A new proof of the Howe Conjecture

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.