Francesco Brenti
(Universita degli Studi di Tor Vergata)
Parabolic Kazhdan-Lusztig and $R$-polynomials for Hermitian symmetric pairs: the hyperoctahedral case.
Abstract: In 1987 V. Deodhar introduced the so-called parabolic Kazhdan-Lusztig and $R$-polynomials. These polynomials are closely related to the classical Kazhdan-Lusztig and $R$-polynomials, and have applications to representation theory and to geometry.
In [Pacific J. Math., 207 (2002), 257-286] the author determined these polynomials for the Hermitian symmetric pairs of type $A$. In this talk we study type $B$. In this case the polynomials are naturally indexed by pairs of shifted partitions contained in a shifted staircase. Our main results are combinatorial product formulas for these polynomials. These formulas allow their computation almost ``by inspection'' and show, in particular, that they are combinatorial invariants. Namely, they depend only on the interval determined by the two indexing shifted partitions as an abstract partially ordered set. The formula for the parabolic Kazhdan-Lusztig polynomials involves a new class of shifted partitions which may be considered the "shifted analogue" of the Dyck partitions which are involved in the computation of these polynomials in type $A$.