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Basic Representation Theory [ Draft ] Basic representation theory [ draft ] Incomplete notes for a mini-course given at UCAS, Summer 2017 Wen-Wei Li Version: December 10, 2018 Contents 1 Topological groups 2 1.1 Definition of topological groups .............................. 2 1.2 Local fields ......................................... 4 1.3 Measures and integrals ................................... 5 1.4 Haar measures ........................................ 6 1.5 The modulus character ................................... 9 1.6 Interlude on analytic manifolds ............................... 11 1.7 More examples ....................................... 13 1.8 Homogeneous spaces .................................... 14 2 Convolution 17 2.1 Convolution of functions .................................. 17 2.2 Convolution of measures .................................. 19 3 Continuous representations 20 3.1 General representations ................................... 20 3.2 Matrix coefficients ..................................... 23 3.3 Vector-valued integrals ................................... 25 3.4 Action of convolution algebra ................................ 26 3.5 Smooth vectors: archimedean case ............................. 28 4 Unitary representation theory 28 4.1 Generalities ......................................... 29 4.2 External tensor products .................................. 32 4.3 Square-integrable representations .............................. 33 4.4 Spectral decomposition: the discrete part .......................... 37 4.5 Usage of compact operators ................................. 39 4.6 The case of compact groups ................................ 40 4.7 Basic character theory .................................... 41 5 Unitary representation of compact Lie groups (a sketch) 42 5.1 Review of basic notions ................................... 43 5.2 Algebraic preparations ................................... 44 5.3 Weyl character formula: semisimple case ......................... 47 5.4 Extension to non-semisimple groups ............................ 50 1 References 50 1 Topological groups 1.1 Definition of topological groups op For any group 퐺, we write 1 = 1퐺 for its identity element and write 퐺 for its opposite group; in other words, 퐺op has the same underlying set as 퐺, but with the new multiplication (푥, 푦) ↦ 푦푥. .Denote by 핊 the group {푧 ∈ ℂ× ∶ |푧| = 1} endowed with its usual topological structure Definition 1.1. A topological group is a topological space 퐺 endowed with a group structure, such that the maps (푥, 푦) ↦ 푥푦 and 푥 ↦ 푥− are continuous. Unless otherwise specified, we always assume that 퐺 is Hausdorff. -By a homomorphism 휑 ∶ 퐺 → 퐺 between topological groups, we mean a continuous homomor phism. This turns the collection of all topological groups into a category TopGrp, and it makes sense to talk about isomorphisms, etc. We write Hom(퐺, 퐺) and Aut(퐺) for the sets of homomorphisms and automorphisms in TopGrp. Let 퐺 be a topological group. As is easily checked, (i) 퐺op is also a topological group, (ii) inv ∶ 푥 ↦ ∼ ,푥− is an isomorphism of topological groups 퐺 −→퐺op with inv ∘ inv = id, and (iii) for every 푔 ∈ 퐺 the translation map 퐿푔 ∶ 푥 ↦ 푔푥 (resp. 푅푔 ∶ 푥 ↦ 푥푔) is a homeomorphism from 퐺 to itself: indeed, .퐿푥− 퐿푥 = id = 푅푥− 푅푥 -By translating, we see that if 풩 is the set of open neighborhoods of 1, then the set of open neigh .{borhoods of 푥 ∈ 퐺 equals {푥푈 ∶ 푈 ∈ 풩}; it also equals {푈푥 ∶ 푈 ∈ 풩 We need a few elementary properties of topological groups. Notation: for subsets 퐴, 퐵 ⊂ 퐺, we .{write 퐴퐵 = {푎푏 ∶ 푎 ∈ 퐴, 푏 ∈ 퐵} ⊂ 퐺; similarly for 퐴퐵퐶 and so forth. Also put 퐴− = {푎− ∶ 푎 ∈ 퐴 Proposition 1.2. For any topological group 퐺 (not presumed Hausdorff), the following are equivalent: (i) 퐺 is Hausdorff; (ii) the intersection of all open neighborhoods containing 1 is {1}; (iii) {1} is closed in 퐺. ⋂ .Proof. It is clear that (i) ⟹ (ii), (iii). Also (ii) ⟺ (iii) since it is routine to check that {1} = 푈∋ 푈 Let us prove (ii) ⟹ (i) as follows. Let 휈 ∶ 퐺 × 퐺 → 퐺 be the function (푥, 푦) ↦ 푥푦−. Then 퐺 is − Hausforff if and only if the diagonal Δ퐺 ⊂ 퐺 × 퐺 is closed, whilst Δ퐺 = 휈 (1). Now (ii) implies (iii) which in turn implies Δ퐺 is closed. Proposition 1.3. Let 퐺 be a topological group (not necessarily Hausdorff) and 퐴, 퐵 ⊂ 퐺 be subsets. 1. If one of 퐴, 퐵 is open, then 퐴퐵 is open. 2. If both 퐴, 퐵 are compact, then so is 퐴퐵. 3. If one of 퐴, 퐵 is compact and the other is closed, then 퐴퐵 is closed. ⋃ Proof. Suppose that 퐴 is open. Then 퐴퐵 = 푏∈퐵 퐴푏 is open as well. Suppose that 퐴, 퐵 are compact. The image 퐴퐵 of 퐴 × 퐵 under multiplication is also compact. Finally, suppose 퐴 is compact and 퐵 is closed. If 푥 ∉ 퐴퐵, then 푥퐵− is closed and disjoint from 퐴. If we can find an open subset 푈 ∋ 1 such that 푈퐴 ∩ 푥퐵− = ∅, then 푈−푥 ∋ 푥 will be an open neighborhood disjoint from 퐴퐵, proving that 퐴퐵 is closed. 2 ′ − To find 푈, set 푉 ∶= 퐺 ∖ 푥퐵 . For every 푎 ∈ 퐴 ⊂ 푉 there exists an open subset 푈푎 ∋ 1 with ′ ′ 푈푎푎 ⊂ 푉. Take an open 푈푎 ∋ 1 with 푈푎푈푎 ⊂ 푈푎. Compactness furnishes a finite subset 퐴 ⊂ 퐴 such that 퐴 ⊂ ⋃ 푈 푡. We claim that 푈 ∶= ⋂ 푈 satisfies the requirements. Indeed, 푈 ∋ 1 and each 푡∈퐴 푡 푡∈퐴 푡 푎 ∈ 퐴 belongs to 푈푡푡 for some 푡 ∈ 퐴, hence ′ 푈푎 ⊂ 푈푈푡푡 ⊂ 푈푡푈푡푡 ⊂ 푈푡 푡 ⊂ 푉, thus 푈퐴 ⊂ 푉. Recall that a topological space is locally compact if every point has a compact neighborhood. Proposition 1.4. Let 퐻 be a subgroup of a topological group 퐺 (not necessarily Hausdorff). Endow 퐺/퐻 with the quotient topology. 1. The quotient map 휋 ∶ 퐺 → 퐺/퐻 is open and continuous. 2. If 퐺 is locally compact, so is 퐺/퐻. 3. 퐺/퐻 is Hausdorff (resp. discrete) if and only if 퐻 is closed (resp. open). The same holds for 퐻\퐺. ,Proof. The quotient map is always continuous. If 푈 ⊂ 퐺 is open, then 휋−(휋(푈)) = 푈퐻 is open by 1.3 hence 휋(푈) is open. Suppose 퐺 is locally compact. By homogeneity, it suffices to argue that the coset 퐻 = 휋(1) has compact neighborhood in 퐺/퐻. Let 퐾 ∋ 1 be a compact neighborhood in 퐺. Choose a neighborhood 푈 ∋ 1 such that 푈−푈 ⊂ 퐾. Claim: 휋(푈) ⊂ 휋(퐾). Indeed, if 푔퐻 ∈ 휋(푈), then the neighborhood 푈푔퐻 .(intersects 휋(푈); that is, 푢푔퐻 = 푢′퐻 for some 푢, 푢′ ∈ 푈. Hence 푔퐻 = 푢−푢′퐻 ∈ 휋(푈−푈) ⊂ 휋(퐾 Therefore 휋(푈) ∋ 휋(1) is an open neighborhood with 휋(푈) compact, since 휋(퐾) is compact. If 퐺/퐻 is Hausdorff (resp. discrete) then 퐻 = 휋−(휋(1)) is closed (resp. open). Conversely, suppose that 퐻 is closed in 퐺. Given cosets 푥퐻 ≠ 푦퐻, choose an open neighborhood 푉 ∋ 1 in 퐺 such that 푉푥 ∩ 푦퐻 = ∅; equivalently 푉푥퐻 ∩ 푦퐻 = ∅. Then choose an open 푈 ∋ 1 in 퐺 such that 푈−푈 ⊂ 푉. It follows that 푈푥퐻 ∩ 푈푦퐻 = ∅, and these are disjoint open neighborhoods of 휋(푥) and 휋(푦). All in all, 퐺/퐻 is Hausdorff. On the other hand, 퐻 is open implies 휋(퐻) is open, thus all singletons in 퐺/퐻 are open, whence the discreteness. As for 퐻\퐺, we pass to 퐺op. Lemma 1.5. Let 퐺 be a locally compact group (not necessarily Hausdorff) and let 퐻 ⊂ 퐺 be a subgroup. Every compact subset of 퐻\퐺 is the image of some compact subset of 퐺. Proof. Let 푈 ∋ 1 be an open neighborhood in 퐺 with compact closure 푈. For every compact subset 퐾♭ ♭ ⋃ of 퐻\퐺, we have an open covering 퐾 ⊂ 푔∈퐺 휋(푔푈), hence there exists 푔, … , 푔푛 ∈ 퐺 such that ⎛ ⎞ ⎛ ⎞ 푛 ⎜ 푛 ⎟ ⎜ 푛 ⎟ 퐾♭ ⊂ ﾌ 휋(푔 푈) = 휋 ⎜ﾌ 푔 푈⎟ ⊂ 휋 ⎜ﾌ 푔 푈̄ ⎟ 푖 ⎝⎜ 푖 ⎠⎟ ⎝⎜ 푖 ⎠⎟ 푖= 푖= 푖= ♭ ♭ 푛 ̄ −⋃ . Take the compact subset 퐾 ∶= 푖= 푔푖푈 ∩ 휋 (퐾 ) and observe that 휋(퐾) = 퐾 Remark 1.6. The following operations on topological groups are evident. 1. Let 퐻 ⊂ 퐺 be a closed normal subgroup, then 퐺/퐻 is a topological group by Proposition 1.4, locally compact if 퐺 is. We leave it to the reader to characterize 퐺/퐻 by universal properties in TopGrp. 3 If 퐺, 퐺 are topological groups, then 퐺 × 퐺 is naturally a topological group. Again, it has .2 an outright categorical characterization: the product in TopGrp. More generally, one can form fibered products in TopGrp. 3. In a similar vein, lim exist in TopGrp: their underlying abstract groups are just the usual lim. ←−− ←−− Harmonic analysis, in its classical sense, applies mainly to locally compact topological groups. Here- after we adopt the shorthand locally compact groups. Remark 1.7. The family of locally compact groups is closed under passing to closed subgroups, Hausdorff quotients and finite direct products. However, infinite direct products usually yield non-locally compact groups. This is one of the motivation for introducing restrict products into harmonic analysis. Example 1.8. Discrete groups are locally compact; finite groups are compact. Example 1.9. The familiar groups (ℝ, +), (ℂ, +) are locally compact; so are (ℂ×, ⋅) and (ℝ×, ⋅). The × × identity connected component ℝ> of ℝ is isomorphic to (ℝ, +) through the logarithm. The quotient .group (ℝ/ℤ, +) ≃ 핊 (via 푧 ↦ 푒휋푖푧) is compact 1.2 Local fields Just as the case of groups, a locally compact field is a field 퐹 with a locally compact Hausdorff topology, .such that (퐹, +) is a topological group, and that (푥, 푦) ↦ 푥푦 and 푥 ↦ 푥− (on 퐹×) are both continuous Definition 1.10. A local field is a locally compact field that is not discrete. A detailed account of local fields can be found in any textbook on algebraic number theory. The topology on a local field 퐹 is always induced by an absolute value | ⋅ |퐹 ∶ 퐹 → ℝ≥ satisfying |푥|퐹 = 0 ⟺ 푥 = 0, |1|퐹 = 1; |푥 + 푦|퐹 ≤ |푥|퐹 + |푦|퐹; |푥푦|퐹 = |푥| ⋅ |푦|퐹. Furthermore 퐹 is complete with respect to | ⋅ |.
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