Orbital integrals on p-adic groups: A proof of the Howe conjecture.

*(English)*Zbl 0675.22007Let F denote a p-adic field of characteristic 0, and G the F-points of a connected reductive group defined over F. Let \(\omega\) be a compact subset of G, and \(\omega^ G\) the invariant subset \(\{gxg^{-1}:\) \(g\in G\), \(x\in \omega \}\) of G. The purpose of this article is to prove the following theorem concerning invariant distributions on G: Suppose \(\Omega\) is a closed invariant subset of G, compact modulo conjugation (i.e. \(\Omega \subset \omega^ G\) for some compact set \(\omega)\), and let K denote a compact open subset of G (with corresponding Hecke algebra denoted by \({\mathcal H}_ K)\); then the space of all linear functions on \({\mathcal H}_ K\) which are restrictions of invariant distributions supported in \(\Omega\) is finite-dimensional. This result was conjectured by R. Howe [Proc. Symp. Pure Math. 26, 377-380 (1973; Zbl 0284.22004)]. It was proved by the author in an earlier paper assuming a certain finiteness property of discrete series representations [Compos. Math. 56, 87-110 (1985; Zbl 0599.22015)]. (The finiteness assumption is known for GL(n), and therefore implies Howe’s Conjecture in this case.) In the present paper, the author gives a direct proof of Howe’s conjecture for general G, side-stepping altogether the finiteness assumption just alluded to. The idea is to truncate in a new way the expression
\[
<trace \pi,f>\equiv \int_{G}trace(\pi (g)f(g))dg
\]
for \(\pi\) a representation of G and f a smooth compactly supported function on G; the result is a formula for \(<trace \pi,f>\) in terms of the traces of certain Jacquet modules of \(\pi\) on the “compact part” of the Levi subgroups of G. In his introduction, the author also makes a very interesting remark concerning the global analogue of these results. Here the analogue of the finiteness assumption (which was alluded to above and apparently remains unproved in general) is the conjecture that the poles of Eisenstein series constructed from cusp forms on a given parabolic subgroup should lie in a fixed finite set independent of the inducing cusp form. On the one hand, this assertion can now be proved for GL(n) [cf. the preprint of C. Moeglin and J. Waldspurger “Sur le spectre residuel de GL(N)”]; on the other hand, as in the local theory, a suitable truncation process (in this case due to J. Arthur) makes it possible to obtain information on the discrete part of the trace formula (analogous to Howe’s conjecture) without proving the above Eisenstein series assertion!

Reviewer: S.Gelbart

##### MSC:

22E50 | Representations of Lie and linear algebraic groups over local fields |

11S40 | Zeta functions and \(L\)-functions |

11F70 | Representation-theoretic methods; automorphic representations over local and global fields |

11F33 | Congruences for modular and \(p\)-adic modular forms |