Character sheaves on exotic symmetric spaces and Kostka polynomials

Abstract

This paper is a survey on a joint work with K. Sorlin concerning the theory of character sheaves on the exotic symmetric space. After explaining the historical background, we introduce character sheaves on this variety. By using those character sheaves, we show that modified Kostka polynomials, indexed by a pair of double partitions, can be interpreted in terms of the intersection cohomology of the orbits in the exotic nilpotent cone, as conjectured in Achar-Henderson.

A characteristic map for the symmetric space of symplectic forms over a finite field

“…A further connection was made by Shoji and Sorlin between a related space and the modified Kostka polynomials which are the change of basis from the Hall-Littlewood polynomials to the Schur functions. This work also owes much to the reformulation of the results in [1] in terms of character sheaves, due to Grojnowski [9] and Henderson [11] some of which was later extended in the work of Shoji and Sorlin [19][20][21] which can also be found in the survey [22].…”

“…For more details on this section, see [11,Ch. 5,6] or [22], noting that any sheaf on G/H × V can be restricted to G/H × {0}.…”

“…When both have an F q structure, then K (T −ι ,L) does as well and their characteristic functions are invariant under H F . See [11] or [22] for details on their construction and properties.…”