INTRODUCTION TO REPRESENTATIONS OF LIE GROUPS AND LIE ALGEBRAS Course M2 Fall 2013 Vadim Schechtman CONTENTS Names 2 Chapter 1. Lie groups and Lie algebras. Main examples 3 Fields: F1, Fq, R, C, p-adic. GLn(F ). Lie groups and their Lie algebras. Main examples of Lie groups and their Lie algebras. Haar measure. Examples. Exercise. The Haar measure on Qp and on GL2(Qp). Chapter 2. Representations of compact Lie groups 12 Schur lemma. Complete reducibility. Schur orthogonality relations. Chapter 3. Structure and representations of complex semisimple Lie algebras 16 Example: sl(n) Root systems. The Weyl groups. Verma modules and finite-domensional representations. BGG resolution and the Weyl character formula. References 27 1 2 NAMES Wilhelm KILLING 1847 - 1923 Elie CARTAN 1869 - 1951 Hermann WEYL 1885 - 1955 Harold Scott MacDonald COXETER 1907 - 2003 Israel GELFAND 1913 - 2009 Eugene DYNKIN b. 1924 Joseph BERNSTEIN b. 1945 3 Chapter 1. Lie groups and Lie algebras. Main examples. A. Lie groups 1.1. Haar measure. Topological groups. All our toplogies will be Hausdorff. A topological group G is called locally compact if for every open neighbourhood U e there exist an open U ′ and a compact K such that e ∈ U ′ ⊂ K ⊂ U. Let G be a locally compact group. Theorem. (A. Haar and Von Neumann) There exists a unique up to a multi- plicative constant Borel measure µL on G which is left invariant and regular. Alfr´ed Haar (1885, Budapest, - 1933 Szeged) a Hungarian mathematician. John von Neumann (1903, Budapest - 1955, Princeton) a famous Hungarian born American mathematician. Left invariance: for each measurable X ⊂ G µl(X)= µl(gX) Regularity: µ(X) = inf{µ(U)|U ⊃ X, U open } = sup{µ(K)|K ⊂ X,K compact } The same for a right invariant measure, µR. × Define δ : G −→ R+ by −1 f(g hg)dµL(h)= δ(g) f(h)dµL(h). ZG ZG Proposition. (i) δ is a quasicharacter, i.e. a continuous homomorphism. (ii) The measure δ(h)µL(h) is right invariant. Corollary. A compact group is unimodular, i.e. a left invariant Haar measure is right invariant. 1.2. Local fields. Let F be a field (all fields will be commutative). An absolute value on F is a map |.| : F −→ R≥0 such that (i) |x| = 0 iff x = 0; (ii) |xy| = |x||y|; (iii) |x + y|≤|x| + |y|. Example: all absolute values on Q. An absolute value defines a metrics, d(x, y)= |x, y| on F and hence a topology. 4 We say that |.| is non-trivial if there exists x with |x|=0 6 , 1. In that case the topology is not discrete. A field with a non-trivial absolute value, complete wrt the corresponding met- rics and locally compact is called a local field. Examples: R, C, Qp, Fq((t)) and their finite extensions. That is all in fact. 1.2.1. The Haar measure on Qp. Explicitly, ∞ i Qp = { aip , n ∈ Z, ai ∈{0, 1,...,p − 1}} ⊃ Xi=n ∞ i Zp = { aip , n ∈ Z, ai ∈{0, 1,...,p − 1}}. Xi=0 n −n By definition, µ(Zp) = 1 it follows from Qp-invariance that µ(p Zp) = p (ex- plain this). 1.2.1.1. Exercice. (a) Show that da/|a|p is a Haar measure on the multi- × plicative group Qp . n n −n (b) Show that for a linear map f : Qp −→ Qp , f∗µ = | det(f)|p µ. 1.3. Lie groups. Let F be a local field. Then a notion of an analytic variety over F is defined. A Lie group over F is a group and an analytic variety G with both structures compatible. This means that the the multiplication and taking the inverse maps m : G × G −→ G, Inverse : G −→ G, Inverse(x)= x−1 are morphisms of analytic varieties. Examples. Classical groups. ”Die K¨onigin” (Her Majesty). Let V be a finite dimensional vector space over F . GL(V ). n GLn(F )= GL(F ). All other Lie groups are its subgroups. Suppose the F is equipped by a symmetric (resp. antisymmetric) bilinear form (x, y). Then the group G = {g ∈ GL(V )| (gx,gy)=(x, y)} is called orthogonal O(V ) (resp. symplectic Sp(V )). The classical series: An = SLn+1, n ≥ 1. Bn = SO(2n + 1), n ≥ 2, 5 Cn = Sp(2n), n ≥ 2, Dn = SO(2n), n ≥ 3,D3 = A3. Here ”S” means ”with det = 1”. In this course, if not specified otherwise, the base field F = R. In exercises we will probably discuss a little the p-adic case which is important for the number theory. Compact and non-compact Lie groups. Exercice. (i) Let F = R, C, Zp or Qp. Show that n | dgij| dg = i,j=1 Q| det(g)|n is a left- and right invariant measure on GLn(F ). Cf. [A. Knightly, Ch. Li, Traces of Hecke operators], 7.6. (ii) Consider the group a b G = SL (F )= { } 2 c d Consider a three form dbdcdd dadcdd dadbdd dadbdc ω = = = − = − ∈ Ω3(G) d c b a Show that dadbdcdd = d(ab − cd)ω. Show that ω is left and right G-invariant. Deduce that dg =(i/2)3ωω¯ is a Haar measure on G(C). Cf. [I.M.Gelfand, M.I.Graev, A.N.Vilenkin, Integral geometry and representa- tion theory, Generalized functions, v. 5], Ch. IV, Appendix. B. Lie algebras 1.4. Lie algebras. Motivation: let X,Y ∈ End(V ). For small ǫ 1+ ǫX ∈ GL(V ). (1 + Xǫ)(1 + Y ǫ)(1 + Xǫ)−1 = 1 + [X,Y ]ǫ + O(ǫ2) where [X,Y ]= XY − YX. Definition. A Lie algebra over a field F is a vector space g equipped with a bilinear pairing [., ]: g × g −→ F satisfying two axioms: 6 (i) skew symmetry: [x, y]= −[y, x]; (ii) the Jacobi identity: [x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0. Example. A an associative ring; [x, y] = xy − yx. This Lie algebra will be denoted ALie. In this course, unless specified otherwise, ”a Lie algebra” will mean ”a finite dimensional Lie algebra”. Example: gl(V ). 1.5. The Lie algebra of a Lie group. Exponential map. Let X be a smooth variety (over R); C(X): the algebra of smooth functions X −→ R. The Lie algebra of vector fields: T(X)= Der(C(X)), Lie bracket = the commutator (check that a commutator of two derivations is a derivation). Local form. Each vector field τ ∈ T(X) gives rise to a tangent vector τ(x) in the tangent space TxX at each point x ∈ X (explain in the exercises). Let G be a Lie group. For each g ∈ G let ∼ Lg : G −→ G, Lg(h)= gh, whence ∼ Lg∗ : T(G) −→ T(G) A vector field τ ∈ T(G) is called left invariant if for each g ∈ G, Lg∗(τ)= τ. By definition, Lie(G) ⊂ T(G) is the Lie subalgebra of left invariant vector fields (one has to verify that it is closed wrt to the commutator). As a vector space, g = Lie(G)= TeG For g ∈ G define −1 Adg : G −→ G, Adg(h)= ghg . Let X ∈ g. There exists a unique curve g(t) = exp(tX) ∈ G such that the tangent vector to g(t) at t = t0 is equal to X(g(t0)). 7 Thus we get a map exp : g −→ G, exp(X)= eX := g(1). 1.6. The classical Lie algebras. Let V be a finite-dimensional vector space with a non-degenerate bilinear form (., .): V × V −→ F . Consider the following subspace {g ∈ gl(V )| ∀x, y ∈ V (gx,y)+(x,gy)=0} ⊂ gl(V ). Exercise: show that it is a Lie subalgebra. When the form is symmetric (resp. antisymmetric), this Lie subalgebra is denoted by o(V ) (resp. sp(V )). An = sln+1, n ≥ 1. Bn = so(2n + 1), n ≥ 2, Cn = sp(2n), n ≥ 2, B2 = C2. Dn = so(2n), n ≥ 3,D3 = A3. Here ”s” means ”with tr = 0”. 1.6.1. Language of categories. Categories, functors, natural transformations. Exercise: Ioneda’s lemma. Adjoint functors. 1.7. Enveloping algebras. Abstract definition. Let g be a Lie algebra over a field F . Its envelopping algebra Ug is an associative algebra Ug over F together with a map of Lie algebras Lie iU : g −→ Ug having the following universal property: Lie for any associative F -algebra A and a map of Lie algebras iA : g −→ A there exists a unique morphism of associative algebras f : Ug −→ A such that iA = f ◦ iU . A concrete definition. Let V be a vector space. Its tensor algebra: · ∞ ⊗n n ∞ n T V = ⊕n=0V = T ⊕n=0 T V. 8 Its symmetric algebra: S·V = T ·V/I where I is the two-sided (homogeneous) ideal generated by all elements xy − yx, x, y ∈ V . The enveloping algebra Ug of a Lie algebra g: Ug = T ·g/I where I is the two-sided ideal in T ·g generated by all elements xy − yx − [x, y]. Canonical filtration. Let ≥n FnUg = Im(T g −→ Ug) ⊂ Ug, n ≥ 0. The associated graded ∞ gr Ug := FnUg/Fn−1Ug, n ≥ 0 Xn=0 (where F−1g := 0) is commutative, whence the morphism of alegebras S·g −→ gr Ug (1.7.1) extending the identity on g. The Poincar´e- Birkhoff - Witt theorem. The map (1.7.1) is an isomor- phism. 1.7.1. Exercise. The Poisson structure on S·g. 1.7.1. Exercise. The Casimir element. Let g be a finite dimensonal Lie algebra, and (, ): g ⊗ g −→ C a nondegenerate symmetric bilinear form. Let {xi}, {yi} be the dual bases of g, (xi,yj)= δij.