On Certain Degenerate Whittaker Models for GL(n) over Finite Fields [master thesis]

64 Pages·2016·0.583 MB·English

by  Zahi Hazan

The Raymond and Beverly Sackler
Faculty of Exact Sciences
School of Mathematical Sciences
On Certain Degenerate Whittaker
Models for GL(n) over Finite Fields
Thesis submitted in partial fulfillment of the requirements for the
M. Sc. degree in the School of Mathematical Sciences, Tel-Aviv
University
by
Zahi Hazan
Under the supervision of
Prof. David Soudry
October 2016

Abstract
Let F be a finite field and let Fn be the degree n field extension of F. This thesis
deals with certain degenerate Whittaker models of cuspidal representations of
GLn(F). Given aregular character ofF∗n, there existsa corresponding irreducible
cuspidal representation of GLn(F). D. Prasad proved in [Pra00] that for such a
cuspidal representation π(cid:48) of GL2n(F), associated with a regular F∗2n-character
θ(cid:48) and for a nontrivial character ψ0 of F, the following GLn(F)-representation
π ,
N(cid:48) ,ψ
(cid:48)
I X
V = v V π n v = ψ (X)v ,
πN(cid:48) ,ψ(cid:48) (cid:26) ∈ π(cid:48) (cid:12) (cid:48)(cid:18)0 In(cid:19) 0 (cid:27)
(cid:12)
is equivalent to the induced represe(cid:12)(cid:12)ntation IndGF∗nLn(F)(θ(cid:48) (cid:22)F∗n).
We generalize Prasad’s work by considering the GLn(F)-representation πN,ψ,
I X Y
n
Vπ = v V π 0 I Z v = ψ (tr(X +Z))v ,
N,ψ π n 0
 ∈ (cid:12)   
(cid:12) 0 0 I
 n 
(cid:12)
(cid:12)  
(cid:12)
where π is an irreducible cu(cid:12)spidal representation of GL3n(F), associated with
a regular F∗3n-character θ and ψ0 is a nontrivial character of F. We give an
exact formula for the character of π and a nice description of π by all
N,ψ N,ψ
IndGLn(F)(θ (cid:22) ), where (cid:96) n.
F∗(cid:96) F∗(cid:96) |

Acknowledgments
First and foremost I would like to express my sincere gratitude to my supervisor
Professor David Soudry. I am indebted to him for providing me with invaluable
support, encouragement, scientific guidance and most importantly for intriguing
my curiosity over and over again.
I would like to thank my fellow student Ofir Gorodetsky who managed to
prove a necessary identity for this thesis (Theorem B) and Professor Chan Heng
Huat for useful discussions and advice regarding this identity.
I would like to thank my fellow students Or Baruch and Elad Zelingher for
many useful discussions.
Last but not least, I would like to thank my wife Ravit and my kids Orya and
Noya for their constant support, encouragement and patience during all these
years.
5

Contents
1 Introduction 11
1.1 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2 Preliminaries 16
2.1 Cuspidal Representations . . . . . . . . . . . . . . . . . . . . . . . 16
2.2 Characters Induced from Subfields . . . . . . . . . . . . . . . . . . 18
2.3 Some Linear Algebra . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3.1 Number of matrices with same sank and trace . . . . . . . 20
2.3.2 Number of non-square matrices with a given rank . . . . . 22
2.4 On Some Conjugacy Classes of GLn(F) . . . . . . . . . . . . . . . 23
2.4.1 Analog of Jordan form . . . . . . . . . . . . . . . . . . . . 23
2.4.2 Conjugating an arbitrary matrix . . . . . . . . . . . . . . . 25
2.4.3 Trace under conjugation . . . . . . . . . . . . . . . . . . . 27
2.5 Arithmetic properties of certain polynomials . . . . . . . . . . . . 28
3 Calculation of dimπ 32
N,ψ
4 Calculation of the Character Θ 39
N,ψ
4.1 Character Calculation at a Non-Semisimple Element . . . . . . . 40
4.1.1 Case g = λu (λ F,u = In) . . . . . . . . . . . . . . . . . 40
∈ (cid:54)
4.1.2 Case g = s u, (s comes from Fn but not from F) . . . . . 43
·
4.2 Character Calculation at a Semisimple Element . . . . . . . . . . 46
5 Concluding the Main Theorem 51
Appendix - Proof of the Dimension Identity 53
Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
Auxiliary Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
Conclusion of Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
7

List of Notations
F A finite field of cardinality q ......................................... 11
ψ0 A nontrivial character of F...........................................11
Fn The unique degree n field extension of F ............................. 11
Gk The group of invertible k k matrices over F ........................ 11
×
G The group of invertible 3n 3n matrices over F......................12
×
∆r(G ) The diagonal subgroup of (G )r....................................11
k k
P The parabolic subgroup of GL3n(F) of type (n,n,n)..................12
M The Levi subgroup of P..............................................12
N The unipotent radical of P ...........................................12
ψ A nontrivial character of N ..........................................12
V The (N,ψ)-isotypic subspace of V ...................................12
πN,ψ π
πθ An irreducible, cuspidal representation of GL3n(F) associated to a regular
character θ of F∗3n....................................................13
Θ The character of π ................................................14
N,ψ N,ψ
πN,ψ The representation of GLn(F), identified with ∆3(Gn), in VπN,ψ.......13
θ A regular character of F∗3n............................................16
Θ The character of π ..................................................16
θ θ
ΘInd(cid:96) The character of IndFG∗(cid:96)Ln(F)(θ (cid:22)F∗(cid:96)).....................................18
Yα The number of square matrices of order m+k over F with a fixed rank k
m,k
and a fixed trace α...................................................20
Gr(n,r) The Grassmannian of r-dimensional subspace of Fn...............20
Zs,t,k The number of matrices of order s t over F with fixed rank k.......22
×
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