On generalized Steinberg theory for type AIII

Source abstract

The multiple flag variety 𝔛 = Gr ( ℂ p + q , r ) × ( Fl ( ℂ p ) × Fl ( ℂ q ) ) can be considered as a double flag variety associated to the symmetric pair ( G , K ) = ( GL p + q ( ℂ ) , GL p ( ℂ ) × GL q ( ℂ ) ) of type AIII. We consider the diagonal action of K on 𝔛 . There is a finite number of orbits for this action, and our first result is a description of these orbits: parametrization (by a certain set of graphs), dimensions, closure relations and cover relations. In [5], we defined two generalized Steinberg maps from the K -orbits of 𝔛 to the nilpotent K -orbits in 𝔨 and those in the Cartan complement of 𝔨 , respectively. The main result in the present paper is a complete, explicit description of these two Steinberg maps by means of a combinatorial algorithm which extends the classical Robinson–Schensted correspondence.