Orthogonality with Respect to the Hermite Product, KP Wave Functions, and the Bispectral Involution

Abstract

It is well known that for any wave function $\psi(x,z)$ of the KP hierarchy, there is another wave function called its ''adjoint'' such that the path integral of their product with respect to $z$ around any sufficiently large closed path is zero. For the wave functions in the adelic Grassmannian ${\rm Gr}^{\rm ad}$, the bispectral involution which exchanges the role of $x$ and $z$ also implies the existence of an ''$x$-adjoint wave function'' $\psi^{\star}(x,z)$ so that the product of the wave function, the $x$-adjoint, and the Hermite weight ${\rm e}^{-x^2/2}$ has no residue. Utilizing this, we show that the sequences of coefficient functions in the power series expansion of any KP wave function in ${\rm Gr}^{\rm ad}$ and its image under the bispectral involution at $t_2=-\frac{1}{2}$ are always ''almost bi-orthogonal'' with respect to the Hermite product. Whether the sequences have the stronger properties of being (almost) orthogonal can easily be determined in terms of KP flows and the bispectral involution. As a special case, the exceptional Hermite orthogonal polynomials can be recovered in this way. This provides both a generalization of and an explanation of the fact that the generating functions of the exceptional Hermites are certain special wave functions of the KP hierarchy. In addition, one new surprise is that the same KP wave function which generates the sequences of functions is also a generating function for the norms when evaluated at $t_1=1$ and $t_2=0$. The main results are proved using Calogero-Moser matrices satisfying a rank one condition. The same results also apply in the case of ''spin-generalized'' Calogero-Moser matrices, which produce instances of matrix orthogonality.