Spin Whittaker function orthogonality conjecture

Let $mathpzc{W}_N$ be the space of admissible $N$-tuples $\underline{L}_N=(L_{N,1},\dots,L_{N,N})$ and let $\mathfrak{f}_{\underline{Z}}$ denote the spin Whittaker functions, with $\underline{Z}=(Z_1,\dots,Z_N)in(\mathrm{i}\mathbb{R})^N$. Define the $S$-deformed Sklyanin measure by

$$\mathfrak{M}^N_S(\underline{Z})=\frac{1}{N!(2pimathrm{i})^N}\prod_{1le ine jle N}\frac{\Gamma(S+Z_i)\Gamma(S-Z_i)}{\Gamma(2S)\Gamma(Z_i-Z_j)}.$$

Spin Whittaker orthogonality conjecture. For all $\underline{L}_N,\underline{L}'_Ninmathpzc{W}_N$,

$$\int_{(\mathrm{i}\mathbb{R})^N}\mathfrak{f}_{\underline{Z}}(\underline{L}_N)\mathfrak{f}_{-\underline{Z}}(\underline{L}'_N)\mathfrak{M}^N_S(\underline{Z}),dZ_1\dots dZ_N=\prod_{i=1}^{N-1}\left(1-\frac{L_{N,i+1}}{L_{N,i}}\right)^{1-2S}\delta_{\underline{L}_N-\underline{L}'_N},$$

where $\delta_{\underline{L}_N-\underline{L}'_N}$ is a delta function. This is presented as the scaling-limit analogue of the spin $q$-Whittaker orthogonality conjecture. It generalizes the orthogonality of classical Whittaker functions and is part of the expected spectral theory for spin Whittaker functions.