scholarly journals The Bernstein center of a p-adic unipotent group

2020 ◽  
Vol 560 ◽  
pp. 521-537
Author(s):  
Justin Campbell
Keyword(s):  
Unipotent Group

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.

A UNIPOTENT GROUP ACTION ON A FLAG MANIFOLD AND "GAP SEQUENCES" OF PERMUTATIONS

2003 ◽  
Vol 02 (02) ◽  
pp. 215-222 ◽  
Author(s):  
BARBARA A. SHIPMAN
Keyword(s):  
Hamiltonian System ◽  
Toda Lattice ◽  
Initial Conditions ◽  
Flag Manifold ◽  
Unipotent Group ◽  
Unipotent Subgroup ◽  
Bruhat Cells ◽  
Generic Orbit ◽  
Gap Sequence ◽  
Gap Sequences

There is a unipotent subgroup of Sl(n, C) whose action on the flag manifold of Sl(n, C) completes the flows of the complex Kostant–Toda lattice (a Hamiltonian system in Lax form) through initial conditions where all the eigenvalues coincide. The action preserves the Bruhat cells, which are in one-to-one correspondence with the elements of the permutation group Σn. A generic orbit in a given cell is homeomorphic to Cm, where m is determined by the "gap sequence" of the permutation, which lists the number inversions of each degree.

Geometric Quotients of Unipotent Group Actions

1993 ◽  
Vol s3-67 (1) ◽  
pp. 75-105 ◽  
Author(s):  
Gert-Martin Greuel ◽  
Gerhard Pfister
Keyword(s):  
Group Actions ◽  
Unipotent Group

scholarly journals Equidistribution of singular measures on nilmanifolds and skew products

2011 ◽  
Vol 31 (6) ◽  
pp. 1785-1817
Author(s):  
FABRIZIO POLO
Keyword(s):  
Haar Measure ◽  
Uniform Density ◽  
Unipotent Group ◽  
Affine Transformations ◽  
Skew Products ◽  
Singular Measures ◽  
Ergodic Properties ◽  
Push Forward ◽  
Multiplicative Ergodic Theorem ◽  
Uniquely Ergodic

AbstractWe prove that for a minimal rotationTon a two-step nilmanifold and any measureμ, the push-forwardTn⋆μofμunderTntends toward Haar measure if and only ifμprojects to Haar measure on the maximal torus factor. For an arbitrary nilmanifold we get the same result along a sequence of uniform density one. These results strengthen Parry’s result [Ergodic properties of affine transformations and flows on nilmanifolds.Amer. J. Math.91(1968), 757–771] that such systems are uniquely ergodic. Extending the work of Furstenberg [Strict ergodicity and transformations of the torus.Amer. J. Math.83(1961), 573–601], we get the same result for a large class of iterated skew products. Additionally we prove a multiplicative ergodic theorem for functions taking values in the upper unipotent group. Finally we characterize limits ofTn⋆μfor some skew product transformations with expansive fibers. All results are presented in terms of twisting and weak twisting, properties that strengthen unique ergodicity in a way analogous to that in which mixing and weak mixing strengthen ergodicity for measure-preserving systems.

scholarly journals Infinitesimal unipotent group schemes of complexity 1

2001 ◽  
Vol 89 (2) ◽  
pp. 179-192 ◽  
Author(s):  
Rolf Farnsteiner ◽  
Gerhard Röhrle ◽  
Detlef Voigt
Keyword(s):  
Unipotent Group ◽  
Group Schemes

scholarly journals Unitary representations of unipotent groups associated with theta series

1992 ◽  
Vol 127 ◽  
pp. 167-174
Author(s):  
Hisasi Morikawa
Keyword(s):  
Heisenberg Group ◽  
Theta Series ◽  
Unitary Representations ◽  
Unipotent Group ◽  
Split Extension ◽  
Unipotent Groups

1. Unipotent group of real (g + 2) × (g + 2) -matricesmay be regarded as a split extension of Ng (R) by Heisenberg group of real (g + 2) × (g + 2)-matrices

ON THE FIXED-POINT SETS OF TORUS ACTIONS ON FLAG MANIFOLDS

2002 ◽  
Vol 01 (03) ◽  
pp. 255-265 ◽  
Author(s):  
BARBARA A. SHIPMAN
Keyword(s):  
Fixed Point ◽  
Toda Lattice ◽  
Flag Manifold ◽  
Weyl Groups ◽  
Flag Manifolds ◽  
Unipotent Group ◽  
Point Sets ◽  
Torus Actions ◽  
Maximal Tori ◽  
Fixed Point Sets

This paper takes a detailed look at a subject that occurs in various contexts in mathematics, the fixed-point sets of torus actions on flag manifolds, and considers it from the (perhaps nontraditional) perspective of moment maps and length functions on Weyl groups. The approach comes from earlier work of the author where it is shown that certain singular flows in the Hamiltonian system known as the Toda lattice generate the action of a group A on a flag manifold, where A is a direct product of a non-maximal torus and unipotent group. As a first step in understanding the orbits of A in connection with the Toda lattice, this paper seeks to understand the fixed points of the non-maximal tori in this setting.

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