The Local Langlands Conjecture for GL₂

Mishra, Rajat Kumar (2019) The Local Langlands Conjecture for GL₂. Masters thesis, Indian Institute of Science Education and Research Kolkata.

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Abstract

In the thesis we try to introduce the local Langland correspondence for GL₂(F), for any non-archimeadian local field F which asserts that the 2-dimensional smooth representations of Weil Groups of F corresponds to the smooth irreducible representations of GL₂(F), in a natural way. This naturality is expressed by associating to representations of the Weil group, as well as representations of GL₂(F), couple of functions called the L-function and ε-factor. Then the Local Langlands correspondence says that there is a bijection between G₂(F) i.e. isomorphism classes of a particular kind of smooth representations of theWeil group and A₂(F) i.e. the isomorphism classes of irreducible smooth representations of GL₂(F) preserving the L-function and ε-factor. In the 1st chapter we discuss about the smooth representations on any general locally profinite group, then we focus on the special case of GL₂(F). and in the first two chapters we look at all the isomorphism classes of irreducible smooth representations of GL₂(F). Then in the third chapter we talk about the Weil group associated to F and some special smooth representations of it and after that we discuss about the L-function and ε-factors of both GL₂(F) and the weil group. Finally in the last chapter we state the correspondence and try to justify it.

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