A comparison of Landau-Ginzburg models for odd-dimensional Quadrics

In [Rie08], the second author defined a Landau-Ginzburg model for homogeneous spaces G/P, as a regular function on an affine subvariety of the Langlands dual group. In this paper, we reformulate this LG model (X^, W_t) in the case of the odd-dimensional quadric, as a rational function on a Langlands dual projective space, in the spirit of work by R. Marsh and the second author for type A Grassmannians and by both authors for Lagrangian Grassmannians. We also compare this LG model with the one obtained independently by Gorbounov and Smirnov, and we use this comparison to deduce part of a conjecture of the second author for odd-dimensional quadrics.