Schub ert varieties and generalizations
T A Springer
Mathematisch Instituut Universiteit Utrecht
Budap estlaan CD Utrecht
the Netherlands
email springer math ruu nl
Abstract This contribution reviews the main results on Schub ert varieties and their gener
alizations It covers more or less the material of the lectures at the Seminar These were
partly exp ository intro ducing material needed by other lecturers In particular Section
reviews classical material used in several of the other contributions
Intro duction
The aim of this pap er is to give a review of the main results on Schub ert varieties and
their generalizations In the rst section Schub ert varieties over C areintro duced in the
setting of the theory of reductive groups and their Bruhat decomp osition Some geometric
results are discussed The Steinb erg variety asso ciated to a reductive group is also intro duced
Most of the material of this section is classical
In section examples are given of constructions of algebraic ob jects based on the geometry
discussed in Section For example gives an elementary geometric construction of the
Weyl group W of a reductivegroupG It uses corresp ondences on the ag variety X of G
Using machinery from algebraic top ology a calculus of corresp ondences on X pro duces the
Hecke algebra H of W This is discussed in and
Section discusses generalizations of Schub ert varieties These o ccur for example in the con
text of spherical varieties A closed subgroup H of G is spherical if a Borel subgroup B of G
has nitely many orbits on G H Then G H is a homogeneous spherical variety The orbit
closures generalize Schub ert varieties which one recovers for H B An imp ortant sp ecial
case is the case of symmetric varieties where H is the xed p oint group of an involutorial
automorphism of G
The combinatorial prop erties of the set of orbits are discussed in A calculus of corre
sp ondences gives rise to a representation of H discussed in The last part of Section
reviews sp ecial features of the the case of symmetric varieties
I am grateful to Cathy Krilo for help in the preparation of these notes