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Math 210C. Compact Lie Groups MATH 210C. COMPACT LIE GROUPS LECTURES BY BRIAN CONRAD, NOTES BY AARON LANDESMAN This document consists of lectures notes from a course at Stanford University in Spring quarter 2018, along with appendices written by Conrad as supplements to the lectures. The goal is to cover the structure theory of semisimple compact connected Lie groups, an excellent “test case” for the general theory of semisimple connected Lie groups (avoiding many analytic difficulties, thanks to compactness). This includes a tour through the “clas- sical (compact) groups”, the remarkable properties of maximal tori (especially conjugacy thereof), some basic ideas related to Lie algebras, and the combinatorial concepts of root system and Weyl group which lead to both a general structure theory in the compact connected case as well as a foothold into the finite-dimensional representation theory of such groups. For a few topics (such as the manifold structure on G/H and basic properties of root systems) we give a survey in the lectures and refer to the course text [BtD] for omitted proofs. The homework assignments work out some important results (as well as instructive examples), and also develop some topological background (such as covering space theory and its relation to p1), so one should regard the weekly homework as an essential part of the course. We largely avoid getting into the precise structure theory of semisimple Lie algebras, and develop what we need with sl2 at one crucial step. The main novelty of our approach compared to most other treatments of compact groups is using techniques that adapt to linear algebraic groups over general fields: (i) much more emphasis on the character and cocharacter lattices attached to a maximal torus T than on their scalar extensions to R (identified with Lie(T) and its dual), (ii) extensive use of torus centralizers and commutator subgroups, (iii) developing root systems over a general field of characteristic 0 (especially allowing Q, not just R), as in [Bou2]. Please send any errata (typos, math errors, etc.) to [email protected]. CONTENTS 1. Basics of topological groups 3 2. Further classes of compact groups 9 3. Connectedness and vector fields 14 4. Lie algebras 18 5. Maximal compact and 1-parameter subgroups 23 6. The exponential map 27 7. Subgroup-subalgebra correspondence 32 8. Complete reducibility 36 9. Character theory 41 10. Weyl’s Unitary Trick 46 11. Weyl’s Conjugacy Theorem: Applications 49 Thanks to Dan Dore and Vivian Kuperberg for note-taking on days when Aaron Landesman was away. 1 2 BRIAN CONRAD AND AARON LANDESMAN 12. Integration on Coset Spaces 53 13. Proof of Conjugacy Theorem I 58 14. Proof of the Conjugacy Theorem II 61 15. Weyl Integration Formula and Roots 64 16. Root systems and Clifford algebras 69 17. Torus centralizers and SU(2) 75 18. Commutators and rank-1 subgroups 79 19. Reflections in Weyl groups 84 20. Root systems 88 21. Weyl groups and coroots for root systems 94 22. Equality of two notions of Weyl group I 98 23. Equality of two notions of Weyl group II 102 24. Bases of root systems 105 25. Dynkin diagrams 109 26. Normal subgroup structure 114 27. Return to representation theory 118 28. Examples and Fundamental Representations 121 Appendix A. Quaternions 128 Appendix B. Smoothness of inversion 130 Appendix C. The adjoint representation 132 Appendix D. Maximal compact subgroups 135 Appendix E. ODE 140 Appendix F. Integral curves 152 Appendix G. More features and applications of the exponential map 182 Appendix H. Local and global Frobenius theorems 186 Appendix I. Elementary properties of characters 203 Appendix J. Representations of sl2 206 Appendix K. Weyl groups and character lattices 214 Appendix L. The Weyl Jacobian formula 224 Appendix M. Class functions and Weyl groups 227 Appendix N. Weyl group computations 228 Appendix O. Fundamental groups of Lie groups 231 Appendix P. Clifford algebras and spin groups 233 Appendix Q. The remarkable SU(2) 248 Appendix R. Existence of the coroot 254 Appendix S. The dual root system and the Q-structure on root systems 256 Appendix T. Calculation of some root systems 259 Appendix U. Irreducible decomposition of root systems 266 Appendix V. Size of fundamental group 268 Appendix W. A non-closed commutator subgroup 277 Appendix X. Simple factors 282 Appendix Y. Centers of simply connected semisimple compact groups 287 Appendix Z. Representation ring and algebraicity of compact Lie groups 290 References 297 MATH 210C. COMPACT LIE GROUPS 3 1. BASICS OF TOPOLOGICAL GROUPS 1.1. First definitions and examples. Definition 1.1. A topological group is a topological space G with a group structure such that the multiplication map m : G × G ! G and inversion map i : G ! G are continuous. Example 1.2. The open subset GLn(R) ⊂ Matn(R) can be given the structure of a topological group via the usual multiplication and in- version of matrices. We topologize it with the subspace topology (with Matn(R) given the Euclidean topology as a finite-dimensional R-vector space). Matrix multiplication is given in terms of polynomials, and similarly for inversion with rational functions whose denominator is a power of the determinant (Cramer’s Formula). Likewise, GLn(C) ⊂ Matn(C) is a topological group. In general, to give a coordinate-free description, for V a finite-dimensional vector space over R or C, the open subset GL(V) ⊂ End(V) of invertible linear endomorphisms is a topological group with the subspace topology. The following lemma is elementary to check: Lemma 1.3. For topological groups G and G0, G × G0 with the product topology is a topological group. Example 1.4. We have an isomorphism of topological groups 1 × R>0 × S ! C = GL1(C) (r, w) 7! rw. Example 1.5. For G a topological group, any subgroup H ⊂ G is a topological group with the subspace topology. For us, this will be most useful for closed H ⊂ G. Example 1.6. For G = GLn(R), the special linear group SLn(R) := fg 2 G : det g = 1g is a closed subset of Matn(R). Example 1.7. The orthogonal group n n T o O(n) := g 2 G : hgv, gwi = hv, wi for all v, w 2 R , or equivalently g g = 1n . T Note that if g g = 1n then det(g) = ±1. Thus, the subgroup SO(n) := O(n)det=1 ⊂ O(n) has finite index, and we even have O(n) = SO(n) o Z/2Z 4 BRIAN CONRAD AND AARON LANDESMAN where the nontrivial element of Z/2Z corresponds to the matrix 0−1 1 B 1 C B C . @ ... A 1 Remark 1.8. In general, for g 2 O(n) each g(ei) is a unit vector, so g has bounded entries as a matrix. Hence, O(n) is bounded in Matn(R), and it is also closed in Matn(R) (not only T in GLn(R)!) since the description g g = 1n implies invertibilty of g, so O(n) is compact. 1.2. Non-compact analogues of O(n). Recall that a quadratic form q : V ! R on a finite- dimensional R-vector space V is a function given in linear coordinates by a degree-2 homogeneous polynomial ∑i≤j aijxixj (a condition that is clearly insensitive to linear change of coordinates). Lemma 1.9. For V a finite-dimensional vector space over R, we have a bijection fsymmetric bilinear B : V × V ! Rg (1.1) fquadratic forms q : V ! Rg by the assignments B 7! (qB : v 7! B(v, v)) and 1 q 7! (B : (v, w) 7! (q(v + w) − q(v) − q(w))).(1.2) q 2 The main point of the proof is to check that each proposed construction lands in the asserted space of objects and that they’re inverse to each other. This is left as an exercise. The astute reader will check that this works well over any field not of characteristic 2. Applying the preceding lemma with B taken to be the standard dot product on Rn, for which qB is the usual norm-square, we have: Corollary 1.10. n o O(n) = g 2 GL(V) : jjg(v)jj2 = jjvjj2 for all v 2 V = Rn . Definition 1.11. A quadratic form q is non-degenerate if the associated bilinear form Bq is a perfect pairing, or equivalently its associated symmetric matrix (for a choice of basis of V) is invertible. Example 1.12. Let V = Rn. We can consider the form 2 2 2 2 qr,n−r(x) = x1 + ··· + xr − xr+1 − · · · − xn. The associated matrix B is a diagonal matrix whose first r entries are 1 and last n − r entries are −1. The Gram-Schmidt process (adapted to non-degenerate quadratic forms that might have some values ≤ 0) plus a bit of care implies that every (V, q) over R that is non-degenerate is n isomorphic to (R , qr,n−r) for a unique 0 ≤ r ≤ n. The idea is to build an orthonormal-like basis using that all positive real numbers have square roots, and the uniqueness needs MATH 210C. COMPACT LIE GROUPS 5 some care since there is no intrinsic “maximal positive-definite subspace” (there are many such). We call the resulting pair (r, n − r) the signature of the associated quadratic form q. Definition 1.13. For q a non-degenerate quadratic form on a finite-dimensional R-vector space V, we define O(q) := fg 2 GL(V) : q(gv) = q(v)g ⊂ GL(V), which is a closed subset of GL(V). Remark 1.14. When q has signature (r, n − r), we denote O(q) by O(r, n − r).
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