Teaching Statement and Statement of Research Plans - GWDG

Index 7
where O is the ring of integers of K, M its maximal ideal with the generator
π.
N := the group of monomial matrices
S :=
{
s 1 :=
( 0 −1
1 0
)
, s 2 :=
( 0 −
1
π
π 0
)}
W := The Weyl group
N/B ∩ N
W i := the subgroup generated by s i
Recall the following definition of Gyoja’s for the L-function of any Coxeter
system (W, S) with respect to a finite dimensional representation φ :
H q (W, s) −→ End(V ) (where H q (W, s) is the associated Iwahori-Hecke algebra
with the generator set {e w } (for all these definitions see [4].))
L(W, S, φ, u) := ∑ w∈W
φ(e w )u l(w)
In the above l is the length function defined on W . L(W i , S, φ, u) could be
defined similarly by replacing W with W i . My main goal is to generalise this
beautiful formula ( as mentioned in [8]) to the higher dimensional case based
on definitions discussed in 1) and 2) above. With such a suitable definition
of the Zeta function Z( ˜X, u) for the finite regular hypergraph ˜X : Γ\X . , X .
being the Bruhat-Tits building associated to GL(d, K ν ), we must have:
Z( ˜X, u) = ∏ L(W I , S, ρ Γ , u) (−1)|I|
I⊂S
The proof of the above formula and the definitions it needs are being written
up in detail and will appear as a paper in arXiv soon.
Some interesting research Projects for the near future
In [18] the so-called discrete quantum Fourier-transforms of the number states
i.e. the superpositions:
q−1
∑
|θ p >:= q −1/2
n=0
exp( 2iπpn )|n >
q
I believe that such states live in Ramanujan graphs, especially those associated
to finite groups Z/NZ for suitable N, but the question is :
How can one in general interpret such superpositions as vertices of suitable
Ramanujan graphs? How can one use the deep results from representation
theory and arithmetical group theory lying behind the Ramanujan property
to the study of Cyclotomy or in general to associated physical phenomena?
I also have ideas to generalise these connections even to higher dimensions.