A Class of Supercuspidal Representations of G2(k)

1999 ◽  
Vol 42 (3) ◽  
pp. 393-400 ◽  
Author(s):  
Gordan Savin
Keyword(s):  
Division Algebra ◽  
Dual Pair ◽  
Adjoint Group ◽  
Minimal Representation ◽  
Supercuspidal Representations ◽  
Interesting Class ◽  
Rank 2 ◽  
Split Rank

AbstractLet H be an exceptional, adjoint group of type E6 and split rank 2, over a p-adic field k. In this article we discuss the restriction of the minimal representation of H to a dual pair PD× × G2(k), where D is a division algebra of dimension 9 over k. In particular, we discover an interesting class of supercuspidal representations of G2(k).

The Dual Pair PGL3 × G2

1997 ◽  
Vol 40 (3) ◽  
pp. 376-384
Author(s):  
Benedict H. Gross ◽  
Gordan Savin
Keyword(s):  
Closed Subgroup ◽  
Dual Pair ◽  
Adjoint Group ◽  
Minimal Representation

AbstractLet H be the split, adjoint group of type E6 over a p-adic field. In this paper we study the restriction of the minimal representation of H to the closed subgroup PGL3 × G2.

Characters of Depth-Zero, Supercuspidal Representations of the Rank-2 Symplectic Group

2000 ◽  
Vol 52 (2) ◽  
pp. 306-331 ◽  
Author(s):  
Clifton Cunningham
Keyword(s):  
Lie Algebras ◽  
Symplectic Group ◽  
Orbital Integrals ◽  
Finite Linear Combination ◽  
Supercuspidal Representations ◽  
Rank 2 ◽  
The Fourier Transform ◽  
The Stability ◽  
Springer Correspondence ◽  
Finite Linear

AbstractThis paper expresses the character of certain depth-zero supercuspidal representations of the rank-2 symplectic group as the Fourier transform of a finite linear combination of regular elliptic orbital integrals—an expression which is ideally suited for the study of the stability of those characters. Building on work of F. Murnaghan, our proof involves Lusztig’s Generalised Springer Correspondence in a fundamental way, and also makes use of some results on elliptic orbital integrals proved elsewhere by the author using Moy-Prasad filtrations of p-adic Lie algebras. Two applications of the main result are considered toward the end of the paper.

scholarly journals Some matrix groups over finite-dimensional division algebras

1990 ◽  
Vol 33 (1) ◽  
pp. 97-111 ◽  
Author(s):  
B. A. F. Wehrfritz
Keyword(s):  
Division Algebra ◽  
Finite Dimension ◽  
Linear Groups ◽  
Matrix Groups ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Skew Fields ◽  
Rank 2 ◽  
Soluble Subgroup ◽  
Free Subgroups

Let n be a positive integer and D a division algebra of finite dimension m over its centre. We describe in detail the structure of a soluble subgroup G of GL(n,D). (More generally we consider subgroups of GL{n,D) with no free subgroup of rank 2.) Of course G is isomorphic to a linear group of degree mn and hence linear theory describes G, but the object here is to reduce as far as possible the dependence of the description on m. The results are particularly sharp if n=l. They will be used in later papers to study matrix groups over certain types of infinite-dimensional division algebra. This present paper was very much inspired by A. I. Lichtman's work: Free subgroups in linear groups over some skew fields, J. Algebra105 (1987), 1–28.

The Dual Pair G2 × PU3 (D) (p-Adic Case)

1999 ◽  
Vol 51 (1) ◽  
pp. 130-146 ◽  
Author(s):  
Gordan Savin ◽  
Wee Teck Gan
Keyword(s):  
Linear Group ◽  
Main Tool ◽  
Dual Pair ◽  
Parabolic Subgroups ◽  
Minimal Representation ◽  
Relative Rank ◽  
Adic Case ◽  
Maximal Parabolic Subgroups

AbstractWe study the correspondence of representations arising by restricting the minimal representation of the linear group of type E7 and relative rank 4. The main tool is computations of the Jacquet modules of the minimal representation with respect to maximal parabolic subgroups of G2 and PU3(D).

Indices for the Exceptional Bruhat-Tits Buildings

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Division Algebra ◽  
Capital Letter ◽  
Coxeter Diagram ◽  
Root Group ◽  
Relative Type ◽  
Quaternion Division Algebra

This chapter considers the affine Tits indices for exceptional Bruhat-Tits buildings. It begins with a few small observations and some notations dealing with the relative type of the affine Tits indices, the canonical correspondence between the circles in a Tits index and the vertices of its relative Coxeter diagram, and Moufang sets. It then presents a proposition about an involutory set, a quaternion division algebra, a root group sequence, and standard involution. It also describes Θ‎-orbits in S which are disjoint from A and which correspond to the vertices of the Coxeter diagram of Ξ‎ and hence to the types of the panels of Ξ‎. Finally, it shows how it is possible in many cases to determine properties of the Moufang set and the Tits index for all exceptional Bruhat-Tits buildings of type other than Latin Capital Letter G with Tilde₂.

Existence

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Division Algebra ◽  
Quadratic Forms ◽  
Structure Theorem ◽  
Quadratic Extension ◽  
Quadratic Space ◽  
Division Algebras ◽  
Separable Extension ◽  
Valued Fields ◽  
Quaternion Division Algebra

This chapter proves that Bruhat-Tits buildings exist. It begins with a few definitions and simple observations about quadratic forms, including a 1-fold Pfister form, followed by a discussion of the existence part of the Structure Theorem for complete discretely valued fields due to H. Hasse and F. K. Schmidt. It then considers the generic unramified cases; the generic semi-ramified cases, the generic ramified cases, the wild unramified cases, the wild semi-ramified cases, and the wild ramified cases. These cases range from a unique unramified quadratic space to an unramified separable quadratic extension, a tamely ramified division algebra, a ramified separable quadratic extension, and a unique unramified quaternion division algebra. The chapter also describes ramified quaternion division algebras D₁, D₂, and D₃ over K containing a common subfield E such that E/K is a ramified separable extension.

Totally Wild Quadratic Forms of Type E7

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Division Algebra ◽  
Quadratic Forms ◽  
Quadratic Space ◽  
Trace Map ◽  
Quaternion Division Algebra

This chapter assumes that (K, L, q) is a totally wild quadratic space of type E₇. The goal is to prove the proposition that takes into account Λ‎ of type E₇, D as the quaternion division algebra over K whose image in Br(K) is the Clifford invariant of q, and the trace and trace map. The chapter also considers two other propositions: the first states that if the trace map is not equal to zero, then the Moufang residues R₀ and R₁ are not indifferent; the second states that if the trace map is equal to zero, then the Moufang residues R₀ and R₁ are both indifferent.

Linked Tori, II

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Division Algebra ◽  
Quadratic Forms ◽  
Integral Domain ◽  
Quadratic Space ◽  
Small Observation ◽  
Quaternion Division Algebra

This chapter proves several more results about weak isomorphisms between Moufang sets arising from quadratic forms and involutory sets. It first fixes a non-trivial anisotropic quadratic space Λ‎ = (K, L, q) before considering two proper anisotropic pseudo-quadratic spaces. It then describes a quaternion division algebra and its standard involution, a second quaternion division algebra and its standard involution, and an involutory set with a quaternion division algebra and its standard involution. It concludes with one more small observation regarding a pointed anisotropic quadratic space and shows that there is a unique multiplication on L that turns L into an integral domain with a multiplicative identity.

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