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.

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).

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.

Residues

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Euclidean Plane ◽  
Complete System ◽  
Quadratic Space ◽  
Affine Building ◽  
Root Group ◽  
Tits Building ◽  
Moufang Quadrangle

This chapter deals with the residues of a Bruhat-Tits building whose building at infinity is an exceptional quadrangle. It begins with the remark that if Λ‎ is an arbitrary quadratic space of type Eℓ for ℓ = 6, 7 or 8 or of typeF₄ over a field K that is complete with respect to a discrete valuation, and if in the F4-case the subfield F is closed with respect to this valuation and if Δ‎ is the corresponding Moufang quadrangle of type Eℓ or F₄, then there always exists a unique affine building Ξ‎ such that Δ‎ is the building at infinity of Ξ‎ with respect to its complete system of apartments. The chapter also considers the standard embedding of the apartment A in the Euclidean plane which takes the intersection of A and R to the set of eight triangles containing the origin. Finally, it describes a Moufang polygon with two root group sequences.

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.

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.

Moufang Structures

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Fixed Point ◽  
Unipotent Group ◽  
Spherical Building ◽  
Unipotent Element ◽  
Moufang Condition ◽  
Descent Group

This chapter uses the notion of a Moufang structure to show that if Δ‎ is a spherical building satisfying the Moufang condition and Γ‎ is a descent group of Δ‎, then the fixed point building Δ‎Γ‎ also satisfies the Moufang condition. The discussion begins with the notation: Let (W, S) denote the type of Δ‎, let G = Aut(Δ‎) and let G° denote the group of type-preserving elements of G. The chapter then presents the conditions for an element g of G to be unipotent and for a subgroup U of G to be unipotent. It also describes a unipotent group U stabilizing a residue R and a unipotent element fixing two chambers x and y. Finally, it considers the set of extensions that forms a group acting faithfully on R.

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
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