The Langlands software summarizes these components of mathematical learn belonging to the illustration concept of reductive teams and to type box conception. those issues are attached by means of the imaginative and prescient that, approximately conversing, the irreducible representations of the overall linear team may function parameters for the outline of all quantity fields. within the neighborhood case, the bottom box is a given $p$-adic box $K$ and the extension thought of $K$ is noticeable as decided via the irreducible representations of absolutely the Galois staff $G_K$ of $K$. nice development has been made in constructing correspondence among the supercuspidal representations of $GL(n,K)$ and people irreducible representations of $G_K$ whose levels divide $n$. regardless of those advances, no ebook or paper has provided the assorted tools used or perhaps gathered identified effects.
This quantity includes the court cases of the convention "Representation conception and quantity concept in reference to the neighborhood Langlands Conjecture," held in December 1985 on the college of Augsburg. this system of the convention was once divided into elements: (i) the illustration thought of neighborhood department algebras and native Galois teams, and the Langlands conjecture within the tame case; and (ii) new effects, corresponding to the case $n=p$, the matching theorem, critical orders, tame Deligne representations, category of representations of $GL(n)$, and the numerical Langlands conjecture. the gathering of papers during this quantity offers a great account of the present country of the neighborhood Langlands Program.
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