Spherical functions on spherical varieties (Q2856611)

Let \(\mathbb{X} = \mathbb{H} \backslash \mathbb{G}\) be a homogeneous spherical variety over a nonarchimedean local field \(k\), where \(\mathbb{G}\) is a split connected reductive \(k\)-group. Assume moreover that everything in sight has nice model over the ring of integers \(\mathfrak{o}\) in \(k\). Use the ordinary letters \(X\), \(G\), etc. to denote the set of \(k\)-points of these schemes and set \(K = \mathbb{G}(\mathfrak{o})\). The main aim of this paper is to give explicit formulas for the eigenvectors of the right regular representation of the spherical Hecke algebra \(\mathcal{H}(G,K)\) on \(V := C^\infty(X)\), with respect to general eigencharacters \(\eta\) in the maximal spectrum of \(\mathcal{H}(G, K)\). More generally, we may allow twist \(C^\infty(X)\) by some additive character \(\psi\). A prominent example in this direction is the celebrated Casselman-Shalika formula for Whittaker functions.NEWLINENEWLINEOne of the main motivations is a global one: to understand the period integrals along \(H\) of Eisenstein series, or other automorphic forms in general. This has important applications for the Relative Trace Formula for \(\mathbb{G}\). The first step is to look at the unramified places and deduce explicit, combinatorial or algebro-geometric flavoured formulas for the relevant terms. This is what the author endeavours in this impressive paper.NEWLINENEWLINEThe precise (and complicated) statements can be found in the paper. As in the Introduction thereof, we make simplifying assumptions such as that the chosen Borel subgroup \(B\) has a unique open orbit in \(X\). Using the general theory of spherical varieties, one defines the torus \(\mathbb{A}_X\), its Langlands dual \(A^*_X\), a parabolic subgroup \(\mathbb{P}(X) \supset \mathbb{B}\) and the little Weyl group \(W_X\), etc. The general formula (Theorem 1.2.1) furnishes a basis of the \(\eta\)-eigenspace, for general eigencharacter \(\eta\), consisting of functions NEWLINE\[NEWLINE \Omega_{\delta_{(X)}^{1/2} \chi}(x_{\check{\lambda}}) = e^{-\check{\lambda}}(\delta_{P(X)}^{1/2}) \sum_{w \in W_X} B_w(\chi) e^{\check{\lambda}} ({}^w \chi) NEWLINE\]NEWLINE where \(\delta_{(X)}^{1/2} \chi\) ranges over a set of representatives of \(\delta_{(X)}^{1/2} A^*_X/W_X\) which map to \(\eta\). The terms \(B_w(\chi)\) are certain cocycles related to natural intertwining operators \(S_\chi: C^\infty_c(X) \to I(\chi)\) into the principal series, as proportionality factors under the standard intertwining operator \(T_w\).NEWLINENEWLINEThis formula is then rephrased, under additional assumptions such as the affineness of \(\mathbb{X}\), in terms of some factors \(\beta(\chi)\) whose definition involves the ``colors'' of \(X\) (Luna-Vust theory). The product \(\beta(\chi)\beta(\chi^{-1})\) is related to a certain \(L\)-value for the dual group \(\check{G}_X\) for \(X\), and intervenes in the unramified Plancherel formula for \(X\). All these phenomena are already familiar in the classical Plancherel formula of Harish-Chandra.NEWLINENEWLINEAs applications, the author mentions three results: (i) The isomorphism \(C^\infty_c(X)^K \simeq \mathbb{C}[\delta_{(X)}^{1/2} A^*_X]^{W_X}\) as \(\mathcal{H}(G,K)\)-modules (Theorem 8.0.2) for affine \(\mathbb{X}\). (ii) The unramified part of the Plancherel formula for \(X\) (Theorem 9.0.1) together with a formula for the Tamagawa number \(\mathrm{vol}(\mathbb{X}(\mathfrak{o}))\). (iii) A formula for the pseudo-Eisenstein series induced from a torus (Theorem 10.0.2); in particular the relevant \(L\)-value here is explicitly described.NEWLINENEWLINEThis paper is a significant step in the study of periods and harmonic analysis on spherical varieties. The geometric techniques employed here, mainly due to Luna, Vust, Knop, Brion et al., will no doubt also turn out to be important. One should also consult the earlier paper [Compos. Math. 144, No. 4, 978--1016 (2008; Zbl 1167.22011)] by the same author.