Duke Mathematical Journal

On special representations of p-adic reductive groups

Elmar Grosse-Klönne

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Let F be a non-Archimedean locally compact field, and let G be a split connected reductive group over F. For a parabolic subgroup QG and a ring L, we consider the G-representation on the L-module C(G/Q,L)/Q'QC(G/Q',L).() Let IG denote an Iwahori subgroup. We define a certain free finite rank-L module M (depending on Q; if Q is a Borel subgroup, then (∗) is the Steinberg representation and M is of rank 1) and construct an I-equivariant embedding of (∗) into C(I,M). This allows the computation of the I-invariants in (∗). We then prove that if L is a field with characteristic equal to the residue characteristic of F and if G is a classical group, then the G-representation (∗) is irreducible. This is the analogue of a theorem of Casselman (which says the same for L=C); it had been conjectured by Vignéras.

Herzig (for G=GLn(F)) and Abe (for general G) have given classification theorems for irreducible admissible modulo p representations of G in terms of supersingular representations. Some of their arguments rely on the present work.

Article information

Duke Math. J., Volume 163, Number 12 (2014), 2179-2216.

First available in Project Euclid: 15 September 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 22E50: Representations of Lie and linear algebraic groups over local fields [See also 20G05]
Secondary: 11S99: None of the above, but in this section


Grosse-Klönne, Elmar. On special representations of $p$ -adic reductive groups. Duke Math. J. 163 (2014), no. 12, 2179--2216. doi:10.1215/00127094-2785697. https://projecteuclid.org/euclid.dmj/1410789513

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  • [1] N. Abe, On a classification of irreducible admissible modulo $p$ representations of a $p$-adic split reductive group, Compos. Math. 149 (2013), 2139–2168.
  • [2] A. Björner and F. Brenti, Combinatorics of Coxeter Groups, Grad. Texts in Math. 231, Springer, New York, 2005.
  • [3] A. Borel and N. Wallach, Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups, 2nd ed., Math. Surveys Monogr. 67, Amer. Math. Soc., Providence, 2000.
  • [4] M. Emerton, Ordinary parts of admissible representations of $p$-adic reductive groups, I: Definition and first properties, Astérisque 331 (2010), 355–402.
  • [5] F. Herzig, The classification of irreducible admissible mod $p$ representations of a $p$-adic $\operatorname{GL}_{n}$, Invent. Math. 186 (2011), 373–434.
  • [6] J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge Stud. Adv. Math. 29, Cambridge Univ. Press, Cambridge, 1990.
  • [7] N. Iwahori and H. Matsumoto, On some Bruhat decomposition and the structure of the Hecke rings of $p$-adic Chevalley groups, Publ. Math. Inst. Hautes Études Sci. 25 (1965), 5–48.
  • [8] J. C. Jantzen, Representations of Algebraic Groups, Pure Appl. Math. 131, Academic Press, Boston, 1987.
  • [9] G. Laumon, Cohomology of Drinfeld Modular Varieties, Part I: Geometry, Counting of Points and Local Harmonic Analysis, Cambridge Stud. Adv. Math. 41, Cambridge Univ. Press, Cambridge, 1996.
  • [10] S. Orlik, On extensions of generalized Steinberg representations, J. Algebra 293 (2005), 611–630.
  • [11] V. Paskunas, Coefficient Systems and Supersingular Representations of $\operatorname{GL}_{2}(F)$, Mém. Soc. Math. Fr. (N.S.) 99, Soc. Math. France, Paris, 2004.
  • [12] P. Schneider and U. Stuhler, The cohomology of $p$-adic symmetric spaces, Invent. Math. 105 (1991), 47–122.
  • [13] P. Schneider and U. Stuhler, Representation theory and sheaves on the Bruhat-Tits building, Publ. Math. Inst. Hautes Études Sci. 85 (1997), 97–191.
  • [14] J. Tits, “Reductive groups over local fields” in Automorphic Forms, Representations and $L$-functions, Part 1 (Corvallis, 1977), Proc. Sympos. Pure Math. 33, Amer. Math. Soc., Providence, 1979.
  • [15] M.-F. Vignéras, Représentations $\ell$-modulaires d’un groupe réductif $p$-adique avec $\ell\neq p$, Progr. Math. 137, Birkhäuser, Boston, 1996.
  • [16] M.-F. Vignéras, Pro-$p$-Iwahori Hecke ring and supersingular $\overline{F}\!_{p}$-representations, Math. Ann. 331 (2005), 523–556. Erratum, Math. Ann. 333 (2005), 699–701.
  • [17] M.-F. Vignéras, Série principale modulo $p$ de groupes réductifs $p$-adiques, Geom. Funct. Anal. 17 (2008), 2090–2112.