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The self-dual spaces of polynomials are related to Bethe vectors in theGaudin model associated to the Lie algebras of types B and C. In this paper, wegive lower bounds for the numbers of real self-dual spaces in intersections ofSchubert varieties related to osculating flags in the Grassmannian. The higherGaudin Hamiltonians are self-adjoint with respect to a nondegenerate indefiniteHermitian form. Our bound comes from the computation of the signature of thisform.

Lower Bounds for Numbers of Real Self-Dual Spaces in Problems of Schubert Calculus