an isogeny$\unicode[STIX]{x1D713}$between$E$and$E^{\prime }$. We propose an algorithm that determines$\unicode[STIX]{x1D713}$from the knowledge of$E$,$E^{\prime }$and of its degree$r$, by using the structure of the$\ell$-torsion of the curves (where $\ell$ is a prime different from the characteristic $p$of the base field). Our approach is inspired by a previous algorithm due to Couveignes, which involved computations using the$p$-torsion on the curves. The most refined version of that algorithm, due to De Feo, has a complexity of $\tilde{O} (r^{2})p^{O(1)}$base field operations. On the other hand, the cost of our algorithm is$\tilde{O} (r^{2})\log (q)^{O(1)}$, for a large class of inputs; this makes it an interesting alternative for the medium- and large-characteristic cases.

scholarly journals Duality for representations of a reductive group over a finite field

1982 ◽  
Vol 74 (1) ◽  
pp. 284-291 ◽  
Author(s):  
Pierre Deligne ◽  
George Lusztig
Keyword(s):  
Finite Field ◽  
Reductive Group

MEANING AND STRUCTURE OF HOSEA X 1-8

2003 ◽  
Vol 53 (3) ◽  
pp. 367-378
Author(s):  
Cornelis Van Leeuwen
Keyword(s):  
The Other ◽  
Similar Relationship ◽  
The Real ◽  
Symmetry Scheme ◽  
Key Word ◽  
Preceding Page

AbstractHos. x 1-8 is generally seen as an accumulation of dierent prophecies composed without any coherence. A few scholars have pointed to the inclusio constituted by the key-word mizbehôt in vv. 1-2 and 8. The other verses show, however, a similar relationship. So Israel’s prosperity (v. 1a) corresponds with the despair in v. 8b, and the indictment of the kings (vv. 3-4) with their condemnation (v. 7). The calf statue is central: its derision (v. 5) and its being carried away (v. 6). So our pericope offers an example of concentric symmetry (scheme on the preceding page). This construction can help us to solve exegetical questions. So the king of v. 7 cannot be an image of the calf (vv. 5-6): in the light of the corresponding vv. 3-4, the king of v. 7 can only be the real king of Israel.

scholarly journals Automorphic vector bundles on the stack of G-zips

2021 ◽  
Vol 9 ◽  
Author(s):  
Naoki Imai ◽  
Jean-Stefan Koskivirta
Keyword(s):  
Vector Bundle ◽  
Finite Field ◽  
Vector Bundles ◽  
Reductive Group ◽  
Levi Subgroup ◽  
Algebraic Subgroup ◽  
Equivalence Of Categories

Abstract For a connected reductive group G over a finite field, we study automorphic vector bundles on the stack of G-zips. In particular, we give a formula in the general case for the space of global sections of an automorphic vector bundle in terms of the Brylinski-Kostant filtration. Moreover, we give an equivalence of categories between the category of automorphic vector bundles on the stack of G-zips and a category of admissible modules with actions of a 0-dimensional algebraic subgroup a Levi subgroup and monodromy operators.

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