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We find bounds for the Hofer–Zehnder capacity of spherically monotone coadjoint orbits of compact Lie groups with respect to the Kostant–Kirillov–Souriau symplectic form in terms of the combinatorics of their Bruhat graphs. We show that our bounds are sharp for coadjoint orbits of the unitary group and equal in that case to the diameter of a weighted Cayley graph.

The depth statistic was defined for every Coxeter group in terms of factorizations of its elements into product of reflections. Essentially, the depth gives the minimal path cost in the Bruaht graph, where the edges have prescribed weights. We present an algorithm for calculating the depth of a signed permutation which yields a simple formula for this statistic. We use our algorithm to characterize signed permutations having depth equal to length. These are the fully commutative top-and-bottom elements defined by Stembridge. We finally give a characterization of the signed permutations in which the reflection length coincides with both the depth and the length.

Let $G$ be a connected reductive linear algebraic group over $ℂ$ with an involution $θ$ . Denote by $K$ the subgroup of fixed points. In certain cases, the $K-orbits$ in the flag variety $G/B$ are indexed by the twisted identities $ι (θ ) = {θ (w^{-1})w | w∈W}$ in the Weyl group $W$. Under this assumption, we establish a criterion for rational smoothness of orbit closures which generalises classical results of Carrell and Peterson for Schubert varieties. That is, whether an orbit closure is rationally smooth at a given point can be determined by examining the degrees in a "Bruhat graph'' whose vertices form a subset of $ι (θ )$. Moreover, an orbit closure is rationally smooth everywhere if and only if its corresponding interval in the Bruhat order on $ι (θ )$ is rank symmetric. In the special case $K=\mathrm{Sp}_{2n}(ℂ), G=\mathrm{SL}_{2n}(ℂ)$, we strengthen our criterion by showing that only the degree of a single vertex, the "bottom one'', needs to be examined. This generalises a result of Deodhar for type A Schubert varieties., Soit $G$ un groupe algébrique connexe réductif sur $ℂ$, équipé d'une involution $θ$ . Soit $K$ le sousgroupe de ses points fixes. Dans certains cas, les orbites des points de la variété de drapeaux $G/B$ sous l'action de $K$ sont indexées par les identités tordues, $ι (θ ) = {θ (w^{-1})w | w∈W}$, du groupe de Weyl $W$. Sous cette hypothèse, on établit un critère pour la lissité rationnelle des adhérences des orbites, qui généralise des résultats classiques de Carrell et Peterson pour les variétés de Schubert. Plus précisément, on peut déterminer si l'adhérence d'une orbite est rationnellement lisse en examinant les degrés dans un "Graphe de Bruhat" dont les sommets forment un sous-ensemble de $ι (θ )$. En outre, l'adhérence d'une orbite est partout rationnellement lisse si et seulement si l'intervalle correspondant dans l'ordre de Bruhat de $ι (θ )$ est symétrique respectivement au rang. Dans le cas particulier $K=\mathrm{Sp}_{2n}(ℂ), G=\mathrm{SL}_{2n}(ℂ)$, nous améliorons notre critère en montrant qu'il suffit d'examiner le degré d'un seul sommet, celui "du bas". Ceci généralise un résultat de Deodhar pour les variétés de Schubert de type A.