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ON A COHEN-MACAULAY PROPERTY OF SPHERICAL VARIETIES

We study the space S(X) of Schwartz functions on a spherical variety X = G/H as a smooth representation of a p-adic group G. We notice that in some cases this is a Cohen-Macaulay object in the category of smooth representations of G. We show that this property is related to several central questions of harmonic analysis on X: (1) Behavior of the multiplicity d(π) = dimHomH(π,C) in families of representation. (2) Density of orbital integrals in the space of invariant distributions. (3) A question of P. Delorme regarding the disjointness of X-cuspidal representations and the unramified spectrum of X. (4) Freeness properties of the module S(X)K over subalgebras of the Hecke algebra H(G, K), for various compact subgroups K ⊂ G. We also show that the Cohen-Macaulay property holds in the following cases: (a) The diagonal case: X = H × H/Δ(H). In this case, the Cohen-Macaulay property is a corollary of an unpublished result of J. Bernstein. (b) X = GL(2)/U(2) We believe that the property of Cohen-Macaulay holds for many spherical spaces but we present cases were it fails. Finally, we formulate a weaker property which we believe holds in general. Along the way, we provide a simple proof of a result of Hironaka ([Hir99]) which refers to the freeness of the module S(X)K0 over H(GL(V ), K0), where X is the space of Hermitian forms on V and K0 is the maximal compact subgroup of G.