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We provide a Lagrangian construction for the fixed-point subalgebra, togetherwith its idempotent form, in a quasi-split symmetric pair of type $A_{n-1}$.This is obtained inside the limit of a projective system of Borel-Moorehomologies of the Steinberg varieties of $n$-step isotropic flag varieties.Arising from the construction are a basis of homological origin for theidempotent form and a geometric realization of rational modules.

Quasi-split symmetric pairs of $U(\mathfrak{sl}_n)$ and Steinberg varieties of classical type