Math 249B. Geometric Bruhat decomposition

Size: px
Start display at page:

Download "Math 249B. Geometric Bruhat decomposition"

Transcription

1 Math 249B. Geometric Bruhat decomposition 1. Introduction Let (G, T ) be a split connected reductive group over a field k, and Φ = Φ(G, T ). Fix a positive system of roots Φ Φ, and let B be the unique Borel k-subgroup of G containing T such that Φ(B, T ) = Φ. Let W = W (G, T )(k) = N G(k) (T )/T (k), and for each W let n N G(k) (T ) be a representative of W. For each a Φ e let r a W be the associated involution, and e let denote the base of Φ. Under the action of B B on G by (b, b ).g = bgb 1, the orbit BgB through any g G(k) is a locally closed subvariety (as for orbits of linear algebraic groups in general). The double coset C() = Bn B (a Bruhat cell) depends only on, not n, and in class e proved that C() decomposes as a direct product scheme under multiplication: for the closed subset the multiplication map Φ = Φ ( Φ ) Φ, U Φ n B C() is an isomorphism of k-schemes. Equivalently, the left translate n 1 C() is identified ith the closed subscheme U 1 (Φ ) B in the open cell Ω = U Φ B G. (Thus, C() is open in G if and only if Φ = Φ, hich is to say that (Φ ) = Φ. There is exactly one such 0, and C( 0 ) is the unique open Bruhat cell.) Remark 1.1. The k-scheme U Φ is a direct product (under multiplication, in any order) of the root groups U a for a Φ, so it is an affine space of dimension #Φ. This cardinality has a simple combinatorial interpretation: it is the length l() relative to the generating set {r a } a of W. (That is, this is the minimal length of a ord in the r a s hose product is equal to.) In particular, l() #Φ ith equality if and only if C() is the unique open Bruhat cell. See Corollary 2 to Proposition 17 in 1.6 of Chapter VI of Bourbaki for a proof. By Corollary 3 to Proposition 17 in 1.6 of Chapter VI of Bourbaki there is a unique longest element 0 W relative to {r a } a, it is given by the product a Φ r a taken in any order, and 0 (Φ ) = Φ. In particular, 0 2 = 1 since 2 0 (Φ ) = Φ. We call 0 the long Weyl elements. The purpose of this handout is to prove that the subvarieties C() are a pairise disjoint covering of G; e call this the geometric Bruhat decomposition since it concerns covering the entire scheme G (and since each C() is locally closed, this is equivalent to a covering statement at the level of k-points). In the setting of possibly non-split connected reductive groups there ill be an analogous covering result (called the Bruhat decomposition) that is only valid at the level of k-points, recovering the geometric version over algebraically closed fields. Remark 1.2. The Zariski closure of each C() is a union of Bruhat cells C( ) for some subset of elements W. Indeed, to prove this it suffices to ork over k (since the formation of the Zariski closure of a locally closed subscheme commutes ith extension of the ground field), and this closure is visibly stable under left and right multiplication by B. Hence, the closure of C()(k) inside G(k) is a union of B(k) double cosets, so it is indeed a union of Bruhat cells. But hich C( ) lie in the closure of C()? This has a nice combinatorial anser: if e fix a reduced decomposition = r a1 r al() (i.e., a minimal length expression for as a product of the r a s, alloing some repetition in the sequence {a j }) then C( ) C() if and only if is obtained as a subproduct of r aj by removing some terms: = r ai1 r ain ith 1

2 2 1 i 1 < < i n l(). We ill not use this important result; see in Springer s book for a proof. 2. Bruhat decomposition We begin by proving the disjointness of the Bruhat cells. This ill ultimately reduce to a general result concerning torus actions on unipotent groups. Proposition 2.1. If then C() C( ) =. Proof. Since any to B(k) double cosets in G(k) are either equal or disjoint, e just have to rule out the possibility that n C( )(k) = U Φ (k)n B(k). That is, for, W e shall prove that an equality n = un b ith u U Φ (k) and b B(k) forces =. Let U = U Φ, and define the closed set of roots Φ = Φ 1 (Φ ) inside Φ, so Φ = Φ Φ and U = U Φ U Φ under multiplication. Similarly define Φ. For u as above, e claim that (1) U Φ = uu Φ u 1. To establish this equality, e first prove the formula U Φ = U n Bn 1 for any W. A point u of U lies in n Bn 1 if and only if n 1 un B. But if e rite u = u u under the decomposition U = U Φ U Φ (via multiplication inside G) then n 1 un = (n 1 u n )(n 1 u n ) ith n 1 u n n 1 U Φ n = U 1 (Φ U ) Φ and likeise n 1 un U Φ. Hence, the direct product structure of the open cell Ω = U Φ B implies that n 1 un B if and only if u = 1, hich is to say u U Φ. But e can compute U n Bn 1 in another ay: since n = un b, clearly n Bn 1 = un n 1 u 1, so Thus, e have proved (1). No comes the key point: U n Bn 1 = u(u n Bn 1 )u 1 = uu Φ u 1. Lemma 2.2. Let U be a unipotent smooth connected k-group equipped ith an action by a split k- torus T such that all T -eights on Lie(U) are non-trivial, pairise linearly independent in X(T ) Q, and have a 1-dimensional eight space. If V, V U are T -stable smooth connected k-subgroups that are U(k)-conjugate then they are equal inside U. Proof. We may and do assume k = k, and e argue by induction on dim U (the case dim U = 0 being trivial). For each a Φ(U, T ), let T a = (ker a) 0 red. The centralizer U a = Z U (T a ) of the T a -action on U is smooth and connected, ith Lie(U a ) = u a since Φ(U, T ) Qa = {a} by the hypotheses. Thus, U a is 1-dimensional, so U a G a. The 1-dimensionality implies by dynamic considerations (!) that U a is the unique nontrivial smooth connected T -stable k-subgroup of U ith T -eight a on its Lie algebra. The k-subgroups U a for a Φ(U, T ) generate U since their Lie

3 algebras span Lie(U). These conclusions also apply to any T -stable smooth connected k-subgroup of U in place of U. The unipotent U is nilpotent (see the handout Nilpotence of unipotent groups ), so (Z U ) 0 red is nontrivial and T -stable. The above reasoning can be applied to (Z U ) 0 red in place of U, so (Z U) 0 red contains U a for each T -eight a on Lie((Z U ) 0 red ). Fix such a eight a, and consider the central quotient U/U a. The images of V and V in this quotient coincide (by dimension induction), so V U a = V U a. If the central U a U is contained in one of V or V then it is contained in both (as V and V are U(k)-conjugate), in hich case V = V U a = V U a = V as desired. Hence, e can assume that U a is not contained in V nor in V. The T -eight a cannot occur on Lie(V ) or Lie(V ) (as otherise the construction of U a could be carried out inside V or V, a contradiction), so the intersections V U a and V U a have vanishing Lie algebra and thus are étale. But U a = G a on hich T acts as t.x = a(t)x for the nontrivial character a of T, so visibly U a has no nontrivial T -stable finite étale k-subgroup. In other ords, the surjective homomorphisms V U a V U a and V U a V U a are isomorphisms. Passing to Lie algebras and comparing T -eights, e see that Φ(V, T ) = Φ(V, T ) inside Φ(U, T ). But the unipotent smooth connected V is generated by the groups U a for a Φ(V, T ), and similarly for V, so V = V inside U as desired. It follos from the lemma that U Φ = U Φ inside U = U Φ, so Φ = Φ inside Φ. Passing to complements in Φ, e also have Φ = Φ. Thus, Φ Φ = Φ Φ. Denoting this set as Ψ, e have (Ψ) = (Φ ) ( Φ ) = (Φ (Φ )) (Φ ( Φ )) = Φ and similarly (Ψ) = Φ. Thus, 1 (Φ ) = Φ, so by simple transitivity of the W -action on the set of positive systems of roots in Φ e have =. Proposition 2.3 (Geometric Bruhat decomposition). The locally closed subvarieties {C()} W cover G. In particular, G(k) is covered by the disjoint subsets C()(k) = U Φ (k)n B(k) and the natural map W (G, T )(k) B(k)\G(k)/B(k) is bijective. Proof. We first treat the case hen the split connected reductive G has semisimple-rank 1 (i.e., D(G) is k-isomorphic to SL 2 or PGL 2 ). Let W be the unique nontrivial element, so C() = Un B for U = U Φ = G a. Thus, C()/B is an affine line that is open in G/B P 1 k, so its complement in G/B is a single k-rational point. The extra point C(1)/B accounts for this, so C() C(1) = G. In general, for every a Φ(G, T ) and the associated codimension-1 torus T a = (ker a) 0 red, G a := Z G (T a ) is split connected reductive ith semisimple-rank 1 and W (G a, T )(k) = {1, r a }. A Borel subgroup of G a containing T is B a = T U a B, so for n a N Ga(k)(T ) representing r a e have (2) G a = B a Ua n a B a. For each a, e claim that (3) G a Bn B Bn B Bn a n B, or equivalently G a C() C() C(r a ). Once this is proved, it follos that W C()(k) is stable under left multiplication by the subgroups G a (k) for a. But G(k) is generated by the subgroups G a (k) for a since T, U ±a G a and the elements n a U a, U a generate W (and W = Φ, so every root group is in the W -orbit of some U a ith a ). Thus, W C()(k) is 3

4 4 stable under left multiplication by G(k) and hence it coincides ith G(k). This implies the Bruhat decomposition (conditional on (3)), as each C() is locally closed in G. To prove (3), e fix a, so Ψ = Φ {a} is a closed set of roots contained in Φ. The associated unipotent smooth connected k-subgroup U Ψ U Φ = U satisfies under multiplication. Thus, U = U Ψ U a = U a U Ψ G a Bn B = G a UT n B = G a U a U Ψ n B = G a U Ψ n B. Lemma 2.4. For each a and Ψ = Φ {a}, G a U Ψ = U Ψ G a. Proof. Since G a is generated by T, U a, U a, it suffices to prove that U Ψ is normalized by each of T, U a, and U a. Normalization by T is obvious, and the cases of U a and U a are equivalent upon replacing the positive system of roots Φ = Ψ {a} in Φ ith a (Φ ) = Ψ { a}. Thus, e focus on U a -conjugation. Fix isomorphisms u c : G a U c for c Φ, so for b ±a in Φ e have u a (x)u b (y)u a (x) 1 u b (y) 1 U ( a b ) Φ. Thus, u a (x)u b (y)u a (x) 1 U ( a b ) Φ U b U Ψ since U Ψ is directly spanned in any order by the root groups U c for c Ψ. This gives that u a (x) conjugates U b into U Ψ for all b Ψ, so U a normalizes U Ψ. The preceding lemma implies (via (2)) that G a Bn B = U Ψ G a n B = U Ψ (B a U a n a B a )n B Bn B Un a B a n B = Bn B Un a U a n B since B a n = U a T n = U a n T ith T B. But Un a U a = UU a n a, so We claim that G a Bn B Bn B BU a n a n B. (4) G a U a n Bn 1 U a n a n Bn 1, from hich it ould follo that U a n a n U a n B U a n a n B, so BU a n a n B Bn B Bn a n B and hence G a Bn B Bn B Bn a n B as desired for (3). To prove (4) it is harmless to replace the Borel subgroup n Bn 1 of G containing T ith its intersection G a (n Bn 1 ). This intersection is a Borel subgroup of G a = Z G (T a ) containing T, so it is equal to one of the groups B ±a = T U ±a. Hence, (4) is equivalent to G a? = Ua B ±a U a n a B ±a for each of the roots ±a. For the case of B a this is the Bruhat decomposition (2) of the semisimplerank 1 group G a relative to (B a, T ), and for the case of B a this is the left n a -translate of the Bruhat decomposition of G a relative to the pair (B a, T ). Here is an important application of the Bruhat decomposition. Proposition 2.5 (Chevalley). Let (G, T ) be a split connected semisimple group over a field k, and assume that G is simply connected. Then G(k) is generated by the subgroups U c (k) for c Φ(G, T ).

5 Proof. Let B be a Borel k-subgroup containing T, Φ = Φ(B, T ) a positive system of roots in Φ = Φ(G, T ), and the base of Φ. By the Bruhat decomposition, G(k) is generated by the subgroups U c (k), T (k), and n a N G(k) (T ) for representatives n a of the generating set {r a } a of W = W (G, T )(k). Since G is simply connected, e have an isomorphism G m T via (λ a ) a a a (λ a ). This direct product structure implies that even on rational points, T (k) is generated by its subgroups a (k ) for a. Hence, G(k) is generated by: the subgroups {U c (k)} c Φ, the subgroups a (k ) for a, and representative elements n a for a. But Φ = W, so in this generating list e can limit the root groups considered to just those associated to roots in. But U a, U a SL 2 in hich U ±a go over to the standard unipotent k-subgroups U ±, a (k ) goes over to the subgroup of diagonal elements, and n a goes over to the standard Weyl element. It is classical that SL 2 (k) is generated by U ± (k), so e are done. The simply connected hypothesis in Proposition 2.5 cannot be dropped. To prove this, consider a split connected semisimple k-group G. Let f : G G be the simply connected central cover and T the split maximal k-torus preimage of T in G. We have seen in class (Example 5.2.6) that the bijection Φ( G, T ) = Φ(G, T ) induced by X(f) : X(T ) Q X( T ) Q yields isomorphisms U c U c for corresponding roots c Φ(G, T ) and c Φ( G, T ). Proposition 2.5 implies that the subgroups U c (k) G(k) generate the image of G(k) G(k), a normal subgroup for hich the cokernel is a generally non-trivial commutative group. (For example, SL n (k) PGL n (k), has cokernel k /(k ) n, and in general the cokernel is a subgroup of H 1 (k, ker f).) More explicitly, the commutator morphism G G G factors uniquely through a morphism G G G since G is a central quotient of G, and this induced morphism is a lift of the commutator morphism of G. Hence, the commutator subgroup of G(k) is contained in the image of G(k) G(k). 3. The long Weyl element As e noted in Remark 1.1, there is a unique longest element 0 W relative to {r a } a, and it is characterized by the property that 0 (Φ ) = Φ. It is therefore natural to onder if perhaps 0 = 1. This holds if and only if 1 W inside GL(X(T )), since if 1 W then it satisfies the property uniquely characterizing 0 W through its effect on Φ. But does 1 belong to W? Inspection of the Bourbaki Plates shos that 1 W precisely for the root systems A 1, B n (n 2), C n (n 3), D 2m (m 2), E 7, E 8, F 4, and G 2. In the other cases (i.e., A n for n 2, D 2m1 for m 2, and E 6 ), the Bourbaki Plates sho that 0 = ι here ι on Q arises from the unique diagram involution of Φ. Remark 3.1. Don t confuse 0 = a Φ r a (independent of the order of multiplication) ith the Coxeter element a r a that does depend on the order of multiplication but has conjugacy class independent of all choices (see Bourbaki Chapter IV, 1.11). Whereas 0 has order 2, the order of the Coxeter element is the Coxeter number h that is generally larger than 2. We finish by describing the long Weyl element and the Coxeter element (up to conjugacy) for the root system A n ith n 1. Recall that the eight lattice is X = Z n1 /diag. Letting ε 1,..., ε n1 be the standard basis of Z n1, a root basis is given by a i = ε i ε i1 modulo the diagonal, for 1 i n. For 1 i n, the effect of the reflection r ai is given by negation of a i, a i1 a i1 a i if i < n, a i 1 a i 1 a i if i > 1, and no effect on any other a j. This is induced by the linear automorphism of Z n1 (preserving the diagonal!) given by sapping ε i and ε i1 and leaving all other ε j unaffected. In terms of the indexed set of residue classes of ε j s in the eight lattice, it follos that the subgroup 5

6 6 S n1 GL(Z n1 ) arising as permutations of the standard basis (preserving the diagonal) maps onto W GL(X). The resulting map S n1 W is an isomorphism since the finite kernel consists of unipotent elements. Under this description of W, the long Weyl element 0 is induced by the sapping of ε i and ε n2 i for all 1 i n 1; i.e., it saps a i and a n1 i for all 1 i n. This shos quite explicitly that 0 1 hen n 2. Likeise, the Coxeter element (up to conjugacy) is induced by the left shift ε i ε i 1 using indexing modulo n 1 (so ε 1 ε n1 ). In terms of, this is induced by a i a i 1 for 1 < i n and a 1 (a 1 a n ). Clearly this latter operation is not an involution hen n > 2.

Math 249B. Tits systems

Math 249B. Tits systems Math 249B. Tits systems 1. Introduction Let (G, T ) be a split connected reductive group over a field k, and Φ = Φ(G, T ). Fix a positive system of roots Φ + Φ, and let B be the unique Borel k-subgroup

More information

Math 249B. Nilpotence of connected solvable groups

Math 249B. Nilpotence of connected solvable groups Math 249B. Nilpotence of connected solvable groups 1. Motivation and examples In abstract group theory, the descending central series {C i (G)} of a group G is defined recursively by C 0 (G) = G and C

More information

U a n w = ( U a )n w. U a n w

U a n w = ( U a )n w. U a n w Math 249B. Tits systems, root groups, and applications 1. Motivation This handout aims to establish in the general case two key features of the split case: the applicability of BN-pair formalism (a.k.a.

More information

descends to an F -torus S T, and S M since S F ) 0 red T F

descends to an F -torus S T, and S M since S F ) 0 red T F Math 249B. Basics of reductivity and semisimplicity In the previous course, we have proved the important fact that over any field k, a non-solvable connected reductive group containing a 1-dimensional

More information

Math 249C. Tits systems, root groups, and applications

Math 249C. Tits systems, root groups, and applications Math 249C. Tits systems, root groups, and applications 1. Motivation This handout aims to establish in the general case two key features of the split case: the applicability of BN-pair formalism (a.k.a.

More information

Lemma 1.3. The element [X, X] is nonzero.

Lemma 1.3. The element [X, X] is nonzero. Math 210C. The remarkable SU(2) Let G be a non-commutative connected compact Lie group, and assume that its rank (i.e., dimension of maximal tori) is 1; equivalently, G is a compact connected Lie group

More information

Parameterizing orbits in flag varieties

Parameterizing orbits in flag varieties Parameterizing orbits in flag varieties W. Ethan Duckworth April 2008 Abstract In this document we parameterize the orbits of certain groups acting on partial flag varieties with finitely many orbits.

More information

Math 249B. Applications of Borel s theorem on Borel subgroups

Math 249B. Applications of Borel s theorem on Borel subgroups Math 249B. Applications of Borel s theorem on Borel subgroups 1. Motivation In class we proved the important theorem of Borel that if G is a connected linear algebraic group over an algebraically closed

More information

FINITE GROUP THEORY: SOLUTIONS FALL MORNING 5. Stab G (l) =.

FINITE GROUP THEORY: SOLUTIONS FALL MORNING 5. Stab G (l) =. FINITE GROUP THEORY: SOLUTIONS TONY FENG These are hints/solutions/commentary on the problems. They are not a model for what to actually write on the quals. 1. 2010 FALL MORNING 5 (i) Note that G acts

More information

Margulis Superrigidity I & II

Margulis Superrigidity I & II Margulis Superrigidity I & II Alastair Litterick 1,2 and Yuri Santos Rego 1 Universität Bielefeld 1 and Ruhr-Universität Bochum 2 Block seminar on arithmetic groups and rigidity Universität Bielefeld 22nd

More information

On Mordell-Lang in Algebraic Groups of Unipotent Rank 1

On Mordell-Lang in Algebraic Groups of Unipotent Rank 1 On Mordell-Lang in Algebraic Groups of Unipotent Rank 1 Paul Vojta University of California, Berkeley and ICERM (work in progress) Abstract. In the previous ICERM workshop, Tom Scanlon raised the question

More information

In class we proved most of the (relative) Bruhat decomposition: the natural map

In class we proved most of the (relative) Bruhat decomposition: the natural map Math 249B. Relative Bruhat decomposition and relative Weyl group 1. Overview Let G be a connected reductive group over a field k, S a maximal k-split torus in G, and P a minimal parabolic k-subgroup of

More information

0 t in the 1 SL2 -case and a (t) = ( )

0 t in the 1 SL2 -case and a (t) = ( ) Math 249B. Root datum for split reductive groups The link between (split) connected reductive groups and combinatorial objects called root data was first discovered in the theory of compact Lie groups

More information

where Σ is a finite discrete Gal(K sep /K)-set unramified along U and F s is a finite Gal(k(s) sep /k(s))-subset

where Σ is a finite discrete Gal(K sep /K)-set unramified along U and F s is a finite Gal(k(s) sep /k(s))-subset Classification of quasi-finite étale separated schemes As we saw in lecture, Zariski s Main Theorem provides a very visual picture of quasi-finite étale separated schemes X over a henselian local ring

More information

Math 210C. Weyl groups and character lattices

Math 210C. Weyl groups and character lattices Math 210C. Weyl groups and character lattices 1. Introduction Let G be a connected compact Lie group, and S a torus in G (not necessarily maximal). In class we saw that the centralizer Z G (S) of S in

More information

ON THE CLASSIFICATION OF RANK 1 GROUPS OVER NON ARCHIMEDEAN LOCAL FIELDS

ON THE CLASSIFICATION OF RANK 1 GROUPS OVER NON ARCHIMEDEAN LOCAL FIELDS ON THE CLASSIFICATION OF RANK 1 GROUPS OVER NON ARCHIMEDEAN LOCAL FIELDS LISA CARBONE Abstract. We outline the classification of K rank 1 groups over non archimedean local fields K up to strict isogeny,

More information

Cover Page. Author: Yan, Qijun Title: Adapted deformations and the Ekedahl-Oort stratifications of Shimura varieties Date:

Cover Page. Author: Yan, Qijun Title: Adapted deformations and the Ekedahl-Oort stratifications of Shimura varieties Date: Cover Page The handle http://hdl.handle.net/1887/56255 holds various files of this Leiden University dissertation Author: Yan, Qijun Title: Adapted deformations and the Ekedahl-Oort stratifications of

More information

Math 210C. A non-closed commutator subgroup

Math 210C. A non-closed commutator subgroup Math 210C. A non-closed commutator subgroup 1. Introduction In Exercise 3(i) of HW7 we saw that every element of SU(2) is a commutator (i.e., has the form xyx 1 y 1 for x, y SU(2)), so the same holds for

More information

Irreducible subgroups of algebraic groups

Irreducible subgroups of algebraic groups Irreducible subgroups of algebraic groups Martin W. Liebeck Department of Mathematics Imperial College London SW7 2BZ England Donna M. Testerman Department of Mathematics University of Lausanne Switzerland

More information

Parabolic subgroups Montreal-Toronto 2018

Parabolic subgroups Montreal-Toronto 2018 Parabolic subgroups Montreal-Toronto 2018 Alice Pozzi January 13, 2018 Alice Pozzi Parabolic subgroups Montreal-Toronto 2018 January 13, 2018 1 / 1 Overview Alice Pozzi Parabolic subgroups Montreal-Toronto

More information

Math 145. Codimension

Math 145. Codimension Math 145. Codimension 1. Main result and some interesting examples In class we have seen that the dimension theory of an affine variety (irreducible!) is linked to the structure of the function field in

More information

GEOMETRIC STRUCTURES OF SEMISIMPLE LIE ALGEBRAS

GEOMETRIC STRUCTURES OF SEMISIMPLE LIE ALGEBRAS GEOMETRIC STRUCTURES OF SEMISIMPLE LIE ALGEBRAS ANA BALIBANU DISCUSSED WITH PROFESSOR VICTOR GINZBURG 1. Introduction The aim of this paper is to explore the geometry of a Lie algebra g through the action

More information

Math 249B. Standard parabolic subgroups: theory and examples 1. Introduction Let G be a connected reductive k-group, P 0 a minimal parabolic

Math 249B. Standard parabolic subgroups: theory and examples 1. Introduction Let G be a connected reductive k-group, P 0 a minimal parabolic Math 249B. Standard parabolic subgroups: theory and examples 1. Introduction Let G be a connected reductive k-group, P 0 a minimal parabolic k-subgroup, and k the basis of k Φ := Φ(G, S) corresponding

More information

Definitions. Notations. Injective, Surjective and Bijective. Divides. Cartesian Product. Relations. Equivalence Relations

Definitions. Notations. Injective, Surjective and Bijective. Divides. Cartesian Product. Relations. Equivalence Relations Page 1 Definitions Tuesday, May 8, 2018 12:23 AM Notations " " means "equals, by definition" the set of all real numbers the set of integers Denote a function from a set to a set by Denote the image of

More information

PART I: GEOMETRY OF SEMISIMPLE LIE ALGEBRAS

PART I: GEOMETRY OF SEMISIMPLE LIE ALGEBRAS PART I: GEOMETRY OF SEMISIMPLE LIE ALGEBRAS Contents 1. Regular elements in semisimple Lie algebras 1 2. The flag variety and the Bruhat decomposition 3 3. The Grothendieck-Springer resolution 6 4. The

More information

12. Projective modules The blanket assumptions about the base ring k, the k-algebra A, and A-modules enumerated at the start of 11 continue to hold.

12. Projective modules The blanket assumptions about the base ring k, the k-algebra A, and A-modules enumerated at the start of 11 continue to hold. 12. Projective modules The blanket assumptions about the base ring k, the k-algebra A, and A-modules enumerated at the start of 11 continue to hold. 12.1. Indecomposability of M and the localness of End

More information

Representations and Linear Actions

Representations and Linear Actions Representations and Linear Actions Definition 0.1. Let G be an S-group. A representation of G is a morphism of S-groups φ G GL(n, S) for some n. We say φ is faithful if it is a monomorphism (in the category

More information

Geometric Structure and the Local Langlands Conjecture

Geometric Structure and the Local Langlands Conjecture Geometric Structure and the Local Langlands Conjecture Paul Baum Penn State Representations of Reductive Groups University of Utah, Salt Lake City July 9, 2013 Paul Baum (Penn State) Geometric Structure

More information

SYMPLECTIC GEOMETRY: LECTURE 5

SYMPLECTIC GEOMETRY: LECTURE 5 SYMPLECTIC GEOMETRY: LECTURE 5 LIAT KESSLER Let (M, ω) be a connected compact symplectic manifold, T a torus, T M M a Hamiltonian action of T on M, and Φ: M t the assoaciated moment map. Theorem 0.1 (The

More information

ALGEBRAIC GROUPS J. WARNER

ALGEBRAIC GROUPS J. WARNER ALGEBRAIC GROUPS J. WARNER Let k be an algebraically closed field. varieties unless otherwise stated. 1. Definitions and Examples For simplicity we will work strictly with affine Definition 1.1. An algebraic

More information

arxiv: v1 [math.rt] 14 Nov 2007

arxiv: v1 [math.rt] 14 Nov 2007 arxiv:0711.2128v1 [math.rt] 14 Nov 2007 SUPPORT VARIETIES OF NON-RESTRICTED MODULES OVER LIE ALGEBRAS OF REDUCTIVE GROUPS: CORRIGENDA AND ADDENDA ALEXANDER PREMET J. C. Jantzen informed me that the proof

More information

MATH 8253 ALGEBRAIC GEOMETRY WEEK 12

MATH 8253 ALGEBRAIC GEOMETRY WEEK 12 MATH 8253 ALGEBRAIC GEOMETRY WEEK 2 CİHAN BAHRAN 3.2.. Let Y be a Noetherian scheme. Show that any Y -scheme X of finite type is Noetherian. Moreover, if Y is of finite dimension, then so is X. Write f

More information

is maximal in G k /R u (G k

is maximal in G k /R u (G k Algebraic Groups I. Grothendieck s theorem on tori (based on notes by S. Lichtenstein) 1. Introduction In the early days of the theory of linear algebraic groups, the ground field was assumed to be algebraically

More information

L(C G (x) 0 ) c g (x). Proof. Recall C G (x) = {g G xgx 1 = g} and c g (x) = {X g Ad xx = X}. In general, it is obvious that

L(C G (x) 0 ) c g (x). Proof. Recall C G (x) = {g G xgx 1 = g} and c g (x) = {X g Ad xx = X}. In general, it is obvious that ALGEBRAIC GROUPS 61 5. Root systems and semisimple Lie algebras 5.1. Characteristic 0 theory. Assume in this subsection that chark = 0. Let me recall a couple of definitions made earlier: G is called reductive

More information

SYMMETRIC SUBGROUP ACTIONS ON ISOTROPIC GRASSMANNIANS

SYMMETRIC SUBGROUP ACTIONS ON ISOTROPIC GRASSMANNIANS 1 SYMMETRIC SUBGROUP ACTIONS ON ISOTROPIC GRASSMANNIANS HUAJUN HUANG AND HONGYU HE Abstract. Let G be the group preserving a nondegenerate sesquilinear form B on a vector space V, and H a symmetric subgroup

More information

GEOMETRIC SATAKE, SPRINGER CORRESPONDENCE, AND SMALL REPRESENTATIONS

GEOMETRIC SATAKE, SPRINGER CORRESPONDENCE, AND SMALL REPRESENTATIONS GEOMETRIC SATAKE, SPRINGER CORRESPONDENCE, AND SMALL REPRESENTATIONS PRAMOD N. ACHAR AND ANTHONY HENDERSON Abstract. For a simply-connected simple algebraic group G over C, we exhibit a subvariety of its

More information

Math 210C. The representation ring

Math 210C. The representation ring Math 210C. The representation ring 1. Introduction Let G be a nontrivial connected compact Lie group that is semisimple and simply connected (e.g., SU(n) for n 2, Sp(n) for n 1, or Spin(n) for n 3). Let

More information

Math 121 Homework 5: Notes on Selected Problems

Math 121 Homework 5: Notes on Selected Problems Math 121 Homework 5: Notes on Selected Problems 12.1.2. Let M be a module over the integral domain R. (a) Assume that M has rank n and that x 1,..., x n is any maximal set of linearly independent elements

More information

On exceptional completions of symmetric varieties

On exceptional completions of symmetric varieties Journal of Lie Theory Volume 16 (2006) 39 46 c 2006 Heldermann Verlag On exceptional completions of symmetric varieties Rocco Chirivì and Andrea Maffei Communicated by E. B. Vinberg Abstract. Let G be

More information

ON BASE SIZES FOR ALGEBRAIC GROUPS

ON BASE SIZES FOR ALGEBRAIC GROUPS ON BASE SIZES FOR ALGEBRAIC GROUPS TIMOTHY C. BURNESS, ROBERT M. GURALNICK, AND JAN SAXL Abstract. For an algebraic group G and a closed subgroup H, the base size of G on the coset variety of H in G is

More information

ALGEBRAIC GROUPS: PART IV

ALGEBRAIC GROUPS: PART IV ALGEBRAIC GROUPS: PART IV EYAL Z. GOREN, MCGILL UNIVERSITY Contents 11. Quotients 60 11.1. Some general comments 60 11.2. The quotient of a linear group by a subgroup 61 12. Parabolic subgroups, Borel

More information

Subgroups of Linear Algebraic Groups

Subgroups of Linear Algebraic Groups Subgroups of Linear Algebraic Groups Subgroups of Linear Algebraic Groups Contents Introduction 1 Acknowledgements 4 1. Basic definitions and examples 5 1.1. Introduction to Linear Algebraic Groups 5 1.2.

More information

Introduction to Arithmetic Geometry Fall 2013 Lecture #18 11/07/2013

Introduction to Arithmetic Geometry Fall 2013 Lecture #18 11/07/2013 18.782 Introduction to Arithmetic Geometry Fall 2013 Lecture #18 11/07/2013 As usual, all the rings we consider are commutative rings with an identity element. 18.1 Regular local rings Consider a local

More information

0 A. ... A j GL nj (F q ), 1 j r

0 A. ... A j GL nj (F q ), 1 j r CHAPTER 4 Representations of finite groups of Lie type Let F q be a finite field of order q and characteristic p. Let G be a finite group of Lie type, that is, G is the F q -rational points of a connected

More information

LECTURE 11: SOERGEL BIMODULES

LECTURE 11: SOERGEL BIMODULES LECTURE 11: SOERGEL BIMODULES IVAN LOSEV Introduction In this lecture we continue to study the category O 0 and explain some ideas towards the proof of the Kazhdan-Lusztig conjecture. We start by introducing

More information

TITS SYSTEMS, PARABOLIC SUBGROUPS, PARABOLIC SUBALGEBRAS. Alfred G. Noël Department of Mathematics Northeastern University Boston, MA

TITS SYSTEMS, PARABOLIC SUBGROUPS, PARABOLIC SUBALGEBRAS. Alfred G. Noël Department of Mathematics Northeastern University Boston, MA TITS SYSTEMS, PARABOLIC SUBGROUPS, PARABOLIC SUBALGEBRAS Alfred G. Noël Department of Mathematics Northeastern University Boston, MA In this paper we give a brief survey of the basic results on B-N pairs

More information

On lengths on semisimple groups

On lengths on semisimple groups On lengths on semisimple groups Yves de Cornulier May 21, 2009 Abstract We prove that every length on a simple group over a locally compact field, is either bounded or proper. 1 Introduction Let G be a

More information

ELEMENTARY SUBALGEBRAS OF RESTRICTED LIE ALGEBRAS

ELEMENTARY SUBALGEBRAS OF RESTRICTED LIE ALGEBRAS ELEMENTARY SUBALGEBRAS OF RESTRICTED LIE ALGEBRAS J. WARNER SUMMARY OF A PAPER BY J. CARLSON, E. FRIEDLANDER, AND J. PEVTSOVA, AND FURTHER OBSERVATIONS 1. The Nullcone and Restricted Nullcone We will need

More information

Math 676. A compactness theorem for the idele group. and by the product formula it lies in the kernel (A K )1 of the continuous idelic norm

Math 676. A compactness theorem for the idele group. and by the product formula it lies in the kernel (A K )1 of the continuous idelic norm Math 676. A compactness theorem for the idele group 1. Introduction Let K be a global field, so K is naturally a discrete subgroup of the idele group A K and by the product formula it lies in the kernel

More information

On a question of B.H. Neumann

On a question of B.H. Neumann On a question of B.H. Neumann Robert Guralnick Department of Mathematics University of Southern California E-mail: guralnic@math.usc.edu Igor Pak Department of Mathematics Massachusetts Institute of Technology

More information

16.2. Definition. Let N be the set of all nilpotent elements in g. Define N

16.2. Definition. Let N be the set of all nilpotent elements in g. Define N 74 16. Lecture 16: Springer Representations 16.1. The flag manifold. Let G = SL n (C). It acts transitively on the set F of complete flags 0 F 1 F n 1 C n and the stabilizer of the standard flag is the

More information

10. The subgroup subalgebra correspondence. Homogeneous spaces.

10. The subgroup subalgebra correspondence. Homogeneous spaces. 10. The subgroup subalgebra correspondence. Homogeneous spaces. 10.1. The concept of a Lie subgroup of a Lie group. We have seen that if G is a Lie group and H G a subgroup which is at the same time a

More information

Tamagawa Numbers in the Function Field Case (Lecture 2)

Tamagawa Numbers in the Function Field Case (Lecture 2) Tamagawa Numbers in the Function Field Case (Lecture 2) February 5, 204 In the previous lecture, we defined the Tamagawa measure associated to a connected semisimple algebraic group G over the field Q

More information

CHEVALLEY S THEOREM AND COMPLETE VARIETIES

CHEVALLEY S THEOREM AND COMPLETE VARIETIES CHEVALLEY S THEOREM AND COMPLETE VARIETIES BRIAN OSSERMAN In this note, we introduce the concept which plays the role of compactness for varieties completeness. We prove that completeness can be characterized

More information

Reducibility of generic unipotent standard modules

Reducibility of generic unipotent standard modules Journal of Lie Theory Volume?? (??)???? c?? Heldermann Verlag 1 Version of March 10, 011 Reducibility of generic unipotent standard modules Dan Barbasch and Dan Ciubotaru Abstract. Using Lusztig s geometric

More information

LECTURE 4: REPRESENTATION THEORY OF SL 2 (F) AND sl 2 (F)

LECTURE 4: REPRESENTATION THEORY OF SL 2 (F) AND sl 2 (F) LECTURE 4: REPRESENTATION THEORY OF SL 2 (F) AND sl 2 (F) IVAN LOSEV In this lecture we will discuss the representation theory of the algebraic group SL 2 (F) and of the Lie algebra sl 2 (F), where F is

More information

Math 210C. Size of fundamental group

Math 210C. Size of fundamental group Math 210C. Size of fundamental group 1. Introduction Let (V, Φ) be a nonzero root system. Let G be a connected compact Lie group that is semisimple (equivalently, Z G is finite, or G = G ; see Exercise

More information

(dim Z j dim Z j 1 ) 1 j i

(dim Z j dim Z j 1 ) 1 j i Math 210B. Codimension 1. Main result and some interesting examples Let k be a field, and A a domain finitely generated k-algebra. In class we have seen that the dimension theory of A is linked to the

More information

V (v i + W i ) (v i + W i ) is path-connected and hence is connected.

V (v i + W i ) (v i + W i ) is path-connected and hence is connected. Math 396. Connectedness of hyperplane complements Note that the complement of a point in R is disconnected and the complement of a (translated) line in R 2 is disconnected. Quite generally, we claim that

More information

BRUHAT-TITS BUILDING OF A p-adic REDUCTIVE GROUP

BRUHAT-TITS BUILDING OF A p-adic REDUCTIVE GROUP Trends in Mathematics Information Center for Mathematical Sciences Volume 4, Number 1, June 2001, Pages 71 75 BRUHAT-TITS BUILDING OF A p-adic REDUCTIVE GROUP HI-JOON CHAE Abstract. A Bruhat-Tits building

More information

Math 250A, Fall 2004 Problems due October 5, 2004 The problems this week were from Lang s Algebra, Chapter I.

Math 250A, Fall 2004 Problems due October 5, 2004 The problems this week were from Lang s Algebra, Chapter I. Math 250A, Fall 2004 Problems due October 5, 2004 The problems this week were from Lang s Algebra, Chapter I. 24. We basically know already that groups of order p 2 are abelian. Indeed, p-groups have non-trivial

More information

Math 210B. Artin Rees and completions

Math 210B. Artin Rees and completions Math 210B. Artin Rees and completions 1. Definitions and an example Let A be a ring, I an ideal, and M an A-module. In class we defined the I-adic completion of M to be M = lim M/I n M. We will soon show

More information

k k would be reducible. But the zero locus of f in A n+1

k k would be reducible. But the zero locus of f in A n+1 Math 145. Bezout s Theorem Let be an algebraically closed field. The purpose of this handout is to prove Bezout s Theorem and some related facts of general interest in projective geometry that arise along

More information

Category O and its basic properties

Category O and its basic properties Category O and its basic properties Daniil Kalinov 1 Definitions Let g denote a semisimple Lie algebra over C with fixed Cartan and Borel subalgebras h b g. Define n = α>0 g α, n = α

More information

Notes on p-divisible Groups

Notes on p-divisible Groups Notes on p-divisible Groups March 24, 2006 This is a note for the talk in STAGE in MIT. The content is basically following the paper [T]. 1 Preliminaries and Notations Notation 1.1. Let R be a complete

More information

ON DISCRETE SUBGROUPS CONTAINING A LATTICE IN A HOROSPHERICAL SUBGROUP HEE OH. 1. Introduction

ON DISCRETE SUBGROUPS CONTAINING A LATTICE IN A HOROSPHERICAL SUBGROUP HEE OH. 1. Introduction ON DISCRETE SUBGROUPS CONTAINING A LATTICE IN A HOROSPHERICAL SUBGROUP HEE OH Abstract. We showed in [Oh] that for a simple real Lie group G with real rank at least 2, if a discrete subgroup Γ of G contains

More information

HARTSHORNE EXERCISES

HARTSHORNE EXERCISES HARTSHORNE EXERCISES J. WARNER Hartshorne, Exercise I.5.6. Blowing Up Curve Singularities (a) Let Y be the cusp x 3 = y 2 + x 4 + y 4 or the node xy = x 6 + y 6. Show that the curve Ỹ obtained by blowing

More information

Definition 9.1. The scheme µ 1 (O)/G is called the Hamiltonian reduction of M with respect to G along O. We will denote by R(M, G, O).

Definition 9.1. The scheme µ 1 (O)/G is called the Hamiltonian reduction of M with respect to G along O. We will denote by R(M, G, O). 9. Calogero-Moser spaces 9.1. Hamiltonian reduction along an orbit. Let M be an affine algebraic variety and G a reductive algebraic group. Suppose M is Poisson and the action of G preserves the Poisson

More information

A NEW APPROACH TO UNRAMIFIED DESCENT IN BRUHAT-TITS THEORY. By Gopal Prasad

A NEW APPROACH TO UNRAMIFIED DESCENT IN BRUHAT-TITS THEORY. By Gopal Prasad A NEW APPROACH TO UNRAMIFIED DESCENT IN BRUHAT-TITS THEORY By Gopal Prasad Dedicated to my wife Indu Prasad with gratitude Abstract. We give a new approach to unramified descent in Bruhat-Tits theory of

More information

Introduction to Arithmetic Geometry Fall 2013 Lecture #17 11/05/2013

Introduction to Arithmetic Geometry Fall 2013 Lecture #17 11/05/2013 18.782 Introduction to Arithmetic Geometry Fall 2013 Lecture #17 11/05/2013 Throughout this lecture k denotes an algebraically closed field. 17.1 Tangent spaces and hypersurfaces For any polynomial f k[x

More information

Math 594. Solutions 5

Math 594. Solutions 5 Math 594. Solutions 5 Book problems 6.1: 7. Prove that subgroups and quotient groups of nilpotent groups are nilpotent (your proof should work for infinite groups). Give an example of a group G which possesses

More information

Lemma 1.1. The field K embeds as a subfield of Q(ζ D ).

Lemma 1.1. The field K embeds as a subfield of Q(ζ D ). Math 248A. Quadratic characters associated to quadratic fields The aim of this handout is to describe the quadratic Dirichlet character naturally associated to a quadratic field, and to express it in terms

More information

Rings and groups. Ya. Sysak

Rings and groups. Ya. Sysak Rings and groups. Ya. Sysak 1 Noetherian rings Let R be a ring. A (right) R -module M is called noetherian if it satisfies the maximum condition for its submodules. In other words, if M 1... M i M i+1...

More information

Thus, the integral closure A i of A in F i is a finitely generated (and torsion-free) A-module. It is not a priori clear if the A i s are locally

Thus, the integral closure A i of A in F i is a finitely generated (and torsion-free) A-module. It is not a priori clear if the A i s are locally Math 248A. Discriminants and étale algebras Let A be a noetherian domain with fraction field F. Let B be an A-algebra that is finitely generated and torsion-free as an A-module with B also locally free

More information

ALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 2: HILBERT S NULLSTELLENSATZ.

ALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 2: HILBERT S NULLSTELLENSATZ. ALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 2: HILBERT S NULLSTELLENSATZ. ANDREW SALCH 1. Hilbert s Nullstellensatz. The last lecture left off with the claim that, if J k[x 1,..., x n ] is an ideal, then

More information

Reductive subgroup schemes of a parahoric group scheme

Reductive subgroup schemes of a parahoric group scheme Reductive subgroup schemes of a parahoric group scheme George McNinch Department of Mathematics Tufts University Medford Massachusetts USA June 2018 Contents 1 Levi factors 2 Parahoric group schemes 3

More information

Lie Algebras. Shlomo Sternberg

Lie Algebras. Shlomo Sternberg Lie Algebras Shlomo Sternberg March 8, 2004 2 Chapter 5 Conjugacy of Cartan subalgebras It is a standard theorem in linear algebra that any unitary matrix can be diagonalized (by conjugation by unitary

More information

The Grothendieck-Katz Conjecture for certain locally symmetric varieties

The Grothendieck-Katz Conjecture for certain locally symmetric varieties The Grothendieck-Katz Conjecture for certain locally symmetric varieties Benson Farb and Mark Kisin August 20, 2008 Abstract Using Margulis results on lattices in semi-simple Lie groups, we prove the Grothendieck-

More information

4. Images of Varieties Given a morphism f : X Y of quasi-projective varieties, a basic question might be to ask what is the image of a closed subset

4. Images of Varieties Given a morphism f : X Y of quasi-projective varieties, a basic question might be to ask what is the image of a closed subset 4. Images of Varieties Given a morphism f : X Y of quasi-projective varieties, a basic question might be to ask what is the image of a closed subset Z X. Replacing X by Z we might as well assume that Z

More information

On the Irreducibility of the Commuting Variety of the Symmetric Pair so p+2, so p so 2

On the Irreducibility of the Commuting Variety of the Symmetric Pair so p+2, so p so 2 Journal of Lie Theory Volume 16 (2006) 57 65 c 2006 Heldermann Verlag On the Irreducibility of the Commuting Variety of the Symmetric Pair so p+2, so p so 2 Hervé Sabourin and Rupert W.T. Yu Communicated

More information

SOME GOOD-FILTRATION SUBGROUPS OF SIMPLE ALGEBRAIC GROUPS CHUCK HAGUE AND GEORGE MCNINCH

SOME GOOD-FILTRATION SUBGROUPS OF SIMPLE ALGEBRAIC GROUPS CHUCK HAGUE AND GEORGE MCNINCH SOME GOOD-FILTRATION SUBGROUPS OF SIMPLE ALGEBRAIC GROUPS CUCK AGUE AND GEORGE MCNINC ABSTRACT. Let G be a connected and reductive algebraic group over an algebraically closed field of characteristic p

More information

Chow rings of Complex Algebraic Groups

Chow rings of Complex Algebraic Groups Chow rings of Complex Algebraic Groups Shizuo Kaji joint with Masaki Nakagawa Workshop on Schubert calculus 2008 at Kansai Seminar House Mar. 20, 2008 Outline Introduction Our problem (algebraic geometory)

More information

arxiv:math/ v1 [math.rt] 1 Jul 1996

arxiv:math/ v1 [math.rt] 1 Jul 1996 SUPERRIGID SUBGROUPS OF SOLVABLE LIE GROUPS DAVE WITTE arxiv:math/9607221v1 [math.rt] 1 Jul 1996 Abstract. Let Γ be a discrete subgroup of a simply connected, solvable Lie group G, such that Ad G Γ has

More information

Math 429/581 (Advanced) Group Theory. Summary of Definitions, Examples, and Theorems by Stefan Gille

Math 429/581 (Advanced) Group Theory. Summary of Definitions, Examples, and Theorems by Stefan Gille Math 429/581 (Advanced) Group Theory Summary of Definitions, Examples, and Theorems by Stefan Gille 1 2 0. Group Operations 0.1. Definition. Let G be a group and X a set. A (left) operation of G on X is

More information

Maximal non-commuting subsets of groups

Maximal non-commuting subsets of groups Maximal non-commuting subsets of groups Umut Işık March 29, 2005 Abstract Given a finite group G, we consider the problem of finding the maximal size nc(g) of subsets of G that have the property that no

More information

EKT of Some Wonderful Compactifications

EKT of Some Wonderful Compactifications EKT of Some Wonderful Compactifications and recent results on Complete Quadrics. (Based on joint works with Soumya Banerjee and Michael Joyce) Mahir Bilen Can April 16, 2016 Mahir Bilen Can EKT of Some

More information

Essays on the structure of reductive groups. Root systems. X (A) = Hom(A, G m ) t t t t i,

Essays on the structure of reductive groups. Root systems. X (A) = Hom(A, G m ) t t t t i, 9:51 p.m. November 11, 2006 Essays on the structure of reductive groups Root systems Bill Casselman University of British Columbia cass@math.ubc.ca Suppose G to be a reductive group defined over an algebraically

More information

On the centralizer of a regular, semi-simple, stable conjugacy class. Benedict H. Gross

On the centralizer of a regular, semi-simple, stable conjugacy class. Benedict H. Gross On the centralizer of a regular, semi-simple, stable conjugacy class Benedict H. Gross Let k be a field, and let G be a semi-simple, simply-connected algebraic group, which is quasi-split over k. The theory

More information

Characteristic classes in the Chow ring

Characteristic classes in the Chow ring arxiv:alg-geom/9412008v1 10 Dec 1994 Characteristic classes in the Chow ring Dan Edidin and William Graham Department of Mathematics University of Chicago Chicago IL 60637 Let G be a reductive algebraic

More information

Notes 10: Consequences of Eli Cartan s theorem.

Notes 10: Consequences of Eli Cartan s theorem. Notes 10: Consequences of Eli Cartan s theorem. Version 0.00 with misprints, The are a few obvious, but important consequences of the theorem of Eli Cartan on the maximal tori. The first one is the observation

More information

arxiv:math/ v1 [math.ag] 23 Oct 2006

arxiv:math/ v1 [math.ag] 23 Oct 2006 IS THE LUNA STRATIFICATION INTRINSIC? arxiv:math/0610669v1 [math.ag] 23 Oct 2006 J. KUTTLER AND Z. REICHSTEIN Abstract. Let G GL(V ) will be a finite-dimensional linear representation of a reductive linear

More information

Bruhat order, rationally smooth Schubert varieties, and hyperplane arrangements

Bruhat order, rationally smooth Schubert varieties, and hyperplane arrangements FPSAC 2010, San Francisco, USA DMTCS proc. AN, 2010, 833 840 Bruhat order, rationally smooth Schubert varieties, and hyperplane arrangements Suho Oh 1 and Hwanchul Yoo Department of Mathematics, Massachusetts

More information

FALTINGS SEMINAR TALK. (rf)(a) = rf(a) = f(ra)

FALTINGS SEMINAR TALK. (rf)(a) = rf(a) = f(ra) FALTINGS SEMINAR TALK SIMON RUBINSTEIN-SALZEDO 1. Cartier duality Let R be a commutative ring, and let G = Spec A be a commutative group scheme, where A is a finite flat algebra over R. Let A = Hom R mod

More information

INVARIANT PROBABILITIES ON PROJECTIVE SPACES. 1. Introduction

INVARIANT PROBABILITIES ON PROJECTIVE SPACES. 1. Introduction INVARIANT PROBABILITIES ON PROJECTIVE SPACES YVES DE CORNULIER Abstract. Let K be a local field. We classify the linear groups G GL(V ) that preserve an probability on the Borel subsets of the projective

More information

ALGEBRAIC GROUPS JEROEN SIJSLING

ALGEBRAIC GROUPS JEROEN SIJSLING ALGEBRAIC GROUPS JEROEN SIJSLING The goal of these notes is to introduce and motivate some notions from the theory of group schemes. For the sake of simplicity, we restrict to algebraic groups (as defined

More information

6 Orthogonal groups. O 2m 1 q. q 2i 1 q 2i. 1 i 1. 1 q 2i 2. O 2m q. q m m 1. 1 q 2i 1 i 1. 1 q 2i. i 1. 2 q 1 q i 1 q i 1. m 1.

6 Orthogonal groups. O 2m 1 q. q 2i 1 q 2i. 1 i 1. 1 q 2i 2. O 2m q. q m m 1. 1 q 2i 1 i 1. 1 q 2i. i 1. 2 q 1 q i 1 q i 1. m 1. 6 Orthogonal groups We now turn to the orthogonal groups. These are more difficult, for two related reasons. First, it is not always true that the group of isometries with determinant 1 is equal to its

More information

On the Self-dual Representations of a p-adic Group

On the Self-dual Representations of a p-adic Group IMRN International Mathematics Research Notices 1999, No. 8 On the Self-dual Representations of a p-adic Group Dipendra Prasad In an earlier paper [P1], we studied self-dual complex representations of

More information

Half the sum of positive roots, the Coxeter element, and a theorem of Kostant

Half the sum of positive roots, the Coxeter element, and a theorem of Kostant DOI 10.1515/forum-2014-0052 Forum Math. 2014; aop Research Article Dipendra Prasad Half the sum of positive roots, the Coxeter element, and a theorem of Kostant Abstract: Interchanging the character and

More information

Stab(t) = {h G h t = t} = {h G h (g s) = g s} = {h G (g 1 hg) s = s} = g{k G k s = s} g 1 = g Stab(s)g 1.

Stab(t) = {h G h t = t} = {h G h (g s) = g s} = {h G (g 1 hg) s = s} = g{k G k s = s} g 1 = g Stab(s)g 1. 1. Group Theory II In this section we consider groups operating on sets. This is not particularly new. For example, the permutation group S n acts on the subset N n = {1, 2,...,n} of N. Also the group

More information

Groups of Prime Power Order with Derived Subgroup of Prime Order

Groups of Prime Power Order with Derived Subgroup of Prime Order Journal of Algebra 219, 625 657 (1999) Article ID jabr.1998.7909, available online at http://www.idealibrary.com on Groups of Prime Power Order with Derived Subgroup of Prime Order Simon R. Blackburn*

More information