Abstract
The work presented in this talk is joint with Toshie Takata.
I will first present compact expressions for a certain family
of representations $R_{r}^{frg}$ indexed by a complex finite
dimensional simple Lie algebra $frg$ and a positive integer $r$ bigger than or equal to the dual coxeter number of $frg$
(the so-called shifted quantum level).
These expressions are obtained using a certain (multidimensional)
reciprocity formula together with certain symmetries.
The representations $R_{r}^{frg}$ are known from
the study of theta functions and modular forms in connection with the study of affine Lie algebras. They also play a fundamental role in conformal field theory and thereby in the Chern--Simons path integral TQFT's of Witten.
I will explain how these representations enter surgery fomulas
for the rigorously defined quantum invariants of $3$--manifolds, the so-called
Reshetikhin--Turaev invariants. In the final part of the talk I will pay
special attention to the Seifert manifolds. I will present compact
expressions for the quantum invariants of these spaces using our expressions
for the representations $R_{r}^{frg}$. In particular, we obtain hereby
a formula for the large $r$ asymptotic expansion of the quantum invariants
of lens spaces. This result is in accordance with the asymptotic expansion
conjecture of J.E.Andersen. If time permits I will explain some of
the problems encountered when trying to calculate these asymptotics for more
general Seifert manifolds.
Contact person:Jørgen Ellegaard Andersen.