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Schubert Varieties and Generalizations Schub ert varieties and generalizations T A Springer Mathematisch Instituut Universiteit Utrecht Budap estlaan CD Utrecht the Netherlands email springermathruunl 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 are intro 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 reductive group G 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 GH Then GH 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 Flag manifolds and Schub ert varieties The origin of the Schub ert varieties lies in the Schub ert calculus devised by H Schu th b ert at the end of the century which gives recip es to determine not always rigorously numb ers of solutions of geometric problems see Sch A simple example of such a problem determine the numb er of lines in P C intersecting lines in general p osition the answer is A more general example determine the numb er n of dplanes in P C intersecting d n d planes of dimension n d in general p osition the answer is dd n d n dn d n n Let G b e the set of ddimensional subspaces of C It is a pro jective algebraic variety dn n co ordinatized by Pluc ker co ordinates It is also the variety of d planes in P C n Fix a basis e e of V C and let V b e the subspace of V spanned by e e with n i i V fg Then F V V V V is a complete ag in V Let W b e a ddimensional n n subspace and put J W fj j V W V W g j j This is an increasing sequence of d integers the jump sequence of W It determines the p osition of W relative to F For example for a subspace in general p osition we have J W n d n Let Y b e the set of W Y G with J W J a given J dn sequence Then Y is the disjoint union of the Y Moreover one shows that each Y is lo cally J J closed in Y and is isomorphic to an ane space After ordering the set of J comp onentwise the closures b oth in the Zariski top ology and the complex top ology are describ ed by 0 Y Y J J 0 J J These closures are the Schubert varieties in G dn In the Schub ert calculus one deals with intersections of Schub ert varieties and their multi plicities This is b est done in terms of the Chow ring of Y spanned by equivalence classes of subvarieties of Y This leads into the theory of symmetric functions See Fu I shall not go into this I shall concentrate on the group theoretical asp ects The group G GL C acts alge n braically on Y The action is transitive so Y is a homogeneous space of G and is of the form Y GP where P is the parab olic subgroup of the g g G with g for i d and ij ij j d It is not hard to see that the Y are precisely the orbits of G J We shall consider a more general situation which will englob e the sp ecial case of Grassman nians Notations The notions and results from the theory of algebraic groups which we use without further reference can b e found in Bo or Hu For ro ot systems and the Weyl group see Bou G is a connected reductive linear algebraic group over C one could work over an arbitrary algebraically closed eld but I wont do this We x a maximal torus T of G and a Borel subgroup B T Also N is the normalizer of T and W N T is the Weyl group Fix a section w w of W to N Let R b e the ro ot system of G T and let R b e the system of p ositive ro ots dened by B For R we have a one parameter subgroup U of G normalized by T The unip otent radical U of B is generated by the U with R For w W let U b e the subgroup of w G generated by the subgroups U with R w R For R let s W b e the reection which it denes Let D b e the basis of R dened by R The set S of simple reections s D generates W The corresp onding length function on W is l We have dim U l w w Put G B w B This is a lo cally closed subset of G b eing an orbit of B B The underlying w top ology is the Zariski top ology One might also take the complex top ology Bruhats lemma Prop osition Bruhats lemma i G G w ii u b uwb denes an isomorphism of algebraic varieties U B B w B w In fact G B N S make up the ingredients of a Tits system see Hu no This implies that for w W s S G if l sw l w sw G G s w G G if l sw l w w sw It follows that for s S P G G s e s is a parab olic subgroup of G containing B G We have P B P It also follows that if e s s s s is a reduced decomp osition of w W where s S l l w we have l i G G G G w s s s 1 2 l Lemma P P P is the closure G s s s w 1 2 l If Y and Z are varieties with a right resp ectively left B action we write Y Z for the B quotient of Y Z by the B action by z y b bz It is presupp osed that the quotient exists A similar notation is used for multiple pro ducts Put Z P P P s B s B B s 1 2 l this is an irreducible variety The pro duct map of G induces a morphism Z G which is prop er b ecause all quotients P B are pro jective lines Hence Im is closed and irre s i ducible Moreover G is op en and dense in Z and the restriction of to this set maps it w bijectively onto G The lemma follows from these facts w Bruhat order G is a union of double cosets G Dene an order on W by x w if G G The closure w x x w This is the Bruhat order originally intro duced by Chevalley It follows from the lemma of that there is the following combinatorial description of the Bruhat order Let s s s b e a reduced decomp osition of w W where s S l l i l w and let x W Then x w if and only if x is a subpro duct of s s In fact on any l Coxeter group there exists an order with this description see Hu Schub ert varieties The quotient X GB is a ag variety It is an irreducible smo oth pro jective homogeneous space for G Let X b e the image of G in X under the canonical map this is a Bruhat cel l w w where w is the longest element of W The Bruhat cell X is a in X The big cel l is X w w 0 lw lo cally closed subvariety of X isomorphic to ane space A as a consequence of ii The big cell is op en and dense in X By i X is a paving of X by ane spaces or a cellular decomp osition The w w W X are the B orbits or U orbits on X w A Schubert variety is a closure S X w W It is an in general nonsmo oth irre w w ducible pro jective variety on which B acts By i and we have a paving S X w x xw n Example Let G GL It acts on V k A ag in V of length s is a sequence of n distinct subspaces V i s of V with V fg V V V V The ag i s is complete if s n in which case dim V i for all i G acts on the set of ags and the i parab olic subgroups of G are the stabilizers of ags The Borel subgroups are the stabilizers of complete ags Let V V V b e the complete ag of Its stabilizer is the Borel group B of upp er n triangular matrices and GB can b e identied with the space of all complete ags Let P B b e the stabilizer of the ag V V V Then GP is the Grassmannian G The canonical d n dn morphism GB GP maps a complete ag onto its ddimensional ingredient A classical Schub ert variety Y as in is the image in GP of a Schub ert variety S in GB or the J w closure of a B orbit in GP By Tits system theory these orbits are parametrized by the cosets of the Weyl group of G mo dulo the Weyl group of P see Bou Ch IV p In the present case this means that the Schub ert varieties in G are indexed by the elements dn of S S S ie by the delement subsets of f ng These are in bijection with n d nd the jump sequences of For w W put O fx y X X j x y G g w w The O are the Gorbits on X X There is a close connection with the Bruhat cells the w rst pro jection X X X denes a b ering O X with b ers X Similarly for the w w closures O It follows that w O O w x xw The T action The torus T acts on X and on all Schub ert varieties The xed p oints of T in X are the images p of the w in X w W so their numb er is nite The xed p oints of T in the w Schub ert variety S are the p with x w w x Let again w b e the longest element of W Lemma
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