Forms of Residually Pseudo-Split Buildings

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Fixed Point ◽  
Galois Action ◽  
Coxeter Diagram ◽  
Tits Building

This chapter deals with forms of residually pseudo-split buildings. The proof rests on the fact that in every case, there is a Galois action of Γ‎ := GalL/K on Δ‎L whose fixed point building is isomorphic to Δ‎. A Tits index = (Π‎, Θ‎, A) is displayed by drawing the Coxeter diagram, bending edges where necessary so that vertices in the same Θ‎-orbit are conspicuously near to each other, and putting a circle around the set of vertices in each orbit of Θ‎ disjoint from A. The chapter presents the main result showing that every exceptional Bruhat-Tits building of rank at least 3 but not of type G˜2 with Tilde₂ is the fixed point building of an unramified group of order 2 or 4 acting on a residually pseudo-split building.

scholarly journals Typical representations via fixed point sets in Bruhat–Tits buildings

2021 ◽  
Vol 25 (36) ◽  
pp. 1021-1048
Author(s):  
Peter Latham ◽  
Monica Nevins
Keyword(s):  
Fixed Point ◽  
Maximal Compact Subgroup ◽  
Sufficient Conditions ◽  
Compact Subgroup ◽  
Supercuspidal Representation ◽  
Content Type ◽  
Branching Rules ◽  
Irreducible Components ◽  
Tits Building ◽  
Fixed Point Sets

For a tame supercuspidal representation π \pi of a connected reductive p p -adic group G G , we establish two distinct and complementary sufficient conditions, formulated in terms of the geometry of the Bruhat–Tits building of G G , for the irreducible components of its restriction to a maximal compact subgroup to occur in a representation of G G which is not inertially equivalent to π \pi . The consequence is a set of broadly applicable tools for addressing the branching rules of π \pi and the unicity of [ G , π ] G [G,\pi ]_G -types.

Unramified Galois Involutions

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Fixed Point ◽  
Complete System ◽  
Canonical Isomorphism ◽  
Descent Group ◽  
Tits Building

This chapter describes the fixed point building of an automorphism of a Bruhat-Tits building Ξ‎ which induces an unramified Galois involution on the building at infinity Ξ‎∞. An element of G (for example, a Galois involution of Δ‎) is unramified if the subgroup of G it generates is unramified. Before presenting the main result, the chapter presents the notation stating that Δ‎ = Ξ‎∞ is the building at infinity of Ξ‎ with respect to its complete system of apartments and G = Aut(Δ‎), followed by definitions. The central theorem shows how an unramified Galois involution of Δ‎ is obtained. Here Γ‎ := τ‎ is a descent group of both Δ‎ and Ξ‎, there is a canonical isomorphism from Δ‎Γ‎ to (Ξ‎Γ‎), where Ξ‎Γ‎ and Ξ‎Γ‎ are the fixed point buildings.

Fixed Point Buildings

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Fixed Point ◽  
Fundamental Theorem ◽  
Coxeter System ◽  
Coxeter Diagram ◽  
Vertex Set ◽  
Descent Group

This chapter presents the proof for the Fundamental Theorem of Descent in buildings: that if Γ‎ is a descent group, the set of residues of a building Δ‎ that are stabilized by a subgroup Γ‎ of Aut(Γ‎) forms a thick building. It begins with the hypothesis: Let Π‎ be an arbitrary Coxeter diagram, let S be the vertex set of Π‎ and let (W, S) be the corresponding Coxeter system. It then defines a Γ‎-residue and a Γ‎-chamber as well as a descent group of Δ‎ before concluding with the main result about the fixed point building of Γ‎.

Residually Pseudo-Split Buildings

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Fixed Point ◽  
Division Algebra ◽  
Residue Field ◽  
Quadratic Extension ◽  
Splitting Field ◽  
Quadratic Space ◽  
Descent Group ◽  
Tits Building ◽  
Quaternion Division Algebra

This chapter presents results about a residually pseudo-split Bruhat-Tits building Ξ‎L. It begins with a case for some quadratic space of type E⁶, E₇, and E₈ in order to identify an unramified extension such that the residue field is a pseudo-splitting field. It then considers a wild quaternion or octonion division algebra and the existence of an unramified quadratic extension L/K such that L is a splitting field of the quaternion division algebra. It also discusses the properties of an unramified extension L/K and shows that every exceptional Bruhat-Tits building is the fixed point building of a strictly semi-linear descent group of a residually pseudo-split building.

Affine Fixed Point Buildings

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Fixed Point ◽  
Metric Spaces ◽  
Canonical Isomorphism ◽  
Affine Building ◽  
Coxeter Diagram ◽  
Descent Group

This chapter shows that if Ξ‎ is an affine building and Γ‎ is a finite descent group of Ξ‎, then Γ‎ is a descent group of Ξ‎∞ and (Ξ‎∞) is congruent to (Ξ‎∞). Ξ‎Γ‎ and Ξ‎ can be viewed as metric spaces. The chapter first considers the assumptions that Π‎ is an irreducible affine Coxeter diagram, Ξ‎ is a thick building of type Ξ‎, Γ‎is a finite descent group of Ξ‎, and Tits index �� = (Π‎, Θ‎, A). It then describes apartments that are endowed with reflection hyperplanes and reflection half-spaces before concluding with a theorem about a canonical isomorphism from the fixed point building Ξ‎Γ‎ to (Ξ‎Γ‎).

Mental equilibrium: Identifying fixed-point attractors in the stream of thought

2003 ◽  
Author(s):  
Robin R. Vallacher ◽  
Andrzej Nowak ◽  
Matthew Rockloff
Keyword(s):  
Fixed Point

Bifurcation in a Simple Model of the Cardiovascular System

2000 ◽  
Vol 39 (02) ◽  
pp. 118-121 ◽  
Author(s):  
S. Akselrod ◽  
S. Eyal
Keyword(s):  
Nonlinear Dynamics ◽  
Blood Pressure ◽  
Heart Rate ◽  
Fixed Point ◽  
Cardiovascular System ◽  
Modified Model ◽  
Mayer Waves ◽  
Sustained Oscillations ◽  
Human Cardiovascular System ◽  
Physiological Values

Abstract:A simple nonlinear beat-to-beat model of the human cardiovascular system has been studied. The model, introduced by DeBoer et al. was a simplified linearized version. We present a modified model which allows to investigate the nonlinear dynamics of the cardiovascular system. We found that an increase in the -sympathetic gain, via a Hopf bifurcation, leads to sustained oscillations both in heart rate and blood pressure variables at about 0.1 Hz (Mayer waves). Similar oscillations were observed when increasing the -sympathetic gain or decreasing the vagal gain. Further changes of the gains, even beyond reasonable physiological values, did not reveal another bifurcation. The dynamics observed were thus either fixed point or limit cycle. Introducing respiration into the model showed entrainment between the respiration frequency and the Mayer waves.

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