Spectrum of the $\bar{\partial}$-Laplace operator on zero forms for the quantum quadric $\mathcal{O}_q(\textbf{Q}_N)$

Abstract

We study the Laplacian operator $\Delta_{\bar{\partial}}$ associated to a K\"ahler structure $(\Omega^{(\bullet, \bullet)}, \kappa)$ for the Heckenberger--Kolb differential calculus of the quantum quadrics $\mathcal{O}_q(\textbf{Q}_N)$, which is to say, the irreducible quantum flag manifolds of types $B_n$ and $D_n$. We show that the eigenvalues of $\Delta_{\bar{\partial}}$ on zero forms tend to infinity and have finite multiplicity.