Lifting the dual immaculate functions

Dual immaculate creation operators and a dendriform algebra structure on the quasisymmetric functions

The dual immaculate functions are a basis of the ring QSym of quasisymmetric functions, and form one of the most natural analogues of the Schur functions. The dual immaculate function corresponding to a composition is a weighted generating function for immaculate tableaux in the same way as a Schur function is for semistandard Young tableaux; an “immaculate tableau” is defined similarly to a se...

The immaculate basis of the non - commutative symmetric functions ( Extended Abstract )

We introduce a new basis of the non-commutative symmetric functions whose elements have Schur functions as their commutative images. Dually, we build a basis of the quasi-symmetric functions which expand positively in the fundamental quasi-symmetric functions and decompose Schur functions according to a signed combinatorial formula. Résumé. Nous introduisons une nouvelle base des fonctions symé...

Multiplicative structures of the immaculate basis of non-commutative symmetric functions

Geometric theta-lifting for the dual pair GSp2n, GO2m

Let X be a smooth projective curve over an algebraically closed field of characteristic > 2. Consider the dual pair H = GO2m, G = GSp2n over X , where H splits over an étale two-sheeted covering π : X̃ → X . Write BunG and BunH for the stacks of G-torsors and H-torsors on X . We show that for m ≤ n (respectively, for m > n) the theta-lifting functor FG : D(BunH) → D(BunG) (respectively, FH : D(B...

Geometric theta-lifting for the dual pair SO2m, Sp2n

Let X be a smooth projective curve over an algebraically closed field of characteristic > 2. Consider the dual pair H = SO2m, G = Sp2n over X with H split. Write BunG and BunH for the stacks of G-torsors and H-torsors on X . The theta-kernel AutG,H on BunG ×BunH yields the theta-lifting functors FG : D(BunH) → D(BunG) and FH : D(BunG) → D(BunH) between the corresponding derived categories. Assu...