scholarly journals Gross–Prasad periods for reducible representations

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
David Loeffler
Keyword(s):  
Local Field ◽  
Irreducible Representations ◽  
Total Degree ◽  
Field Extensions

Abstract We study GL 2 ⁡ ( F ) {\operatorname{GL}_{2}(F)} -invariant periods on representations of GL 2 ⁡ ( A ) {\operatorname{GL}_{2}(A)} , where F is a non-archimedean local field and A / F {A/F} a product of field extensions of total degree 3. For irreducible representations, a theorem of Prasad shows that the space of such periods has dimension ⩽ 1 {\leqslant 1} , and is non-zero when a certain ε-factor condition holds. We give an extension of this result to a certain class of reducible representations (of Whittaker type), extending results of Harris–Scholl when A is the split algebra F × F × F {F\times F\times F} .

Ramification Groups of Abelian Local Field Extensions

1971 ◽  
Vol 23 (2) ◽  
pp. 271-281 ◽  
Author(s):  
Murray A. Marshall
Keyword(s):  
Prime Ideal ◽  
Local Field ◽  
Galois Extension ◽  
Residue Class ◽  
Abelian Extensions ◽  
Ring Of Integers ◽  
Field Extensions ◽  
Valued Field ◽  
Residue Class Field ◽  
Image Position

Let k be a local field; that is, a complete discrete-valued field having a perfect residue class field. If L is a finite Galois extension of k then L is also a local field. Let G denote the Galois group GL|k. Then the nth ramification group Gn is defined bywhere OL, denotes the ring of integers of L, and PL is the prime ideal of OL. The ramification groups form a descending chain of invariant subgroups of G:1In this paper, an attempt is made to characterize (in terms of the arithmetic of k) the ramification filters (1) obtained from abelian extensions L\k.

Computing embedding numbers and branches of orders via extensions of the Bruhat-Tits tree

2019 ◽  
Vol 15 (10) ◽  
pp. 2067-2088
Author(s):  
Luis Arenas-Carmona ◽  
Claudio Bravo
Keyword(s):  
Local Field ◽  
Matrix Algebra ◽  
Explicit Representation ◽  
Past Work ◽  
Field Extensions ◽  
The Past ◽  
Commutative Subalgebra ◽  
Maximal Orders

Let [Formula: see text] be a local field and let [Formula: see text] be the two-by-two matrix algebra over [Formula: see text]. In our previous work, we developed a theory that allows the computation of the set of maximal orders in [Formula: see text] containing a given suborder. This set is given as a subgraph of the Bruhat–Tits (BT)-tree that is called the branch of the order. Branches have been used to study the global selectivity problem and also to compute local embedding numbers. They can usually be described in terms of two invariants. To compute these invariants explicitly, the strategy in our past work has been visualizing branches through the explicit representation of the BT-tree in terms of balls in [Formula: see text]. This is easier for orders spanning a split commutative subalgebra, i.e. an algebra isomorphic to [Formula: see text]. In fact, we have successfully used this idea in the past to compute embedding numbers for the split algebra. In the present work, we develop a theory of branches over field extensions that can be used to extend our previous computations to orders spanning a field. We use the same idea to compute branches for orders generated by arbitrary pairs of non-nilpotent pure quaternions, generalizing previous results due to the first author and Saavedra. We assume throughout that [Formula: see text].

scholarly journals Conjectures and results about parabolic induction of representations of $${\text {GL}}_n(F)$$

2020 ◽  
Vol 222 (3) ◽  
pp. 695-747
Author(s):  
Erez Lapid ◽  
Alberto Mínguez
Keyword(s):  
Linear Group ◽  
Local Field ◽  
General Linear Group ◽  
Irreducible Representations ◽  
General Linear ◽  
Parabolic Induction ◽  
Geometric Conditions ◽  
Irreducible Components ◽  
Special Cases

Abstract In 1980 Zelevinsky introduced certain commuting varieties whose irreducible components classify complex, irreducible representations of the general linear group over a non-archimedean local field with a given supercuspidal support. We formulate geometric conditions for certain triples of such components and conjecture that these conditions are related to irreducibility of parabolic induction. The conditions are in the spirit of the Geiss–Leclerc–Schröer condition that occurs in the conjectural characterization of $$\square $$ □ -irreducible representations. We verify some special cases of the new conjecture and check that the geometric and representation-theoretic conditions are compatible in various ways.

scholarly journals k-Isomorphism classes of local field extensions

2015 ◽  
Vol 153 ◽  
pp. 97-106
Author(s):  
Duc Van Huynh ◽  
Kevin Keating
Keyword(s):  
Local Field ◽  
Isomorphism Classes ◽  
Field Extensions

scholarly journals Multiplicity Free Jacquet Modules

2012 ◽  
Vol 55 (4) ◽  
pp. 673-688 ◽  
Author(s):  
Avraham Aizenbud ◽  
Dmitry Gourevitch
Keyword(s):  
Finite Field ◽  
Natural Number ◽  
Parabolic Subgroup ◽  
Local Field ◽  
Classical Method ◽  
Irreducible Representations ◽  
Levi Subgroup ◽  
Unipotent Subgroup ◽  
Multiplicity Free

AbstractLet F be a non-Archimedean local field or a finite field. Let n be a natural number and k be 1 or 2. Consider G := GLn+k(F) and let M := GLn(F) × GLk(F) < G be a maximal Levi subgroup. Let U < G be the corresponding unipotent subgroup and let P = MU be the corresponding parabolic subgroup. Let be the Jacquet functor, i.e., the functor of coinvariants with respect toU. In this paper we prove that J is a multiplicity free functor, i.e., dim HomM(J(π), ρ) ≤ 1, for any irreducible representations π of G and ρ of M. We adapt the classical method of Gelfand and Kazhdan, which proves the “multiplicity free” property of certain representations to prove the “multiplicity free” property of certain functors. At the end we discuss whether other Jacquet functors are multiplicity free.

Genericity of Representations of p-Adic Sp2n and Local Langlands Parameters

2011 ◽  
Vol 63 (5) ◽  
pp. 1107-1136 ◽  
Author(s):  
Baiying Liu
Keyword(s):  
Local Field ◽  
Symplectic Group ◽  
Rational Points ◽  
Descent Method ◽  
Irreducible Representations ◽  
Local Factors ◽  
Langlands Parameters ◽  
Repre Sentations

Abstract Let G be the F-rational points of the symplectic group Sp2n, where F is a non-Archimedean local field of characteristic 0. Cogdell, Kim, Piatetski-Shapiro, and Shahidi constructed local Lang- lands functorial lifting from irreducible generic representations of G to irreducible representations of GL2n+1(F). Jiang and Soudry constructed the descent map from irreducible supercuspidal repre- sentations of GL2n+1(F) to those of G, showing that the local Langlands functorial lifting from the irreducible supercuspidal generic representations is surjective. In this paper, based on above results, using the same descent method of studying SO2n+1 as Jiang and Soudry, we will show the rest of local Langlands functorial lifting is also surjective, and for any local Langlands parameter , we construct a representation such that and ¾ have the same twisted local factors. As one application, we prove the G-case of a conjecture of Gross-Prasad and Rallis, that is, a local Langlands parameter is generic, i.e., the representation attached to is generic, if and only if the adjoint L-function of is holomorphic at s = 1. As another application, we prove for each Arthur parameter , and the corresponding local Langlands parameter , the representation attached to is generic if and only if is tempered.

scholarly journals Cuspidal ℓ-modular representations of p-adic classical groups

2020 ◽  
Vol 2020 (764) ◽  
pp. 23-69 ◽  
Author(s):  
Robert Kurinczuk ◽  
Shaun Stevens
Keyword(s):  
Local Field ◽  
Classical Group ◽  
Irreducible Representations ◽  
Closed Field ◽  
Modular Representations ◽  
Algebraically Closed Field ◽  
Fundamental Step ◽  
Residual Characteristic ◽  
Cuspidal Representations

AbstractFor a classical group over a non-archimedean local field of odd residual characteristic p, we construct all cuspidal representations over an arbitrary algebraically closed field of characteristic different from p, as representations induced from a cuspidal type. We also give a fundamental step towards the classification of cuspidal representations, identifying when certain cuspidal types induce to equivalent representations; this result is new even in the case of complex representations. Finally, we prove that the representations induced from more general types are quasi-projective, a crucial tool for extending the results here to arbitrary irreducible representations.

A Functional Equation for Degree two Local Factors

1985 ◽  
Vol 28 (3) ◽  
pp. 355-371
Author(s):  
Paul Gérardin ◽  
Wen-Ch'ing Winnie Li
Keyword(s):  
Functional Equation ◽  
Local Field ◽  
Fourier Transforms ◽  
Discrete Series ◽  
Irreducible Representations ◽  
Quaternion Group ◽  
Langlands Correspondence ◽  
Local Factors

AbstractWe show that the Fourier transforms of the admissible irreducible representations of the group GL2 over a nonarchimedian local field F are characterized by a functional equation (MF). We also prove that the functions satisfying (MF) and having at most one pole are exactly the Fourier transforms of the irreducible representations of the quaternion group H over F. The Jacquet-Langlands correspondence between irreducible representations of H and discrete series of GL2 then follows immediately from our criteria.

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