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Is it known whether for a generalized complex flag variety $X$ (that is, $G/P$ for a complex semisimple Lie group $G$ and a parabolic $P$), the homology of the free loop space $H_*(\Lambda X, \mathbb{Q})$ is degree-wise finite-dimensional? Is it known at least for type A (i.e. classical) flag varieties?

This is known to be true for some concrete examples, such as complex projective spaces (Ziller et al.), and complete flag varieties of rank 2 (Burfitt-Grbić), but I was unable to find any statements for the general case.

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Serre proved that for any simply-connected $X$, if $X$ has finitely generated homology groups in each degree, then the loop space of $X$ has finitely generated homology groups in each degree. (Proposition 9 of chapter IV of Homologie singulière des espaces fibrés. Applications. Ann. of Math., 54, 1951, p. 425-505.)

So the only fact you need about generalized flag varieties is that they are simply-connected compact manifolds.

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