Let \((N, K)\) be a nilpotent Gelfand pair, that is, \(N\) is a simply connected, connected nilpotent Lie group and \(K\) a compact subgroup of automorphisms of \(N\) such that the convolution algebra \(L^1(N)^K\) of \(K\) invariant functions is commutative. Let \[ \mathcal{D} = (D_1, D_2, \ldots D_d) \] be a \(d\)-tuple of generators of \(\mathbb D (N)^K\) (left \(N\) invariant and \(K\)-invariant differential operators on \(N\)). To each bounded spherical function \(\varphi\) one can associate the tuple \[ (\xi_1(\varphi), \ldots \ldots \xi_d(\varphi) ) \] the eigenvalues of \((D_1, \ldots \ldots D_d).\) These \(d\)-tuples form a closed subset of \(\mathbb R^d\) denoted by \(\Sigma_{\mathcal{D}}.\) Let \(S(N)^K\) denote the \(K\)-invariant Schwartz class functions on \(N\) and \(S(\Sigma_{\mathcal{D}}) = S(\mathbb R^d) / \{f: f \equiv 0 \text{ on } \Sigma_{\mathcal{D}} \}.\)
A question which got a lot of attention in the recent past is whether the spherical transform maps \(S(N)^K\) isomorphically to \(S(\Sigma_{\mathcal{D}}).\) This has been already proved in some cases. The paper under review proves a preliminary result (which will be needed in settling the above question) under the assumption that \(N\) satisfies Vinberg’s condition. This means that \(\mathfrak{n} = \mathfrak{v} \oplus [\mathfrak{n}, \mathfrak{n}] \) and \(K\) acts irreducibly on \(\mathfrak{v}\). (It also follows that the center \(\mathfrak{z} = [\mathfrak{n}, \mathfrak{n}].\) )
Write \(\mathfrak{z} = \mathfrak{z_0} \oplus \check {\mathfrak{z}}\) where \(\check{\mathfrak{z}}\) is the space of \(K\)-invariants. Let \(\check{N} = N/ {}{\text{exp}}_N \mathfrak{z_0}\) and \(\tilde{N}\) equals the direct product of \(\check{N}\) with \(\mathfrak{z_0}.\) Points in \(\tilde{N}\) can be denoted by \((v, \zeta, t)\) where \(v \in \mathfrak{v}, \zeta \in \mathfrak{z_0}\) and \(t \in \check{\mathfrak{z}}.\) The main result in the paper is a Taylor development for Schwartz functions on \(\tilde{N}\) which have some homogeneity in \(\zeta\) variable. In particular, such functions can be written as derivatives of functions in \(S(\check{N})^K\) with control over Schwartz norms.
For the entire collection see [Zbl 1276.00017].