Existence of operators with the conjectured properties on Grassmannians

For a Grassmannian associated with a lattice of signature $(2,b_{-})$, let $K_{\mathbb{R}}+iC$ denote the relevant tube domain, let $Y$ be its imaginary component, and let $R_{m}^{(b_{-})}$ be the weight-raising operator. Let $S_{m}^{(b_{-})}$ act on automorphic forms of weight $m$. The notation $P_{k-m,k,k}$, $g_{k,h,\rho}^{(m),+}$, and the coefficients $B_{k,h}^{+}$ and $a_k$ refer to the local basis functions and Fourier-expansion data used in the paper.

$S_{m}^{(b_{-})}$ operator conjecture. For any dimension $b_{-}$ and any weight $m$, there exists an operator $S_{m}^{(b_{-})}$ satisfying properties (i)--(vii) stated in the source: it raises the weight by $b_{-}$, raises Laplacian eigenvalues by $2mb_{-}$, has order $b_{-}$ and squares in the prescribed way to $\frac{1}{(-4\pi^{2})^{b_{-}}}(R_{m}^{(b_{-})})^{b_{-}}$, obeys the stated conjugation and annihilation properties, maps the specified Fourier and singularity types as given, and annihilates the specified homogeneous functions of $Y$ when $b_{-}>1$.

The paper subsequently proves the conjectured properties for even $b_{-}$ using $S_{m}^{(b_{-})}=\frac{1}{(2\pi i)^{b_{-}}}(R_{m}^{(b_{-})})^{b_{-}/2}$, and notes that the case $b_{-}=1$ follows from the cited work. The conjectural existence for odd $b_{-}$ beyond $1$ remains the part not established here.