turkmath.org

Langlands program is basically a study to associate the representations of Galois groups over local and global fields with the automorphic representations of algebraic groups over local fields and adeles, respectively. I will particularly talk about the Langlands reciprocity principles and related conjectures for the general linear group $GL(n)$ which will be discussed in two parts: local Langlands correspondence and global Langlands reciprocity conjecture for $GL(n)$.

Local Langlands correspondence states a well-defined one-to-one correspondence of the set of equivalence classes of $n$-dimensional (Frobenius) semisimple complex Weil-Deligne representations of the Weil group of $F$ with the set of equivalence classes of irreducible admissible representations of $GL(n)$ over $F$ where $F$ is a local field. The local $L$ and $\epsilon$-factors attached to these certain representations of the Weil group and of $GL(n)$ are preserved under this correspondence. For the global part, I will respectively talk about the global class field theory, automorphic and cuspidal representations of $GL(n)$ over the adeles, global automorphic $L$ and $\epsilon$-factors and hypothetical Langlands group in order to state the global Langlands reciprocity conjecture for $GL(n)$.