Geometric and Unipotent Crystals

Geometric and Unipotent Crystals Ii: from Unipotent Bicrystals to Crystal Bases

For each reductive algebraic group G we introduce and study unipotent bicrystals which serve as a regular version of birational geometric and unipotent crystals introduced earlier by the authors. The framework of unipotent bicrystals allows, on the one hand, to study systematically such varieties as Bruhat cells in G and their convolution products and, on the other hand, to give a new construct...

0 Geometric and Unipotent Crystals

2 9 M ar 2 00 6 GEOMETRIC AND UNIPOTENT CRYSTALS

Affine Geometric Crystals in Unipotent Loop Groups

We study products of the affine geometric crystal of type A corresponding to symmetric powers of the standard representation. The quotient of this product by the R-matrix action is constructed inside the unipotent loop group. This quotient crystal has a semi-infinite limit, where the crystal structure is described in terms of limit ratios previously appearing in the study of total positivity of...

COHOMOLOGY OF DELIGNE-LUSZTIG VARIETIES FOR UNIPOTENT BLOCKS OF GLn(q)

We study the cohomology of parabolic Deligne-Lusztig varieties associated to unipotent blocks of GLn(q). We show that the geometric version of Broué’s conjecture over Q , together with Craven’s formula, holds for any unipotent block whenever it holds for the principal Φ1-block.