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Generic representations of p-adic groups (STEVENSU19SCIC2)

  • Full or part time
  • Application Deadline
    Thursday, April 04, 2019
  • Competition Funded PhD Project (European/UK Students Only)
    Competition Funded PhD Project (European/UK Students Only)

Project Description

The Local Langlands Correspondence has motivated a great deal of Number Theory and Representation Theory over the last fifty years. It connects representations of (roughly) the absolute galois group of a p-adic field to the representations of a matrix group G over the field. In the original formulation (now proved for several families of matrix groups G, including general linear groups) it considered only complex representations but, more recently, representations with coefficients in other rings or fields have been considered also. The correspondence results in a partition of the irreducible representations of G into L-packets (which are singletons for general linear groups), and each packet contains (at least conjecturally) precisely one representation which is “generic.”

One of the difficulties in studying this is that the representations of G are (almost all) infinite-dimensional, so they are quite hard to get a handle on. One way for doing so, which has been very successful, is to restrict them to compact open subgroups K: we then look for representations of K whose presence in this restriction characterizes some property which representations of G might have; this property might be being “unramified”, ”tame”, “generic”,… Such results also have interpretations via the Langlands correspondence, so consequences for the absolute galois group.

The idea in this PhD project will be to look at recognizing the property of being generic in this way. Recent results of Pyvovarov provide the answer for complex representations of general linear groups so that two directions to pursue would be: extending these results to other groups; and extending them to representations with different coefficients.

For more information on the supervisor for this project, please go here: https://people.uea.ac.uk/en/persons/shaun-stevens
The type of programme: PhD
The start date of the project: Oct 2019
Acceptable first degree in Mathematics and minimum entry requirement is 2:1.

Funding Notes

This PhD project is in a Faculty of Science competition for funded studentships. These studentships are funded for 3 years and comprise UK/EU fees, an annual stipend of £15,009 and £1,000 per annum to support research training. Overseas applicants may apply but they are required to fund the difference between home/EU and overseas tuition fees (which are detailed on the University’s fees pages at View Website . Please note tuition fees are subject to an annual increase).

References

i) “The local Langlands conjecture for GL(2),” Colin Bushnell and Guy Henniart, Grundlehren der Mathematischen Wissenschaften 335, Springer-Verlag, Berlin, 2006.
ii) “Generic smooth representations,” Alexandre Pyvovarov, arXiv:1803.02693
iii) “Genericity of supercuspidal representations of p-adic Sp(4),” Corinne Blondel and Shaun Stevens, Compos. Math. 145 (2009), no. 1, 213–246; arXiv:0708.2636
iv) "Représentations l-modulaires d'un groupe réductif p-adique avec l≠p," Marie-France Vignéras, Progress in Mathematics, 137, Birkhäuser Boston, Inc., Boston, MA, 1996.

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