Gamma Factors, Root Numbers, and Distinction

AbstractWe study a relation between distinction and special values of local invariants for representations of the general linear group over a quadratic extension of p-adic fields. We show that the local Rankin–Selberg root number of any pair of distinguished representation is trivial, and as a corollary we obtain an analogue for the global root number of any pair of distinguished cuspidal representations. We further study the extent to which the gamma factor at 1/2 is trivial for distinguished representations as well as the converse problem.

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Kirillov models for distinguished representations

Nagoya Mathematical Journal ◽

10.1017/s0027763000001707 ◽

1989 ◽

Vol 116 ◽

pp. 89-110 ◽

Cited By ~ 1

Author(s):

Courtney Moen

Keyword(s):

Linear Group ◽

General Linear Group ◽

Automorphic Forms ◽

Principal Series ◽

Weil Representation ◽

Local Factors ◽

Distinguished Representations ◽

Group A ◽

Local Representations ◽

Covering Groups

In the theory of automorphic forms on covering groups of the general linear group, a central role is played by certain local representations which have unique Whittaker models. A representation with this property is called distinguished. In the case of the 2-sheeted cover of GL2, these representations arise as the the local components of generalizations of the classical θ-function. They have been studied thoroughly in [GPS]. The Weil representation provides these representations with a very nice realization, and the local factors attached to these representations can be computed using this realization. It has been shown [KP] that only in the case of a certain 3-sheeted cover do we find other principal series of covering groups of GL2 which have a unique Whittaker model. It is natural to ask if these distinguished representations also have a realization analgous to the Weil representation.

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Automorphic Representations and L-Functions for the General Linear Group

10.1017/cbo9780511973628 ◽

2009 ◽

Cited By ~ 6

Author(s):

Dorian Goldfeld ◽

Joseph Hundley

Keyword(s):

Linear Group ◽

General Linear Group ◽

General Linear ◽

Automorphic Representations

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Representations induced from the Zelevinsky segment and discrete series in the half-integral case

Forum Mathematicum ◽

10.1515/forum-2020-0054 ◽

2020 ◽

Vol 0 (0) ◽

Author(s):

Ivan Matić

Keyword(s):

Linear Group ◽

Local Field ◽

General Linear Group ◽

Discrete Series ◽

Series Representation ◽

General Linear ◽

Composition Series ◽

Induced Representations ◽

Discrete Series Representation

AbstractLet {G_{n}} denote either the group {\mathrm{SO}(2n+1,F)} or {\mathrm{Sp}(2n,F)} over a non-archimedean local field of characteristic different than two. We study parabolically induced representations of the form {\langle\Delta\rangle\rtimes\sigma}, where {\langle\Delta\rangle} denotes the Zelevinsky segment representation of the general linear group attached to the segment Δ, and σ denotes a discrete series representation of {G_{n}}. We determine the composition series of {\langle\Delta\rangle\rtimes\sigma} in the case when {\Delta=[\nu^{a}\rho,\nu^{b}\rho]} where a is half-integral.

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On the Nielsen root theory of n-valued maps

Journal of Fixed Point Theory and Applications ◽

10.1007/s11784-021-00887-9 ◽

2021 ◽

Vol 23 (4) ◽

Author(s):

Robert F. Brown

Keyword(s):

Fixed Point ◽

Lower Bound ◽

Lie Group ◽

Hausdorff Distance ◽

Root Number ◽

Root Numbers ◽

Fixed Point Result

AbstractLet $$\phi :X \multimap Y$$ ϕ : X ⊸ Y be an n-valued map of connected finite polyhedra and let $$a \in Y$$ a ∈ Y . Then, $$x \in X$$ x ∈ X is a root of $$\phi $$ ϕ at a if $$a \in \phi (x)$$ a ∈ ϕ ( x ) . The Nielsen root number $$N(\phi : a)$$ N ( ϕ : a ) is a lower bound for the number of roots at a of any n-valued map homotopic to $$\phi $$ ϕ . We prove that if X and Y are compact, connected triangulated manifolds without boundary, of the same dimension, then given $$\epsilon > 0$$ ϵ > 0 , there is an n-valued map $$\psi $$ ψ homotopic to $$\phi $$ ϕ within Hausdorff distance $$\epsilon $$ ϵ of $$\phi $$ ϕ such that $$\psi $$ ψ has finitely many roots at a. We conjecture that if X and Y are q-manifolds without boundary, $$q \ne 2$$ q ≠ 2 , then there is an n-valued map homotopic to $$\phi $$ ϕ that has $$N(\phi : a)$$ N ( ϕ : a ) roots at a. We verify the conjecture when $$X = Y$$ X = Y is a Lie group by employing a fixed point result of Schirmer. As an application, we calculate the Nielsen root numbers of linear n-valued maps of tori.

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