Research

I work on the representation theory of reductive groups over finite rings, such as quotients of rings of integers of local fields. I am interested in a generalisation of Deligne-Lusztig theory to this setting, as well as purely algebraic (non-cohomological) constructions of representations, and their relations to supercuspidal representations of p-adic groups. 

I also work on representation zeta functions of nilpotent and compact p-adic groups. In addition, I have done work on group schemes over local rings and commutators in matrix rings.



Papers


My papers can be found on arXiv and on the Durham Research Online repository. The arXiv versions are often, but not always, identical to the published versions.


  • (with N. Monteiro) The conjugation representation of GL_2 and SL_2 over finite local rings
    arXiv:2412.08539, 40 pages.

  • (with Z. Chen) The algebraisation of higher level Deligne--Lusztig representations II: odd levels
    arXiv:2311.05354, 21 pages, submitted.

  • (with M. Zordan) Rationality of twist representation zeta functions of compact p-adic analytic groups
    arXiv:2212.05825, to appear in Trans. Amer. Math. Soc.

  • (with M. Zordan) Rationality of representation zeta functions of compact p-adic analytic groups
    arXiv:2007.10694, to appear in Amer. J. Math.

  • (with L. Knight) Representatives of similarity classes of matrices over PIDs corresponding to ideal classes 
    • Glasg. Math. J., 1-16, (2023), open access.

  • Representations of SL_n over finite local rings of length two
    • J. Algebra, 566 (2021), 119-135.

  • A uniform proof of the finiteness of the class group of a global field
    Amer. Math. Monthly, 128(3) (2021), 239–249.

  • (with A. Vera-Gajardo) Representations of reductive groups over finite local rings of length two
    J. Algebra, 525 (2019), 171-190.

  • (with J. Häsä) Representation growth of compact linear groups
    • Trans. Amer. Math. Soc., 372(2) (2019), 925–980.

  • Commutators of trace zero matrices over principal ideal rings
    • Israel J. Math. 228(1) (2018), 211–227.

  • (with S. Stevens) The regular representations of GL_N over finite local principal ideal rings
    • Bull. London Math. Soc., 49 (2017), 1066-1084.

  • (with Z. Chen) The algebraisation of higher Deligne—Lusztig representations
    • Selecta Math. (N.S.), 23(4) (2017), 2907-2926. Open Access.

  • Representations of GL_N over finite local principal ideal rings - An overview
    • In "Around Langlands Correspondences", Contemp. Math., 691 (2017), 337-358.

  • (with C. Voll) Representation zeta functions of some nilpotent groups associated to prehomogeneous vector spaces
    Forum Math., 29(3) (2017), 717-734.

  • Similarity and commutators of matrices over principal ideal rings
    • Trans. Amer. Math. Soc., 368 (2016), 2333-2354.
    368 (2016), 2333-2354
    368 (2016), 2333-2354

  • (with C. Voll) Representation zeta functions of nilpotent groups and generating functions for Weyl groups of type B
    • Amer. J. Math., 136(2) (2014), 501-550.

  • (with C. Voll) A new statistic on the hyperoctahedral groups
    • Electron. J. Combin., 20(3) (2013), #P50 (23 pages).

  • Reductive group schemes, the Greenberg functor, and associated algebraic groups
    •  J. Pure Appl. Algebra, 216 (2012), 1092-1101. (errata)

  • Extended Deligne-Lusztig varieties for general and special linear groups
    •  Adv. Math., 226 (2011), 2825-2853.

  • (with A.-M. Aubert, U. Onn and A. Prasad) On cuspidal representations of general linear groups over discrete valuation rings
    • Israel J. Math., 175 (2010), 391-420.

  • The smooth representations of GL_2(o)
    • Comm. Algebra, 37 (2009), 4416-4430.

  • Unramified representations of reductive groups over finite rings
    • Represent. Theory 13 (2009), 636-656.


    Unpublished


    • Representations of reductive groups over finite rings and extended Deligne-Lusztig varieties, math.RT/0403487.

    • Representations of reductive groups over quotients of local rings, math.RT/0311243.