Alladi Ramakrishnan Hall

Hecke algebras and the Langlands program

Manish Mishra

University of Heidelberg

Given an irreducible polynomial f(x) with integer coefficients
and a prime number p, one wishes to determine whether f(x) is a product of
distinct linear factors modulo p. When f(x) is a solvable polynomial, this
question is satisfactorily answered by the Class Field Theory. Attempts to
find a non-abelian Class Field Theory lead to the development of an area of
mathematics called the Langlands program.
The Langlands program, roughly speaking, predicts a natural correspondence
between the finite dimensional complex representations of the Galois group
of a local or a number field and the infinite dimensional representations
of real, p-adic and adelic reductive groups. I will give an outline of the
statement of the local Langlands correspondence. I will then briefly talk
about two of the main approaches towards the Langlands program - the type
theoretic approach relying on the theory of types developed by
Bushnell-Kutzko and others; and the endoscopic approach relying on the
trace formula and endoscopy. I will then state some of my results involving
these two approaches.