Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)



A new proof of the Howe Conjecture

Authors: Dan Barbasch and Allen Moy
Journal: J. Amer. Math. Soc. 13 (2000), 639-650
MSC (2000): Primary 22E35
Published electronically: April 26, 2000
MathSciNet review: 1758757
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The Howe Conjecture, which has formulations for both a reductive $p$-adic group $\mathcal G$ and its Lie algebra, is a statement about the finite dimensionality of certain spaces of $\mathcal G$-invariant distributions. Howe proved the algebra version of the conjecture for $GL(n)$ via a method of descent. Harish-Chandra extended Howe's method, when the characteristic is zero, to arbitrary reductive Lie algebras. Harish-Chandra then used the conjecture, in both its Lie algebra and group formulations, as a fundamental underpinning of his approach to harmonic analysis on the group and Lie algebra. Many properties of $\mathcal G$-invariant distributions, which for real Lie groups follow from differential equations, in the $p$-adic case are consequences of the Howe Conjecture and other facts, e.g. properties of orbital integrals. Clozel proved the group Howe Conjecture in characteristic zero via a method very different than Howe's and Harish-Chandra's descent methods. We give a new proof of the group Howe Conjecture via the Bruhat-Tits building. A key tool in our proof is the geodesic convexity of the displacement function. Highlights of the proof are that it is valid in all characteristics, it has similarities to Howe's and Harish-Chandra's methods, and it has similarities to the existence proof of an unrefined minimal K-type.

References [Enhancements On Off] (What's this?)

  • [A1] James Arthur, Some problems in local harmonic analysis, Harmonic analysis on reductive groups (Brunswick, ME, 1989) Progr. Math., vol. 101, Birkhäuser Boston, Boston, MA, 1991, pp. 57–78. MR 1168477
  • [A2] James Arthur, On the Fourier transforms of weighted orbital integrals, J. Reine Angew. Math. 452 (1994), 163–217. MR 1282200, 10.1515/crll.1994.452.163
  • [A3] James Arthur, Intertwining operators and residues. I. Weighted characters, J. Funct. Anal. 84 (1989), no. 1, 19–84. MR 999488, 10.1016/0022-1236(89)90110-9
  • [A4] James Arthur, The trace Paley Wiener theorem for Schwartz functions, Representation theory and analysis on homogeneous spaces (New Brunswick, NJ, 1993) Contemp. Math., vol. 177, Amer. Math. Soc., Providence, RI, 1994, pp. 171–180. MR 1303605, 10.1090/conm/177/01916
  • [BT] F. Bruhat and J. Tits, Groupes réductifs sur un corps local, Inst. Hautes Études Sci. Publ. Math. 41 (1972), 5–251 (French). MR 0327923
  • [C] Laurent Clozel, Orbital integrals on 𝑝-adic groups: a proof of the Howe conjecture, Ann. of Math. (2) 129 (1989), no. 2, 237–251. MR 986793, 10.2307/1971447
  • [HC] Harish-Chandra, Admissible invariant distributions on reductive p-adic groups, Preface and notes by S. DeBacker and P. Sally, University Lecture Series, vol. 16, American Mathematical Society, Providence, RI, 1999. CMP 99:16
  • [Ho1] Roger Howe, Two conjectures about reductive 𝑝-adic groups, Harmonic analysis on homogeneous spaces (Proc. Sympos. Pure Math., Vol. XXVI, Williams Coll., Williamstown, Mass., 1972) Amer. Math. Soc., Providence, R.I., 1973, pp. 377–380. MR 0338278
  • [Ho2] Roger Howe, The Fourier transform and germs of characters (case of 𝐺𝑙_{𝑛} over a 𝑝-adic field), Math. Ann. 208 (1974), 305–322. MR 0342645
  • [M] A. Moy, Displacement functions on the Bruhat-Tits building, preprint.
  • [MP] Allen Moy and Gopal Prasad, Jacquet functors and unrefined minimal 𝐾-types, Comment. Math. Helv. 71 (1996), no. 1, 98–121. MR 1371680, 10.1007/BF02566411

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (2000): 22E35

Retrieve articles in all journals with MSC (2000): 22E35

Additional Information

Dan Barbasch
Affiliation: Department of Mathematics, Cornell University, Malott Hall, Ithaca, New York 14853-4201

Allen Moy
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1109

Received by editor(s): April 20, 1999
Received by editor(s) in revised form: March 7, 2000
Published electronically: April 26, 2000
Additional Notes: The authors were supported in part by the National Science Foundation grants DMS-9706758 and DMS-9801264.
Article copyright: © Copyright 2000 American Mathematical Society