LAGRANGIAN FIBERS OF GELFAND–CETLIN SYSTEMS OF SO(n)-TYPE

The topology of Gelfand-Zeitlin fibers

“…The description of the subgroups H k and L k is derived in a similar manner to that in the unitary case. We note that the description of the subgroups H k and L k given here diverges slightly from that of Cho and Kim, due to an error in their statement of their Lemma 6.5 [CK20]. This minor error does not affect their main results.…”

“…In particular, we recall results of Cho-Kim-Oh which describe the bundles of homogeneous spaces that occur at each stage in terms of the associated GZ pattern [CKO20,CK20]. We note a minor correction to Cho and Kim's description [CK20] of the bundles in the orthogonal case (although this correction does not impact their final results, it does play an important role in our cohomology computations). Section 3 contains our description of the GZ fibers as biquotients (Theorem 3.8).…”

“…For further details regarding GZ systems, we direct the reader to the original papers of Guillemin and Sternberg [GS83a,GS83b] and the recent papers of Cho-Kim-Oh describing the GZ fibers as towers of bundles of homogeneous spaces [CK20,CKO20]. As mentioned in the introduction, an alternative description of the GZ fibers is given by Bouloc et al [BouMZ18].…”

“…In recent work, Cho, Kim and Oh observed that both unitary and orthogonal GZ fibers are total spaces of certain towers of fiber bundles [CKO20,CK20]. They used this description to show that every unitary GZ fiber decomposes as a direct product of a torus and a space whose first and second homotopy groups are trivial.…”