## A new proof of the Howe Conjecture

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- by Dan Barbasch and Allen Moy
- J. Amer. Math. Soc.
**13**(2000), 639-650 - DOI: https://doi.org/10.1090/S0894-0347-00-00336-2
- Published electronically: April 26, 2000
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## 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

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## Bibliographic Information

**Dan Barbasch**- Affiliation: Department of Mathematics, Cornell University, Malott Hall, Ithaca, New York 14853-4201
- MR Author ID: 30950
- Email: barbasch@math.cornell.edu
**Allen Moy**- Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1109
- MR Author ID: 127665
- Email: moy@math.lsa.umich.edu
- 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.
- © Copyright 2000 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**13**(2000), 639-650 - MSC (2000): Primary 22E35
- DOI: https://doi.org/10.1090/S0894-0347-00-00336-2
- MathSciNet review: 1758757