Full Record

Author | Dexter, Kathleen D. |

Title | Some Results on the Representation Theory of the Symplectic Similitude Group of Order Four |

URL | http://hdl.handle.net/10027/9591 |

Publication Date | 2012 |

Date Accessioned | 2012-12-13 22:08:58 |

University/Publisher | University of Illinois – Chicago |

Abstract | We compute the asymptotic expansion of Whittaker functions of an element of the maximal torus for principal series representations. We consider irreducible, reducible, and singular representations. |

Subjects/Keywords | Whittaker; asymptotic expansion; principal series; singular |

Contributors | Takloo-Bighash, Ramin (advisor); Shipley, Brooke (committee member); Shalen, Peter (committee member); Radford, David (committee member); Varma, Sandeep (committee member) |

Language | en |

Rights | Copyright 2012 Kathleen D. Dexter |

Country of Publication | us |

Record ID | handle:10027/9591 |

Repository | uic |

Date Retrieved | 2019-12-17 |

Date Indexed | 2019-12-17 |

Issued Date | 2012-12-13 00:00:00 |

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…of *Whittaker* functions on a proper subgroup of the maximal torus of GSp(4) to
compute the remaining L-functions. The method he created uses the fact that the *Whittaker*
functions appearing in the zeta integrals can be explicitly computed…

…which, in turn, allows
one to explicitly compute the zeta integral and demonstrate that it does, in fact, represent the
desired L-function.
In this work, we use the method developed by Takloo-Bighash to compute the asymptotic
expansion of *Whittaker*…

…to obtain a two variable zeta function involving both the degree four and the degree five Lfunctions of GSp(4).
In Section 2.1, we begin with the description of the asymptotic expansion of a *Whittaker*
function as a certain finite sum. Then…

…of *Whittaker* functions,
intertwining operators, and local coefficients.
x
CHAPTER 1
INTRODUCTION
1.1
The Representation Theory of GSp(4)
1.1.1
GSp(4) and Its Parabolic Subgroups
Let F be a nonarchimedean local field of…

…x7B;π(n)v − ψ(n)v : n ∈ N, v ∈ V }.
1.1.4
Generic Representations and *Whittaker* Models
Let θ be a smooth generic complex character of N . Here generic means that the restriction
of θ to each simple root subgroups of N is non…

…functions
W(π, θ) = {Wv |v ∈ V }
is called a *Whittaker* model of π. The group G acts on this space by right translation
π(h)W (g) = W (gh),
and the map v → Wv intertwines V and W(π, θ).
A…

…*Whittaker* functional is a smooth linear functional Λ : V → C satisfying
Λ(π(n)v) = θ(n)Λ(v).
A *Whittaker* functional gives a *Whittaker* model by setting
Wv (g) = Λ(π(g)v),
8
and a…

…*Whittaker* model gives a *Whittaker* functional by defining
Λ(v) = Wv (e),
where e denotes the identity element of G. (When we need to specify to which representation
Λ belongs, we will write Λπ . Otherwise, the subscript will be…