Comparison of the Weyl integration formula and the equivariant localization formula for a maximal torus of Sp(1)
|
|
- Elijah Robertson
- 5 years ago
- Views:
Transcription
1 Comparison of the Weyl integration formula and the equivariant localization formula for a maximal torus of Sp() Jeffrey D. Carlson January 3, 4 Introduction In this note, we aim to prove the Weyl integration formula for the Lie group Sp() using the Berline Vergne/Atiyah Bott equivariant localization formula. We do not quite succeed. The ideas are due to Loring Tu, and the execution to me. The first four sections are generalities, mostly copied from an earlier document, which should probably be skimmed, and the attempted application to Sp() follows. The Weyl integral formula Let G be a compact, connected Lie group, T its maximal torus, and W the Weyl group. Write Ad g/t : T Aut(g/t) for the adjoint action of T on its Lie algebra modulo the Cartan subalgebra, and : T R for the function ( [ g/t) (t) := det id Ad(t ). If f : G R is a continuous function on G, and dg and dt are Haar measures on G and T respectively, and d(gt) the induced probability measure on G/T, then f (g) dg = [ (t) f (gtg ) d(gt) dt. G W T G/T If f is a class function, meaning it takes one value on each conjugacy class (so that for all g, h G we have f (ghg ) = f (h)), the bracketed integral reduces to f (t), so that G f (g) dg = (t) f (t) dt. W T Note further that if Φ is the set of global roots of G, and Φ + the set of positive roots, then ( = α ) ( = α )( α ). α Φ α Φ +
2 3 The equivariant localization theorem Let T be a torus and M a T-manifold. If X t is a vector in the Lie algebra of T and X Γ(TM) its associated fundamental vector field, write i N for the inclusions into M connected components N of the zero set of X, write ν for the normal bundle to N, and e T (ν) for the T-equivariant Euler class of each ν. Let the orientation on N be induced by that on ν. If α Ω T (M) is an equivariantly closed form, then i α(x) = ) N ( α(x) M N N e T (ν)(x). It would be appropriate in this connection to state what equivariant extensions are. An equivariant extension of an m-form ω Ω (M) is an element ω of the Cartan model ( S(t ) Ω(M) ) T (basically a T-invariant polynomial in a basis u,..., u n for t with coefficients in Ω (M)), of the same degree as ω, and such that the constant term is ω. Here the degree of elements of t is, so we can write ω = ω + I ω I u I, where I = (i,..., i n ) N n is a multiindex, u I = u i u i n has degree I = i k, and ω I Ω m I (M). The integral of such a form is M ω = I u I M ω I; since these summands will vanish unless deg ω I = m I = dim M, this integration will return an element of degree I in S(t ), or in other words a homogeneous degree- I polynomial map t R. In particular, if deg ω = dim M, then M ω(x) = M ω is a real number. It should be independent of X, subject to some genericity condition. So the equivariant localization formula, applied to equivariant extensions of top forms, gives another way to integrate functions on a manifold, assuming those functions have equivariantly closed equivariant extensions. A T-manifold M is called equivariantly formal if all closed forms on M have equivariantly closed extensions, and it is known that T is equivariantly formal under this conjugation action. In the situation we are interested in, the manifold M is a connected, compact Lie group G and the group action we are concerned with is that of a maximal torus T. The fixed points under conjugation by a subgroup are the group s centralizer in this case, the torus T itself. Let f : G R be a smooth function, invariant under conjugation by T. Then ω = f dg is a closed T-invariant m-form on G, so since G is equivariantly formal under conjugation by T, there exists an equivariant extension ω. Putting together the two formulae, we have [ (t) f (gtg ω(x) ) d(gt) dt = f (g) dg = ω = ω(x) = T W T G/T G G G T e T (ν)(x), Like that exp(x) topologically generates a maximal torus of G where e T (ν) is the equivariant Euler class of the normal bundle to T in G; what we would like is some way to see that the left-hand side is the same as the right without the intermediate steps.
3 4 The Equivariant Euler Class e T (ν) The normal bundle ν of T in G fits in the exact sequence T(T) (TG) T ν, and so is a vector bundle of (even) rank m rk T over T, with fiber g/t. We claim it is a trivial bundle, which will follow if (TG) T is a trivial bundle in such a way that T(T) is a trivial subbundle. But TG is parallelizable for any Lie group G: each element of T g G is a left translation (l g ) X for precisely one X g, and by continuity (g, X) (l g ) X is a vector bundle isomorphism G g TG. So ν = T (g/t). Now we want to see how the conjugation action acts on T (g/t) under this identification. For this purpose, we first work in G g, and see what Ad does to TG. Start with (g, X) G g. Write c h : g hgh for conjugation by h G, l h for left multiplication and r h for right multiplication, so c h = l h r h = r h l h. Then (c h ) (l g ) X = (r h ) (l h ) (l g ) X = (r h ) [ (lh ) (l g ) (l h ) (lh ) X = (r h ) (l ch (g)) (l h ) X = (l ch (g)) (r h ) (l h ) X = (l ch (g)) (c h ) X, so we have the diagram (l g ) X (l ch (g)) (c h ) X (g, X) ( ch (g), (c h ) X ), Since conjugation by T on T fixes each fiber, it induces the Ad-action on each fiber of the bundle (TG) T, and so on each g/t fiber of ν = T (g/t). As the conjugation action preserves each fiber, we can factor T out of the homotopy quotient: ν T = (ET ν)/t = ( ET T (g/t) ) /T = T (ET (g/t) ) /T = T (g/t) T, To see continuity of the inverse, given a basis X i of g, the left-invariant vector fields g (l g ) X i form a frame of TG, so that a i (l g ) X i ( g, (a,..., a dim G ) ). The projection TG G is continuous by definition, and this coordinate map a i (l g ) X i (a,..., a dim G ) R dim G is locally continuous over the trivializing neighborhoods U for TG that we already know exist since the left-invariant vector fields are continuous. 3
4 so ν T is a product bundle over T with fiber (g/t) T, which we must further analyze. In particular, it is the pullback of the vector bundle (g/t) T BT under the projection π : T BT BT, which induces a tensor-factor inclusion π : H (BT) H (T) H (BT) in cohomology. Now under the Ad-action, g/t naturally decomposes as a direct sum of two-dimensional representations of T corresponding to the positive roots α of G, which we will view as onedimensional complex representations C α. Each homotopy quotient S α := (C α ) T is then a line bundle over BT, and (g/t) T = α Φ + S α. Since the total Chern class is multiplicative, the Chern total classes of the line bundles S α are all + c (S α ), and the top Chern class of a complex vector bundle is the Euler class of its realization, the equivariant Euler class of ν is e T (ν) = e ( π (g/t) T ) = π ( e ( (g/t) T ) ) = e ( (g/t) T ) = c n rk T 5 Application to Sp() We want to demystify the formula ( α Φ + S α [ (t) f (gtg ) d(gt) dt = W T Sp()/T Sp() f (g) dg = ( f dg)(x) T T e T (ν)(x) ) Cite DFAT. = α Φ + c (S α ) H n rk T (BT). by finding T, W,, dg, dt, d(gt), W, X, f dg, and e T (ν). The compact Lie group Sp() can be seen as the multiplicative subgroup of quaternions of norm. The exponential formula e q := n N q n /n! defines a function on quaternions as well as on complex numbers; we will use this fact and the linearity of conjugation to simplify our identification of the adjoint representation and the Euler class. 5. sp() and T and t and X The standard basis, i, j, k of the quaternions H defines a vector space decomposition into the direct sum of the real part R spanned by and the purely imaginary part Im (H) spanned by i, j, k. Since i, j, k anticommute, if u Im (H) and u = uū =, then u = and u 3 = u and u 4 = and so on, so that e tu = cos t + u sin t for real t. Since any pure imaginary is tu for some real t and unit-norm u Im (H), and any quaternion of unit norm can be written as cos t + u sin t, it follows that the quaternionic exponential is a surjection Im (H) Sp(), and since d dt etu t= = sin + u cos = u, we have a natural identification sp() Im (H). We will consider the conjugation action by the maximal torus T = e ir = { cos t + i sin t : t [, π) }. Then t = ir and we should pick a nonzero X exp {}; we will take X = πi. 4
5 5. Ad and and Φ + and W We can write any q Sp() as z + jz for z, z C := R + ir. We consider the effect of conjugation c t by t = e iθ T on these coordinates. Since i commutes with C and anticommutes with jc, we have iji = j( i) = j, so for z C we have zjz = j zz and for e iθ T we get e iθ (z + jz )e iθ = z + je iθ z. Since multiplication on H is linear, the adjoint representation on Im (H) is given by Ad(t)Y = (c t ) Y = d ds t(esy )t s= d = t ds (esy ) t = tyt = c t (Y). s= In particular, taking t = e iθ, and conjugating the basis i, j, k of sp(), we have c t (i) = i, c t (j) = j(cos θ i sin t) = (cos θ)j + (sin θ)k, c t (k) = k(cos θ i sin t) = ( sin θ)j + (cos θ)k. Since ir = t, on sp()/t, with respect to the basis j, k, we have and so ( ) id Ad(t cos θ sin θ ) =, sin θ cos θ (t) = ( cos θ) + (sin θ) = cos θ + (cos θ + sin θ) = ( cos θ) = ( (cos θ + sin θ) (cos θ sin θ) ) = 4 sin θ. The roots Φ of the adjoint representation are the weights of the complexified representation ( sp()/t ) C. If we take as basis vectors v = j i + k and v = j + k i, we get Ad(t)v = t v and Ad(t)v = t v, so we have one positive root α : t t. So W = S has two elements and e T (ν) = c (S α ) = u H (BT). Here BT CP and u corresponds to the first Chern class of the tautological bundle on CP. Viewing u as an element of the symmetric algebra S(t ), we have u(x) =. 5
6 5.3 dg and dt and d(gt) Writing H = C + jc, we can decompose q Sp() as z + jz, where z + z =. Then we can find η [, π such that z = sin η and z = cos η, and can find complex arguments ξ and ξ of z and z, so that e iξ sin η + je iξ cos η. This identifies Sp() as a quotient space of S S [, π ; on the subspace η (), the ξ -coordinate is irrelevant, and on T = η ( π ), the ξ -coordinate is irrelevant. Thus we can parameterize Sp() minus two circles by E := [, π) (, π ). We will find it helpful to write dg in this parameterization. Writing z = x + iy and z = x + iy for real x, x, y, y gives a vector space isomorphism H R 4 taking Sp() S 3. The natural volume form on R 4 is vol R 4 = dx dy dx dy. If we write Y = x x + x x + y y + y y for the radial vector field on S 3, we get vol S 3 = ι Y (vol R 4) = (x dy y dx ) dx dy + (x dy y dx ) dx dy. Substituting x = sin η cos ξ, y = sin η sin ξ, x = cos η cos ξ, y = cos η sin ξ, much cancellation occurs, and after a few more lines one finds that vol S 3 pulls back to on E. We have ω = cos η sin η dη dξ dξ = vol(s 3 ) = E ω = π sin η π π dξ dξ π = (π) sin ζ d(ζ/) = π ( cos ζ) = π, ζ=π ζ= dη dξ dξ sin η dη so dg = π vol S 3 ω sin η = π 4π dη dξ dξ. Since vol(s ) = π, we have dt π dξ. To find d(gt), we should first understand the quotient space Sp()/T. Since (e iξ sin η + je iξ cos η)e ξ = sin η + je i(ξ ξ ) cos η, a fundamental domain for right translation by T is the set ξ () S (this is the Hopf fibration of S 3 ). Omitting the two points {η =, π } (where the arguments for the j-coordinate are arbitrary), this corresponds to D := {} [, π) (, π ) E. Now ω sin η D = dη dξ, and D ω D = π, so d(gt) = π vol Sp()/T sin η π dη dξ. 6
7 5.4 X and f dg Since the fundamental vector field generated by X = πi under conjugation is ( ) d X q = (c e sx ),q ds s= and c e sx (z + jz ) = z + jz e 4πsi (ξ, ξ + 4πs, η), it follows that on D, we have X = 4π ξ, and a function f on Sp() is T-invariant just if f ξ =. Since dg is a top form, any form f dg is closed, and so, if f is T-equivariant, admits an equivariantly closed extension f dg Ω ( Sp() ) T [u. If we write f dg = f dg + uτ for τ Ω ( Sp() ) T, equivariant closedness means that = (d uι X )( f dg + uτ) = d( f dg) + u ( dτ f ι X (dg) ) u ι X τ, or that dτ = f ι X (dg) and ι X τ =. Since the fundamental vector field generated by X = πi under conjugation is ( ) d X q = (c e sx ),q ds s= and c e sx (z + jz ) = z + jz e 4πsi (ξ, ξ + 4πs, η), it follows that on D, we have X = 4π ξ. Thus we require that ι τ = τ ( ) ξ =. Writing τ = a dξ + b dξ + c dη, this means ξ b =. Since τ is assumed T-invariant, it also follows that ξ a ( a dτ = η c ) dη dξ ξ. On the other hand, so we want f ι X dg = f ι 4π ξ ( sin η = c ξ =, so that dη dξ dξ ) π = π f sin η dη dξ, a η c = f sin η. ξ π Since f is π-periodic in ξ, the most obvious (if not particularly subtle) thing to do is to take c =. Since τ doesn t depend on ξ when η =, it s then easiest to take a η () =. One way to accomplish this is to set a(ξ, η) = π η f (ξ, ζ) sin ζ dζ. 7
8 5.5 Conjugation of a T-invariant function A function on Sp() is invariant under conjugation by T iff it is independent of ξ. The inner integral in the Weyl integral formula is over f (qtq ) for fixed t T and qt Sp()/T. Given an element of Sp()/T, we can find a representative q in the fundamental domain ξ (). Letting t T and q = t q(t ), we see that a T-invariant function will give the same value at qtq and at q t(q ) = t q(t ) tt q (t ) = t qtq (t ), so we may as well assume ξ (q) = as well, or q = sin ζ + j cos ζ. Writing t = e iξ, we get qtq = (cos ξ i sin ξ cos ζ) + ji sin ξ sin ζ. Conjugating to get ξ = again, we get ξ (qtq ) = arctan ( tan ξ cos ζ) and η(qtq ) = arccos(sin ξ sin ζ). So the integral, not particularly transparently, is of f ( arctan ( tan ξ cos η), arccos(sin ξ sin η) ) over η, for fixed ξ. One wonders whether a better choice of coordinates is in order. 5.6 The desired formula, restated Consider a real-valued function f (ξ, η) on R [, π which is π-periodic in ξ and independent of ξ wherever η =. The equivariant localization formula tells us π π π/ Sp() f dg = equals T sin η f (ξ, η) dη dξ dξ π = π π/ f (ξ π, η)sin η dη dξ u(x) u(x) τ {η= π } = π ( a ξ, π ) dξ = π π/ f (ξ π, η) sin η dη dξ. This is at least true, but it s not informative. As for the Weyl integration formula, it tells us that the integrals above equal W π T π π [ (t) 4 sin ξ [ π Sp()/T π/ f (gtg ) d(gt) dt = f ( arctan ( tan ξ cos η), arccos(sin ξ sin η) ) sin η [ π/ sin ξ f ( arctan ( tan ξ cos η), arccos(sin ξ sin η) ) sin η dη This isn t particularly helpful. dξ. dη dξ dξ π π = 8
9 5.7 An easier case Suppose that f is in fact invariant under conjugation by G. Then the bracketed integral simplifies to f (t), so the whole integral becomes π ( 4(sin ξ ) f ξ, π ) dξ. It still isn t clear to me, though, why this should have anything to do with the double integral in the equivariant localization formula. 6 Unsorted thoughts Would the Weyl integration formula be more what we want with different coordinates? Could a less stupid choice of a and c in τ = a dξ + c dη yield the Weyl integral formula? It s a(ξ, π/) dξ that shows up in the equivariant localization formula. It s easy to arrange the functions be constant in ξ at η =, say by giving them sin nη factors or making them integrals from to η, but if we try to select a first and derive c(ξ, η) = ξ ( η a π f sin η) dξ, the resulting function will almost never be π-periodic in ξ. If we assume f is G-invariant, can we make some clever change of coordinates that simplifies the integral over η in the equivariant localization formula? The only part of either formula that involves the roots is (t) in the one formula and e T (ν) in the other. But in the Weyl integration formula, one has to integrate with respect to this function, whereas in the equivariant localization formula, it just cancels u i s. What is the connection between these superficially similar but very different things? The standard proof of the Weyl integration formula starts with the function G T G taking (g, t) gtg, pushes it down to a function (G/T) T G, cuts out the (positive-codimension) set of singular elements, and computes the integral over G by pulling it back to (G/T) T. The function (t) = α Φ ( α) comes out of the Jacobian. On the other hand G/T doesn t even show up in the equivariant localization formula. One can knock the G/T integral out of the Weyl formula, but only at the cost of assuming the function to be integrated is G-invariant, a hypothesis that isn t required for the equivariant localization. Is there a way of proving equivariant localization using G/T, or some way of pulling back both proofs (I realize this is incredibly vague) to some more complicated space, so that both formulae manifest as quotients of the more complicated situation? Given that the Weyl integral formula involves an integral over T (G/T), is it worth considering the action of T by t (t, gt) = (t, tgt), or something similar, and trying to apply equivariant localization? 9
LECTURE 25-26: CARTAN S THEOREM OF MAXIMAL TORI. 1. Maximal Tori
LECTURE 25-26: CARTAN S THEOREM OF MAXIMAL TORI 1. Maximal Tori By a torus we mean a compact connected abelian Lie group, so a torus is a Lie group that is isomorphic to T n = R n /Z n. Definition 1.1.
More informationk=0 /D : S + S /D = K 1 2 (3.5) consistently with the relation (1.75) and the Riemann-Roch-Hirzebruch-Atiyah-Singer index formula
20 VASILY PESTUN 3. Lecture: Grothendieck-Riemann-Roch-Hirzebruch-Atiyah-Singer Index theorems 3.. Index for a holomorphic vector bundle. For a holomorphic vector bundle E over a complex variety of dim
More informationLemma 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 informationAn Invitation to Geometric Quantization
An Invitation to Geometric Quantization Alex Fok Department of Mathematics, Cornell University April 2012 What is quantization? Quantization is a process of associating a classical mechanical system to
More informationMath 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 informationSYMPLECTIC 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 informationREPRESENTATION THEORY WEEK 7
REPRESENTATION THEORY WEEK 7 1. Characters of L k and S n A character of an irreducible representation of L k is a polynomial function constant on every conjugacy class. Since the set of diagonalizable
More informationLECTURE 11: SYMPLECTIC TORIC MANIFOLDS. Contents 1. Symplectic toric manifolds 1 2. Delzant s theorem 4 3. Symplectic cut 8
LECTURE 11: SYMPLECTIC TORIC MANIFOLDS Contents 1. Symplectic toric manifolds 1 2. Delzant s theorem 4 3. Symplectic cut 8 1. Symplectic toric manifolds Orbit of torus actions. Recall that in lecture 9
More informationLecture on Equivariant Cohomology
Lecture on Equivariant Cohomology Sébastien Racanière February 20, 2004 I wrote these notes for a hours lecture at Imperial College during January and February. Of course, I tried to track down and remove
More informationTopics in Representation Theory: Roots and Complex Structures
Topics in Representation Theory: Roots and Complex Structures 1 More About Roots To recap our story so far: we began by identifying an important abelian subgroup of G, the maximal torus T. By restriction
More informationHamiltonian flows, cotangent lifts, and momentum maps
Hamiltonian flows, cotangent lifts, and momentum maps Jordan Bell jordan.bell@gmail.com Department of Mathematics, University of Toronto April 3, 2014 1 Symplectic manifolds Let (M, ω) and (N, η) be symplectic
More informationCHARACTERISTIC CLASSES
1 CHARACTERISTIC CLASSES Andrew Ranicki Index theory seminar 14th February, 2011 2 The Index Theorem identifies Introduction analytic index = topological index for a differential operator on a compact
More informationTHE NEWLANDER-NIRENBERG THEOREM. GL. The frame bundle F GL is given by x M Fx
THE NEWLANDER-NIRENBERG THEOREM BEN MCMILLAN Abstract. For any kind of geometry on smooth manifolds (Riemannian, Complex, foliation,...) it is of fundamental importance to be able to determine when two
More informationON NEARLY SEMIFREE CIRCLE ACTIONS
ON NEARLY SEMIFREE CIRCLE ACTIONS DUSA MCDUFF AND SUSAN TOLMAN Abstract. Recall that an effective circle action is semifree if the stabilizer subgroup of each point is connected. We show that if (M, ω)
More informationNotation. For any Lie group G, we set G 0 to be the connected component of the identity.
Notation. For any Lie group G, we set G 0 to be the connected component of the identity. Problem 1 Prove that GL(n, R) is homotopic to O(n, R). (Hint: Gram-Schmidt Orthogonalization.) Here is a sequence
More informationLecture 6: Classifying spaces
Lecture 6: Classifying spaces A vector bundle E M is a family of vector spaces parametrized by a smooth manifold M. We ask: Is there a universal such family? In other words, is there a vector bundle E
More informationM4P52 Manifolds, 2016 Problem Sheet 1
Problem Sheet. Let X and Y be n-dimensional topological manifolds. Prove that the disjoint union X Y is an n-dimensional topological manifold. Is S S 2 a topological manifold? 2. Recall that that the discrete
More informationLECTURE 5: SOME BASIC CONSTRUCTIONS IN SYMPLECTIC TOPOLOGY
LECTURE 5: SOME BASIC CONSTRUCTIONS IN SYMPLECTIC TOPOLOGY WEIMIN CHEN, UMASS, SPRING 07 1. Blowing up and symplectic cutting In complex geometry the blowing-up operation amounts to replace a point in
More informationNotes 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 informationCohomology and Vector Bundles
Cohomology and Vector Bundles Corrin Clarkson REU 2008 September 28, 2008 Abstract Vector bundles are a generalization of the cross product of a topological space with a vector space. Characteristic classes
More informationSmooth Dynamics 2. Problem Set Nr. 1. Instructor: Submitted by: Prof. Wilkinson Clark Butler. University of Chicago Winter 2013
Smooth Dynamics 2 Problem Set Nr. 1 University of Chicago Winter 2013 Instructor: Submitted by: Prof. Wilkinson Clark Butler Problem 1 Let M be a Riemannian manifold with metric, and Levi-Civita connection.
More informationClifford Algebras and Spin Groups
Clifford Algebras and Spin Groups Math G4344, Spring 2012 We ll now turn from the general theory to examine a specific class class of groups: the orthogonal groups. Recall that O(n, R) is the group of
More informationComplex line bundles. Chapter Connections of line bundle. Consider a complex line bundle L M. For any integer k N, let
Chapter 1 Complex line bundles 1.1 Connections of line bundle Consider a complex line bundle L M. For any integer k N, let be the space of k-forms with values in L. Ω k (M, L) = C (M, L k (T M)) Definition
More informationThe Real Grassmannian Gr(2, 4)
The Real Grassmannian Gr(2, 4) We discuss the topology of the real Grassmannian Gr(2, 4) of 2-planes in R 4 and its double cover Gr + (2, 4) by the Grassmannian of oriented 2-planes They are compact four-manifolds
More informationB 1 = {B(x, r) x = (x 1, x 2 ) H, 0 < r < x 2 }. (a) Show that B = B 1 B 2 is a basis for a topology on X.
Math 6342/7350: Topology and Geometry Sample Preliminary Exam Questions 1. For each of the following topological spaces X i, determine whether X i and X i X i are homeomorphic. (a) X 1 = [0, 1] (b) X 2
More informationEXERCISES IN POISSON GEOMETRY
EXERCISES IN POISSON GEOMETRY The suggested problems for the exercise sessions #1 and #2 are marked with an asterisk. The material from the last section will be discussed in lecture IV, but it s possible
More information8. COMPACT LIE GROUPS AND REPRESENTATIONS
8. COMPACT LIE GROUPS AND REPRESENTATIONS. Abelian Lie groups.. Theorem. Assume G is a Lie group and g its Lie algebra. Then G 0 is abelian iff g is abelian... Proof.. Let 0 U g and e V G small (symmetric)
More informationHyperkähler geometry lecture 3
Hyperkähler geometry lecture 3 Misha Verbitsky Cohomology in Mathematics and Physics Euler Institute, September 25, 2013, St. Petersburg 1 Broom Bridge Here as he walked by on the 16th of October 1843
More informationTHE EULER CHARACTERISTIC OF A LIE GROUP
THE EULER CHARACTERISTIC OF A LIE GROUP JAY TAYLOR 1 Examples of Lie Groups The following is adapted from [2] We begin with the basic definition and some core examples Definition A Lie group is a smooth
More informationMath 868 Final Exam. Part 1. Complete 5 of the following 7 sentences to make a precise definition (5 points each). Y (φ t ) Y lim
SOLUTIONS Dec 13, 218 Math 868 Final Exam In this exam, all manifolds, maps, vector fields, etc. are smooth. Part 1. Complete 5 of the following 7 sentences to make a precise definition (5 points each).
More informationA Problem of Hsiang-Palais-Terng on Isoparametric Submanifolds
arxiv:math/0312251v1 [math.dg] 12 Dec 2003 A Problem of Hsiang-Palais-Terng on Isoparametric Submanifolds Haibao Duan Institute of Mathematics, Chinese Academy of Sciences, Beijing 100080, dhb@math.ac.cn
More informationINSTANTON MODULI AND COMPACTIFICATION MATTHEW MAHOWALD
INSTANTON MODULI AND COMPACTIFICATION MATTHEW MAHOWALD () Instanton (definition) (2) ADHM construction (3) Compactification. Instantons.. Notation. Throughout this talk, we will use the following notation:
More informationQUATERNIONS AND ROTATIONS
QUATERNIONS AND ROTATIONS SVANTE JANSON 1. Introduction The purpose of this note is to show some well-known relations between quaternions and the Lie groups SO(3) and SO(4) (rotations in R 3 and R 4 )
More informationSYMPLECTIC MANIFOLDS, GEOMETRIC QUANTIZATION, AND UNITARY REPRESENTATIONS OF LIE GROUPS. 1. Introduction
SYMPLECTIC MANIFOLDS, GEOMETRIC QUANTIZATION, AND UNITARY REPRESENTATIONS OF LIE GROUPS CRAIG JACKSON 1. Introduction Generally speaking, geometric quantization is a scheme for associating Hilbert spaces
More informationThe Spinor Representation
The Spinor Representation Math G4344, Spring 2012 As we have seen, the groups Spin(n) have a representation on R n given by identifying v R n as an element of the Clifford algebra C(n) and having g Spin(n)
More informationGroup Actions and Cohomology in the Calculus of Variations
Group Actions and Cohomology in the Calculus of Variations JUHA POHJANPELTO Oregon State and Aalto Universities Focused Research Workshop on Exterior Differential Systems and Lie Theory Fields Institute,
More informationMath 550 / David Dumas / Fall Problems
Math 550 / David Dumas / Fall 2014 Problems Please note: This list was last updated on November 30, 2014. Problems marked with * are challenge problems. Some problems are adapted from the course texts;
More informationCOHOMOLOGY OF ARITHMETIC GROUPS AND EISENSTEIN SERIES: AN INTRODUCTION, II
COHOMOLOGY OF ARITHMETIC GROUPS AND EISENSTEIN SERIES: AN INTRODUCTION, II LECTURES BY JOACHIM SCHWERMER, NOTES BY TONY FENG Contents 1. Review 1 2. Lifting differential forms from the boundary 2 3. Eisenstein
More informationRes + X F F + is defined below in (1.3). According to [Je-Ki2, Definition 3.3 and Proposition 3.4], the value of Res + X
Theorem 1.2. For any η HH (N) we have1 (1.1) κ S (η)[n red ] = c η F. Here HH (F) denotes the H-equivariant Euler class of the normal bundle ν(f), c is a non-zero constant 2, and is defined below in (1.3).
More informationGeometry Qualifying Exam Notes
Geometry Qualifying Exam Notes F 1 F 1 x 1 x n Definition: The Jacobian matrix of a map f : N M is.. F m F m x 1 x n square matrix, its determinant is called the Jacobian determinant.. When this is a Definition:
More informationWe then have an analogous theorem. Theorem 1.2.
1. K-Theory of Topological Stacks, Ryan Grady, Notre Dame Throughout, G is sufficiently nice: simple, maybe π 1 is free, or perhaps it s even simply connected. Anyway, there are some assumptions lurking.
More informationEQUIVARIANT COHOMOLOGY. p : E B such that there exist a countable open covering {U i } i I of B and homeomorphisms
EQUIVARIANT COHOMOLOGY MARTINA LANINI AND TINA KANSTRUP 1. Quick intro Let G be a topological group (i.e. a group which is also a topological space and whose operations are continuous maps) and let X be
More informationWEYL INTEGRATION FORMULA
WEYL INERAION FORMULA YIFAN WU, wuyifan@umich.edu Abstract. In this exposition, we will prove the Weyl integration formula, which states that if is a compact connected Lie group with maximal torus, and
More informationExercises on characteristic classes
Exercises on characteristic classes April 24, 2016 1. a) Compute the Stiefel-Whitney classes of the tangent bundle of RP n. (Use the method from class for the tangent Chern classes of complex projectives
More informationThe Weyl Character Formula
he Weyl Character Formula Math 4344, Spring 202 Characters We have seen that irreducible representations of a compact Lie group can be constructed starting from a highest weight space and applying negative
More informationExercises in Geometry II University of Bonn, Summer semester 2015 Professor: Prof. Christian Blohmann Assistant: Saskia Voss Sheet 1
Assistant: Saskia Voss Sheet 1 1. Conformal change of Riemannian metrics [3 points] Let (M, g) be a Riemannian manifold. A conformal change is a nonnegative function λ : M (0, ). Such a function defines
More informationStable generalized complex structures
Stable generalized complex structures Gil R. Cavalcanti Marco Gualtieri arxiv:1503.06357v1 [math.dg] 21 Mar 2015 A stable generalized complex structure is one that is generically symplectic but degenerates
More informationNEW REALIZATIONS OF THE MAXIMAL SATAKE COMPACTIFICATIONS OF RIEMANNIAN SYMMETRIC SPACES OF NON-COMPACT TYPE. 1. Introduction and the main results
NEW REALIZATIONS OF THE MAXIMAL SATAKE COMPACTIFICATIONS OF RIEMANNIAN SYMMETRIC SPACES OF NON-COMPACT TYPE LIZHEN JI AND JIANG-HUA LU Abstract. We give new realizations of the maximal Satake compactifications
More informationRiemann Surfaces and Algebraic Curves
Riemann Surfaces and Algebraic Curves JWR Tuesday December 11, 2001, 9:03 AM We describe the relation between algebraic curves and Riemann surfaces. An elementary reference for this material is [1]. 1
More informationChern forms and the Fredholm determinant
CHAPTER 10 Chern forms and the Fredholm determinant Lecture 10: 20 October, 2005 I showed in the lecture before last that the topological group G = G (Y ;E) for any compact manifold of positive dimension,
More informationChow 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 informationLECTURE 2: SYMPLECTIC VECTOR BUNDLES
LECTURE 2: SYMPLECTIC VECTOR BUNDLES WEIMIN CHEN, UMASS, SPRING 07 1. Symplectic Vector Spaces Definition 1.1. A symplectic vector space is a pair (V, ω) where V is a finite dimensional vector space (over
More informationTopics in Representation Theory: The Weyl Integral and Character Formulas
opics in Representation heory: he Weyl Integral and Character Formulas We have seen that irreducible representations of a compact Lie group G can be constructed starting from a highest weight space and
More informationsatisfying the following condition: If T : V V is any linear map, then µ(x 1,,X n )= det T µ(x 1,,X n ).
ensities Although differential forms are natural objects to integrate on manifolds, and are essential for use in Stoke s theorem, they have the disadvantage of requiring oriented manifolds in order for
More informationAn introduction to cobordism
An introduction to cobordism Martin Vito Cruz 30 April 2004 1 Introduction Cobordism theory is the study of manifolds modulo the cobordism relation: two manifolds are considered the same if their disjoint
More informationNOTES ON FORMAL NEIGHBORHOODS AND JET BUNDLES
NOTES ON FORMAL NEIGHBORHOODS AND JET BUNDLES SHILIN U ABSTRACT. The purpose of this note is to review the construction of smooth and holomorphic jet bundles and its relation to formal neighborhood of
More informationSpin(10,1)-metrics with a parallel null spinor and maximal holonomy
Spin(10,1)-metrics with a parallel null spinor and maximal holonomy 0. Introduction. The purpose of this addendum to the earlier notes on spinors is to outline the construction of Lorentzian metrics in
More informationIntersection of stable and unstable manifolds for invariant Morse functions
Intersection of stable and unstable manifolds for invariant Morse functions Hitoshi Yamanaka (Osaka City University) March 14, 2011 Hitoshi Yamanaka (Osaka City University) ()Intersection of stable and
More informationINTRODUCTION TO ALGEBRAIC GEOMETRY
INTRODUCTION TO ALGEBRAIC GEOMETRY WEI-PING LI 1 Preliminary of Calculus on Manifolds 11 Tangent Vectors What are tangent vectors we encounter in Calculus? (1) Given a parametrised curve α(t) = ( x(t),
More informationThe Based Loop Group of SU(2) Lisa Jeffrey. Department of Mathematics University of Toronto. Joint work with Megumi Harada and Paul Selick
The Based Loop Group of SU(2) Lisa Jeffrey Department of Mathematics University of Toronto Joint work with Megumi Harada and Paul Selick I. The based loop group ΩG Let G = SU(2) and let T be its maximal
More informationPart III Symmetries, Fields and Particles
Part III Symmetries, Fields and Particles Theorems Based on lectures by N. Dorey Notes taken by Dexter Chua Michaelmas 2016 These notes are not endorsed by the lecturers, and I have modified them (often
More informationTwisted Poisson manifolds and their almost symplectically complete isotropic realizations
Twisted Poisson manifolds and their almost symplectically complete isotropic realizations Chi-Kwong Fok National Center for Theoretical Sciences Math Division National Tsing Hua University (Joint work
More informationGEOMETRIC 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 informationL 2 Geometry of the Symplectomorphism Group
University of Notre Dame Workshop on Innite Dimensional Geometry, Vienna 2015 Outline 1 The Exponential Map on D s ω(m) 2 Existence of Multiplicity of Outline 1 The Exponential Map on D s ω(m) 2 Existence
More informationLecture 8: More characteristic classes and the Thom isomorphism
Lecture 8: More characteristic classes and the Thom isomorphism We begin this lecture by carrying out a few of the exercises in Lecture 1. We take advantage of the fact that the Chern classes are stable
More informationarxiv:math/ v2 [math.sg] 19 Feb 2003
THE KIRWAN MAP, EQUIVARIANT KIRWAN MAPS, AND THEIR KERNELS ariv:math/0211297v2 [math.sg] 19 eb 2003 LISA C. JEREY AND AUGUSTIN-LIVIU MARE Abstract. or an arbitrary Hamiltonian action of a compact Lie group
More informationHYPERKÄHLER MANIFOLDS
HYPERKÄHLER MANIFOLDS PAVEL SAFRONOV, TALK AT 2011 TALBOT WORKSHOP 1.1. Basic definitions. 1. Hyperkähler manifolds Definition. A hyperkähler manifold is a C Riemannian manifold together with three covariantly
More informationGENERALIZING THE LOCALIZATION FORMULA IN EQUIVARIANT COHOMOLOGY
GENERALIZING THE LOCALIZATION FORULA IN EQUIVARIANT COHOOLOGY Abstract. We give a generalization of the Atiyah-Bott-Berline-Vergne localization theorem for the equivariant cohomology of a torus action.
More informationThe Atiyah bundle and connections on a principal bundle
Proc. Indian Acad. Sci. (Math. Sci.) Vol. 120, No. 3, June 2010, pp. 299 316. Indian Academy of Sciences The Atiyah bundle and connections on a principal bundle INDRANIL BISWAS School of Mathematics, Tata
More informationNotes 9: Eli Cartan s theorem on maximal tori.
Notes 9: Eli Cartan s theorem on maximal tori. Version 1.00 still with misprints, hopefully fewer Tori Let T be a torus of dimension n. This means that there is an isomorphism T S 1 S 1... S 1. Recall
More informationLECTURE: KOBORDISMENTHEORIE, WINTER TERM 2011/12; SUMMARY AND LITERATURE
LECTURE: KOBORDISMENTHEORIE, WINTER TERM 2011/12; SUMMARY AND LITERATURE JOHANNES EBERT 1.1. October 11th. 1. Recapitulation from differential topology Definition 1.1. Let M m, N n, be two smooth manifolds
More informationTRANSITIVE HOLONOMY GROUP AND RIGIDITY IN NONNEGATIVE CURVATURE. Luis Guijarro and Gerard Walschap
TRANSITIVE HOLONOMY GROUP AND RIGIDITY IN NONNEGATIVE CURVATURE Luis Guijarro and Gerard Walschap Abstract. In this note, we examine the relationship between the twisting of a vector bundle ξ over a manifold
More informationAlgebraic Cobordism Lecture 1: Complex cobordism and algebraic cobordism
Algebraic Cobordism Lecture 1: Complex cobordism and algebraic cobordism UWO January 25, 2005 Marc Levine Prelude: From homotopy theory to A 1 -homotopy theory A basic object in homotopy theory is a generalized
More informationDIFFERENTIAL GEOMETRY 1 PROBLEM SET 1 SOLUTIONS
DIFFERENTIAL GEOMETRY PROBLEM SET SOLUTIONS Lee: -4,--5,-6,-7 Problem -4: If k is an integer between 0 and min m, n, show that the set of m n matrices whose rank is at least k is an open submanifold of
More information16.2. 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 informationComparison for infinitesimal automorphisms. of parabolic geometries
Comparison techniques for infinitesimal automorphisms of parabolic geometries University of Vienna Faculty of Mathematics Srni, January 2012 This talk reports on joint work in progress with Karin Melnick
More informationQualifying Exams I, 2014 Spring
Qualifying Exams I, 2014 Spring 1. (Algebra) Let k = F q be a finite field with q elements. Count the number of monic irreducible polynomials of degree 12 over k. 2. (Algebraic Geometry) (a) Show that
More information10. 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 informationThese notes are incomplete they will be updated regularly.
These notes are incomplete they will be updated regularly. LIE GROUPS, LIE ALGEBRAS, AND REPRESENTATIONS SPRING SEMESTER 2008 RICHARD A. WENTWORTH Contents 1. Lie groups and Lie algebras 2 1.1. Definition
More informationBackground on Chevalley Groups Constructed from a Root System
Background on Chevalley Groups Constructed from a Root System Paul Tokorcheck Department of Mathematics University of California, Santa Cruz 10 October 2011 Abstract In 1955, Claude Chevalley described
More informationMorse Theory and Applications to Equivariant Topology
Morse Theory and Applications to Equivariant Topology Morse Theory: the classical approach Briefly, Morse theory is ubiquitous and indomitable (Bott). It embodies a far reaching idea: the geometry and
More informationfy (X(g)) Y (f)x(g) gy (X(f)) Y (g)x(f)) = fx(y (g)) + gx(y (f)) fy (X(g)) gy (X(f))
1. Basic algebra of vector fields Let V be a finite dimensional vector space over R. Recall that V = {L : V R} is defined to be the set of all linear maps to R. V is isomorphic to V, but there is no canonical
More informationLecture 5 - Lie Algebra Cohomology II
Lecture 5 - Lie Algebra Cohomology II January 28, 2013 1 Motivation: Left-invariant modules over a group Given a vector bundle F ξ G over G where G has a representation on F, a left G- action on ξ is a
More informationIntroduction to Index Theory. Elmar Schrohe Institut für Analysis
Introduction to Index Theory Elmar Schrohe Institut für Analysis Basics Background In analysis and pde, you want to solve equations. In good cases: Linearize, end up with Au = f, where A L(E, F ) is a
More informationPeriod Domains. Carlson. June 24, 2010
Period Domains Carlson June 4, 00 Carlson - Period Domains Period domains are parameter spaces for marked Hodge structures. We call Γ\D the period space, which is a parameter space of isomorphism classes
More informationA MARSDEN WEINSTEIN REDUCTION THEOREM FOR PRESYMPLECTIC MANIFOLDS
A MARSDEN WEINSTEIN REDUCTION THEOREM FOR PRESYMPLECTIC MANIFOLDS FRANCESCO BOTTACIN Abstract. In this paper we prove an analogue of the Marsden Weinstein reduction theorem for presymplectic actions of
More informationIntroduction to Chiral Algebras
Introduction to Chiral Algebras Nick Rozenblyum Our goal will be to prove the fact that the algebra End(V ac) is commutative. The proof itself will be very easy - a version of the Eckmann Hilton argument
More informationA geometric solution of the Kervaire Invariant One problem
A geometric solution of the Kervaire Invariant One problem Petr M. Akhmet ev 19 May 2009 Let f : M n 1 R n, n = 4k + 2, n 2 be a smooth generic immersion of a closed manifold of codimension 1. Let g :
More informationQUALIFYING EXAMINATION Harvard University Department of Mathematics Tuesday September 1, 2015 (Day 1)
QUALIFYING EXAMINATION Harvard University Department of Mathematics Tuesday September 1, 2015 (Day 1) 1. (A) The integer 8871870642308873326043363 is the 13 th power of an integer n. Find n. Solution.
More information4-MANIFOLDS: CLASSIFICATION AND EXAMPLES. 1. Outline
4-MANIFOLDS: CLASSIFICATION AND EXAMPLES 1. Outline Throughout, 4-manifold will be used to mean closed, oriented, simply-connected 4-manifold. Hopefully I will remember to append smooth wherever necessary.
More informationPHYS 4390: GENERAL RELATIVITY NON-COORDINATE BASIS APPROACH
PHYS 4390: GENERAL RELATIVITY NON-COORDINATE BASIS APPROACH 1. Differential Forms To start our discussion, we will define a special class of type (0,r) tensors: Definition 1.1. A differential form of order
More informationCharacteristic 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 informationLECTURE 28: VECTOR BUNDLES AND FIBER BUNDLES
LECTURE 28: VECTOR BUNDLES AND FIBER BUNDLES 1. Vector Bundles In general, smooth manifolds are very non-linear. However, there exist many smooth manifolds which admit very nice partial linear structures.
More informationNOTES ON FIBER BUNDLES
NOTES ON FIBER BUNDLES DANNY CALEGARI Abstract. These are notes on fiber bundles and principal bundles, especially over CW complexes and spaces homotopy equivalent to them. They are meant to supplement
More informationGeometric dimension of stable vector bundles over spheres
Morfismos, Vol. 18, No. 2, 2014, pp. 41 50 Geometric dimension of stable vector bundles over spheres Kee Yuen Lam Duane Randall Abstract We present a new method to determine the geometric dimension of
More informationFibre Bundles. E is the total space, B is the base space and F is the fibre. p : E B
Fibre Bundles A fibre bundle is a 6-tuple E B F p G V i φ i. E is the total space, B is the base space and F is the fibre. p : E B 1 is the projection map and p x F. The last two elements of this tuple
More informationMODULI OF VECTOR BUNDLES ON CURVES AND GENERALIZED THETA DIVISORS
MODULI OF VECTOR BUNDLES ON CURVES AND GENERALIZED THETA DIVISORS MIHNEA POPA 1. Lecture II: Moduli spaces and generalized theta divisors 1.1. The moduli space. Back to the boundedness problem: we want
More informationELEMENTARY 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 informationLecture 4 - Lie Algebra Cohomology I
Lecture 4 - Lie Algebra Cohomology I January 25, 2013 Given a differentiable manifold M n and a k-form ω, recall Cartan s formula for the exterior derivative k dω(v 0,..., v k ) = ( 1) i x i (ω(x 0,...,
More informationBetti numbers of abelian covers
Betti numbers of abelian covers Alex Suciu Northeastern University Geometry and Topology Seminar University of Wisconsin May 6, 2011 Alex Suciu (Northeastern University) Betti numbers of abelian covers
More information