Taiwanese Journal of Mathematics


Andrew Knightly and Charles Li

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Following a construction of Gross and Reeder, we define simple supercuspidal  representations of $\operatorname{GL}(n)$ over a $\mathfrak{p}$-adic field.  We show that they have conductor $\mathfrak{p}^{n+1}$.  We then give a general formula for  the matrix coefficient attached to a new vector, and make it  completely explicit when $n=2$.

Article information

Taiwanese J. Math., Volume 19, Number 4 (2015), 995-1029.

First available in Project Euclid: 4 July 2017

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Zentralblatt MATH identifier

Primary: 22E50: Representations of Lie and linear algebraic groups over local fields [See also 20G05]

simple supercuspidal representation matrix coefficient new vector


Knightly, Andrew; Li, Charles. SIMPLE SUPERCUSPIDAL REPRESENTATIONS OF $\operatorname{GL}(n)$. Taiwanese J. Math. 19 (2015), no. 4, 995--1029. doi:10.11650/tjm.19.2015.3853. https://projecteuclid.org/euclid.twjm/1499133686

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  • M. Adrian and B. Liu, The local Langlands correspondence for simple supercuspidal representations of $\GL_n(F)$, arXiv:1310.2585.
  • C. Bushnell and G. Henniart, The Local Langlands Correspondence for $\GL(2)$, Springer, 2006.
  • –––, Langlands parameters for epipelagic representations of $\text{\rm GL}_n$, Math. Ann., 358(1-2) (2014), 433-463.
  • C. Bushnell and P. Kutzko, The admissible dual of $\GL(N)$ via compact open subgroups, Annals of Mathematics Studies, 129. Princeton University Press, Princeton, NJ, 1993.
  • H. Carayol, Représentations supercuspidales de $\GL_n$, C. R. Acad. Sci. Paris Sr. A-B, 288(1) (1979), A17-A20.
  • –––, Représentations cuspidales du groupe linéaire, Ann. Sci. École Norm. Sup. (4), 17(2) (1984), 191-225.
  • I. M. Gelfand and D. A. Kajdan, Representations of the group GL(n,K) where K is a local field, Lie groups and their representations (Proc. Summer School, Bolyai János Math. Soc., Budapest, 1971), pp. 95-118. Halsted, New York, 1975.
  • B. Gross, Irreducible cuspidal representations with prescribed local behavior, Amer. J. Math., 133(5) (2011), 1231-1258.
  • B. Gross and E. Frenkel, A rigid irregular connection on the projective line, Ann. of Math. (2), 170(3) (2009), 1469-1512.
  • B. Gross and M. Reeder, Arithmetic invariants of discrete Langlands parameters, Duke Math. J., 154(3) (2010), 431-508.
  • G. Henniart, Sur l'unicité des types pour $\GL_2$, Appendix to Duke Math. J., 115(2) (2002), 205-310.
  • R. Howe, Hecke algebras and $p$-adic $\GL_n$, Representation theory and analysis on homogeneous spaces (New Brunswick, NJ, 1993), 65-100, Contemp. Math., \bf177, Amer. Math. Soc., Providence, RI, 1994.
  • J. Heinloth, B. Ngô and Z. Yun, Kloosterman sheaves for reductive groups, Ann. of Math. (2), 177(1) (2013), 241-310.
  • H. Jacquet, A correction to “Conducteur des représentations du groupe linéaire", Pac. J. Math., 260(2) (2012), 515-525.
  • H. Jacquet, I. Pitateskii-Shapiro and J. Shalika, Conducteur des représentations du groupe linéaire, Math. Ann., 256(2) (1981), 199-214.
  • A. Knightly and C. Li, Traces of Hecke operators, Mathematical Surveys and Monographs, 133. Amer. Math. Soc., Providence, RI, 2006.
  • –––, Modular $L$-values of cubic level, Pac. J. Math., 260(2) (2012), 527-563.
  • R. Schmidt, Some remarks on local newforms for $\GL(2)$, J. Ramanujan Math. Soc., 17(2) (2002), 115-147.