scholarly journals Quotients by actions of the derived group of a maximal unipotent subgroup

2012 ◽  
Vol 256 (2) ◽  
pp. 381-405
Author(s):  
Dmitri Panyushev
Keyword(s):  
Unipotent Subgroup

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.

scholarly journals On orbits of unipotent flows on homogeneous spaces, II

1986 ◽  
Vol 6 (2) ◽  
pp. 167-182 ◽  
Author(s):  
S. G. Dani
Keyword(s):  
Invariant Subspace ◽  
Lebesgue Measure ◽  
Compact Subset ◽  
Lie Group ◽  
Diophantine Approximation ◽  
Closed Subgroup ◽  
Homogeneous Spaces ◽  
Semisimple Lie Group ◽  
Unipotent Subgroup ◽  
Unipotent Flows

AbstractWe show that if (ut) is a one-parameter subgroup of SL (n, ℝ) consisting of unipotent matrices, then for any ε > 0 there exists a compact subset K of SL(n, ℝ)/SL(n, ℤ) such that the following holds: for any g ∈ SL(n, ℝ) either m({t ∈ [0, T] | utg SL (n, ℤ) ∈ K}) > (1 – ε)T for all large T (m being the Lebesgue measure) or there exists a non-trivial (g−1utg)-invariant subspace defined by rational equations.Similar results are deduced for orbits of unipotent flows on other homogeneous spaces. We also conclude that if G is a connected semisimple Lie group and Γ is a lattice in G then there exists a compact subset D of G such that for any closed connected unipotent subgroup U, which is not contained in any proper closed subgroup of G, we have G = DΓ U. The decomposition is applied to get results on Diophantine approximation.

scholarly journals On overgroups of the unipotent subgroup of the Сhevalley group of rank 2 over a field

2015 ◽  
Author(s):  
Т.А. Осетрова ◽  
Я.Н. Нужин
Keyword(s):  
Unipotent Subgroup ◽  
Rank 2

Описаны подгруппы группы Шевалле ранга 2 над полем, содержащие ее унипотентную подгруппу.

Crystals and p-adic Integration

2011 ◽  
Author(s):  
Ben Brubaker ◽  
Daniel Bump ◽  
Solomon Friedberg
Keyword(s):  
Partition Function ◽  
Generating Function ◽  
Whittaker Function ◽  
Character Formula ◽  
Whittaker Functions ◽  
Unipotent Subgroup ◽  
Enveloping Algebra ◽  
Quantized Enveloping Algebra ◽  
Kostant Partition Function ◽  
Weyl Character Formula

This chapter describes the properties of Kashiwara's crystal and its role in unipotent p-adic integrations related to Whittaker functions. In many cases, integrations of representation theoretic import over the maximal unipotent subgroup of a p-adic group can be replaced by a sum over Kashiwara's crystal. Partly motivated by the crystal description presented in Chapter 2 of this book, this perspective was advocated by Bump and Nakasuji. Later work by McNamara and Kim and Lee extended this philosophy yet further. Indeed, McNamara shows that the computation of the metaplectic Whittaker function is initially given as a sum over Kashiwara's crystal. The chapter considers Kostant's generating function, the character of the quantized enveloping algebra, and its association with Kashiwara's crystal, along with the Kostant partition function and the Weyl character formula.

scholarly journals The affine part of the Picard scheme

2009 ◽  
Vol 145 (2) ◽  
pp. 415-422 ◽  
Author(s):  
Thomas Geisser
Keyword(s):  
Maximal Torus ◽  
Perfect Field ◽  
Unipotent Subgroup ◽  
Picard Variety ◽  
Picard Scheme ◽  
Affine Part

AbstractWe describe the maximal torus and maximal unipotent subgroup of the Picard variety of a proper scheme over a perfect field.

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