OPERATORS BASED ON DOUBLE COSETS OF $GL_2$
Open Access
- Author:
- Meemark, Yotsanan
- Graduate Program:
- Mathematics
- Degree:
- Doctor of Philosophy
- Document Type:
- Dissertation
- Date of Defense:
- May 04, 2006
- Committee Members:
- Wen Ching Winnie Li, Committee Chair/Co-Chair
Woodrow Dale Brownawell, Committee Member
Yuri G Zarhin, Committee Member
Murat Gunaydin, Committee Member
Nigel David Higson, Committee Member - Keywords:
- Drinfeld modular forms
character sum estimates
Kirillov models
Ramanujan graphs
harmonic cocycles
Hecke operators - Abstract:
- In this thesis, we study several operators based on double cosets of $GL_2$. The first three operators are associated to study the Cayley graphs on $PGL_2(mathbb F_q)$ mod the unipotent subgroup, the split and nonsplit tori, respectively. Using the Kirillov models of the representations of $PGL_2(mathbb F_q)$ of degree greater than one, we obtain explicit eigenvalues of these graphs and the corresponding eigenfunctions. Character sum estimates are then used to conclude that two types of the graphs are Ramanujan, while the third is almost Ramanujan. The graphs arising from the nonsplit torus were previously studied by Terras et al. We give a different approach here. The last two operators are the Hecke operators $T_mathfrak{P}$ on Drinfeld cusp forms for the subgroups $Gamma_1(T)$ and $Gamma(T)$. We study their actions by using Teitelbaum's interpretation as harmonic cocycles on the tree of $GL_2(K_infty)$. We obtain explicit eigenvalues of all Hecke operators on the cusp forms for $Gamma_1(T)$ of small weights $k le q$ and conclude that the Hecke operators are simultaneously diagonalizable for $deg mathfrak{P} = 1$. We also show that the Hecke operators are not diagonalizable for $Gamma_1(T)$ of larger weights $k >q$, and not for $Gamma(T)$ even of small weights.