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Constant term in Harish-Chandra’s limit formula

Mladen Božičević (2008)

Annales mathématiques Blaise Pascal

Let G ℝ be a real form of a complex semisimple Lie group G . Recall that Rossmann defined a Weyl group action on Lagrangian cycles supported on the conormal bundle of the flag variety of G . We compute the signed average of the Weyl group action on the characteristic cycle of the standard sheaf associated to an open G ℝ -orbit on the flag variety. This result is applied to find the value of the constant term in Harish-Chandra’s limit formula for the delta function at zero.

Contact and conformal maps on Iwasawa N groups

Michael Cowling, Filippo De Mari, Adam Korányi, Hans Martin Reimann (2002)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

The action of the conformal group O ⁢ 1 , n + 1 on R n ∪ ∞ may be characterized in differential geometric terms, even locally: a theorem of Liouville states that a C 4 map between domains U and V in R n whose differential is a (variable) multiple of a (variable) isometry at each point of U is the restriction to U of a transformation x → g ⋅ x , for some g in O ⁢ 1 , n + 1 . In this paper, we consider the problem of characterizing the action of a more general semisimple Lie group G on the space G / P , where P is a parabolic subgroup. We solve...

Continued fractions on the Heisenberg group

Anton Lukyanenko, Joseph Vandehey (2015)

Acta Arithmetica

We provide a generalization of continued fractions to the Heisenberg group. We prove an explicit estimate on the rate of convergence of the infinite continued fraction and several surprising analogs of classical formulas about continued fractions.

Continuous measures on compact Lie groups

M. Anoussis, A. Bisbas (2000)

Annales de l'institut Fourier

We study continuous measures on a compact semisimple Lie group G using representation theory. In Section 2 we prove a Wiener type characterization of a continuous measure. Next we construct central measures on G which are related to the well known Riesz products on locally compact abelian groups. Using these measures we show in Section 3 that if C is a compact set of continuous measures on G there exists a singular measure ν such that ν * μ is absolutely continuous with respect to the Haar measure on...

Continuous transformation groups on spaces

K. Spallek (1991)

Annales Polonici Mathematici

A differentiable group is a group in the category of (reduced and nonreduced) differentiable spaces. Special cases are the rationals ℚ, Lie groups, formal groups over ℝ or ℂ; in general there is some mixture of those types, the general structure, however, is not yet completely determined. The following gives as a corollary a first essential answer. It is shown, more generally,that a locally compact topological transformation group, operating effectively on a differentiable space X (which satisfies...

Control systems on semi-simple Lie groups and their homogeneous spaces

Velimir Jurdjevic, Ivan Kupka (1981)

Annales de l'institut Fourier

In the present paper, we consider the class of control systems which are induced by the action of a semi-simple Lie group on a manifold, and we give a sufficient condition which insures that such a system can be steered from any initial state to any final state by an admissible control. The class of systems considered contains, in particular, essentially all the bilinear systems. Our condition is semi-algebraic but unlike the celebrated Kalman criterion for linear systems, it is not necessary. In...

Control Systems on the Orthogonal Group SO(4)

Ross M. Adams, Rory Biggs, Claudiu C. Remsing (2013)

Communications in Mathematics

We classify the left-invariant control affine systems evolving on the orthogonal group S O ( 4 ) . The equivalence relation under consideration is detached feedback equivalence. Each possible number of inputs is considered; both the homogeneous and inhomogeneous systems are covered. A complete list of class representatives is identified and controllability of each representative system is determined.

Controllability of invariant control systems at uniform time

Víctor Ayala, José Ayala-Hoffmann, Ivan de Azevedo Tribuzy (2009)

Kybernetika

Let G be a compact and connected semisimple Lie group and Σ an invariant control systems on G . Our aim in this work is to give a new proof of Theorem 1 proved by Jurdjevic and Sussmann in [6]. Precisely, to find a positive time s Σ such that the system turns out controllable at uniform time s Σ . Our proof is different, elementary and the main argument comes directly from the definition of semisimple Lie group. The uniform time is not arbitrary. Finally, if A = ⋂ t > 0 A ( t , e ) denotes the reachable set from arbitrary...

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