![Basic Representation Theory [ Draft ]](http://data.docslib.org/img/21a052d23f38a2ef485bb05795c77895-1.webp)
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 | ⋅ |. Up to continuous isomorphisms, local fields are classified as follows.
Archimedean The fields ℝ and ℂ, equipped with the usual absolute values;
Non-archimedean, characteristic zero The fields ℚ = ℤ [ ] (the 푝-adic numbers, where 푝 is a prime 푝 푝 푝 number) or their finite extensions;
Non-archimedean, characteristic 푝 > 0 The fields 픽 ((푡)) = 픽 J푡K[ ] of Laurent series in the variable 푞 푞 푡 푡 or their finite extensions, where 푞 is a power of 푝. Here 픽푞 denotes the finite field of 푞 elements. Let 퐹 be a non-archimedean local field. It turns out the ultrametric inequality is satisfied:
|푥 + 푦|퐹 ≤ max{|푥|퐹, |푦|퐹}, with equality when |푥| ≠ |푦|.
Furthermore, 픬퐹 = {푥 ∶ |푥| ≤ 1} is a subring, called the ring of integers, and 픭퐹 = {푥 ∶ |푥| < 1} is its unique maximal ideal. In fact 픭퐹 is of the form (휛퐹); here 휛퐹 is called a uniformizer of 퐹, and
× ℤ × × 퐹 = 휛퐹 × 픬퐹 , 픬퐹 = {푥 ∶ |푥| = 1}.
The normalized absolute value | ⋅ |퐹 for non-archimedean 퐹 is defined by
− |휛퐹| = 푞 , 푞 ∶= |픬퐹/픭퐹|
4 where 휛퐹 is any uniformizer. We will interpret | ⋅ |퐹 in terms of modulus characters in 1.33. If 퐸 is a finite extension of any local field 퐹, then 퐸 is also local and | ⋅ |퐹 admits a unique extension to 퐸 given by [/[퐸∶퐹 | ⋅ | = |푁 (⋅)| 퐸 퐸/퐹 퐹 where 푁퐸/퐹 ∶ 퐸 → 퐹 is the norm map. It defines the normalized absolute value on 퐸 in the non- archimedean case.
Remark 1.11. The example ℝ (resp. ℚ푝) above is obtained by completing ℚ with respect to the usual 푢 absolute value (resp. the 푝-adic one |푥| = 푝−푣푝(푥), where 푣 (푥) = 푘 if 푥 = 푝푘 ≠ 0 with 푢, 푣 ∈ ℤ coprime 푝 푝 푣 to 푝, and 푣푝(0) = +∞). (푣(푥− (Likewise, 픽푞((푡)) is the completion of the function field 픽푞(푡) with respect to |푥| = 푞 where 푣(푥 is the vanishing order of the rational function 푥 at 0. In both cases, the local field 퐹 arises from completing some global field ℚ or 픽푞(푡), or more generally their finite extensions. The adjective “global” comes from geometry, which is manifest in thecase 픽푞(푡): it is the function field of the curve ℙ , and 픽 ((푡)) should be thought as the function field on the “punctured 픽푞 푞 formal disk” at 0.
1.3 Measures and integrals We shall only consider Radon measures on a locally compact Hausdorff space 푋. These measures are by definition Borel, locally finite and inner regular. Define the ℂ-vector space
퐶푐(푋) ∶= 푓 ∶ 푋 → ℂ, continuous, compactly supported�
= ﾌ 퐶푐(푋, 퐾), 퐶푐(푋, 퐾) ∶= {푓 ∈ 퐶푐(푋) ∶ Supp(푓) ⊂ 퐾}. 퐾⊂푋 compact
We write 퐶푐(푋)+ ∶= 푓 ∈ 퐶푐(푋) ∶ 푓 ≥ 0�. Remark 1.12. Following Bourbaki, we identify positive Radon measures 휇 on 푋 and positive linear functionals 퐼 = 퐼휇 ∶ 퐶푐(푋) → ℂ. This means that 퐼(푓) ≥ 0 if 푓 ∈ 퐶푐(푋)+. In terms of integrals, 퐼 (푓) = ∫ 푓 d휇. This is essentially a consequence of the Riesz representation theorem. 휇 푋 Furthermore, to prescribe a positive linear functional 퐼 is the same as giving a function 퐼 ∶ 퐶푐(푋)+ → ℝ≥ that satisfies
,(퐼(푓 + 푓) = 퐼(푓) + 퐼(푓 •
.퐼(푡푓) = 푡퐼(푓) when 푡 ∈ ℝ≥ •
To see this, write 푓 = 푢 + 푖푣 ∈ 퐶푐(푋) where 푢, 푣 ∶ 푋 → ℝ; furthermore, any real-valued 푓 ∈ 퐶푐(푋) can be written as 푓 = 푓+ − 푓− as usual, where 푓± ∈ 퐶푐(푋)+. All these decompositions are canonical.
Remark 1.13. The complex Radon measures correspond to linear functionals 퐼 ∶ 퐶푐(퐶) → ℂ such that is continuous with respect to sup , for all . In other words, is continuous for 퐼|퐶푐(푋,퐾) ‖ ⋅ ‖∞,퐾 ∶= 퐾 | ⋅ | 퐾 퐼 the topology of lim 퐶푐(푋, 퐾) ≃ 퐶푐(푋). −−→퐾 Now let 퐺 be a locally compact group, and suppose that 푋 is endowed with a continuous left 퐺-action. Continuity here means that the action map
푎 ∶ 퐺 × 푋 ⟶ 푋 (푔, 푥) ⟼ 푔푥
5 is continuous. Similarly for right 퐺-actions. This action transports the functions 푓 ∈ 퐶푐(푋) as well:
푓푔 ∶= ￮푥 ↦ 푓(푔푥) , left action, 푔푓 ∶= ￮푥 ↦ 푓(푥푔)] , right action.
푔′ 푔푔′ 푔 푔푔′ 푔 푔′ These terminologies are justified as 푓 = 푓 and 푓 = 푓; they also preserve 퐶푐(푋)+. There- fore, for 퐺 acting on the left (resp. on the right) of 푋, it also acts from the same side on the space of positive Radon measures on 푋 by transport of structure. In terms of positive linear functionals,
퐼 ⟼ ￮푔퐼 ∶ 푓 ↦ 퐼(푓푔) , left action, 퐼 ⟼ ￮퐼푔 ∶ 푓 ↦ 퐼(푔푓) , right action.
This pair of definitions is swapped under 퐺 ⇝ 퐺op. By taking transposes, 퐺 also transports complex Radon measures (Remark 1.13). Mnemonic tech- nique: ,(d(푔휇)(푥) = d휇(푔−푥), d(휇푔)(푥) = d휇(푥푔− since a familiar change of variables yields
transpose ,(ﾙ 푓(푥) d(푔휇)(푥) = 퐼(푓푔) = ﾙ 푓(푔푥) d휇(푥) = ﾙ 푓(푥) d휇(푔−푥 푋 푋 푋 transpose (ﾙ 푓(푥) d(휇푔)(푥) = 퐼(푔푓) = ﾙ 푓(푥푔) d휇(푥) = ﾙ 푓(푥) d휇(푥푔− 푋 푋 푋
and 푓 ∈ 퐶푐(푋) is arbitrary. These formulas extend to all 푓 ∈ 퐿 (푋, 휇) by approximation.
× Definition 1.14. Suppose that 퐺 acts continuous on the left (resp. right) of 푋. Let 휒 ∶ 퐺 → ℝ> by a continuous homomorphism. We say that a complex Radon measure 휇 is quasi-invariant or an eigenmeasure with eigencharacter 휒 if 푔휇 = 휒(푔)−휇 (resp. 휇푔 = 휒(푔)−휇) for all 푔 ∈ 퐺. This can also be expressed as
d휇(푔푥) = 휒(푔) d휇(푥), left action, d휇(푥푔) = 휒(푔) d휇(푥), right action.
When 휒 = 1, we call 휇 an invariant measure.
Observe that 휇 is quasi-invariant with eigencharacter 휒 under left 퐺-action if and only if it is so under the right 퐺op-action. Remark 1.15. Let 휒, 휂 be continuous homomorphisms 퐺 → ℂ×. Suppose that 푓 ∶ 푋 → ℂ is contin- uous with 푓(푔푥) = 휂(푔)푓(푥) (resp. 푓(푥푔) = 휂(푔)푓(푥)) for all 푔 and 푥. Then 휇 is quasi-invariant with eigencharacter 휒 if and only if 푓휇 is quasi-invariant with eigencharacter 휂휒. In particular, we may let 퐺 act on 푋 = 퐺 by left (resp. right) translations. Therefore, it makes sense to talk about left and right invariant measures on 퐺.
1.4 Haar measures In what follows, measures are always nontrivial positive Radon measures.
Definition 1.16. Let 퐺 be a locally compact group. A left (resp. right) invariant measure on 퐺 is called a left (resp. right) Haar measure.
6 × The group ℝ> acts on the set of left (resp. right) Haar measures by rescaling. For commutative groups we make no distinction of left and right.
Example 1.17. If 퐺 is discrete, the counting measure Count(퐸) = |퐸| is a left and right Haar measure. .When 퐺 is finite, it is customary to take the normalized version 휇 ∶= |퐺|−Count ̌ − ̌ Definition 1.18. Let 퐺 be a locally compact. For 푓 ∈ 퐶푐(퐺) we write 푓 ∶ 푥 ↦ 푓(푥 ), 푓 ∈ 퐶푐(퐺). For any Radon measure 휇 on 퐺, let 휇̌ be the Radon measure with d휇(푥)̌ = d휇(푥−); in terms of linear ̌ functionals, 퐼휇̌ (푓) = 퐼휇(푓). Lemma 1.19. In the situation above, 휇 is a left (resp. right) Haar measure if and only if 휇̌ is a right (resp. left) Haar measure.
Proof. An instance of transport of structure, because 푥 ↦ 푥− is an isomorphism of locally compact ∼ groups 퐺 −→퐺op.
Theorem 1.20 (A. Weil). For every locally compact group 퐺, there exists a left (resp. right) Haar mea- × .sure on 퐺. They are unique up to ℝ>-action Proof. The following arguments are taken from Bourbaki [1, VII §1.2]. Upon replacing 퐺 by 퐺op, it suffices to consider the case of left Haar measures. For the existence part, weseekaleft 퐺-invariant positive linear functional on 퐶푐(퐺)+. Write
∗ 퐶푐(퐺)+ ∶= 퐶푐(퐺)+ ∖ {0}.
∗ For any compact subset 퐾 ⊂ 퐺, define 퐶푐(퐺, 퐾)+ and 퐶푐(퐺, 퐾)+ by intersecting 퐶푐(퐺, 퐾) with 퐶푐(퐺)+ ∗ ∗ and 퐶푐(퐺)+. Observe that for all 푓 ∈ 퐶푐(퐺)+ and 푔 ∈ 퐶푐(퐺)+, there exist 푛 ≥ 1, 푐, … , 푐푛 ∈ ℝ≥ and 푠, … , 푠푛 ∈ 퐺 such that 푠 푠푛 푠푖 .(푓 ≤ 푐푔 + … 푐푛푔 , 푔 (푥) ∶= 푔(푠푖푥
Indeed, there is an open subset 푈 ⊂ 퐺 such that inf푈 푔 > 0; now cover Supp(푓) by finitely many .푠푈, … , 푠푛푈 (… ,Given 푓, 푔 as before, define (푓 ∶ 푔) to be the infimum of 푐 +…+푐푛 among all choices of (푐, … , 푠 satisfying the bound above. We contend that
(푓푠 ∶ 푔) = (푓 ∶ 푔), 푠 ∈ 퐺, (1.1)
(푡푓 ∶ 푔) = 푡(푓 ∶ 푔), 푡 ∈ ℝ≥, (1.2)
(푓 + 푓 ∶ 푔) ≤ (푓 ∶ 푔) + (푓 ∶ 푔), (1.3) sup 푓 (푓 ∶ 푔) ≥ , (1.4) sup 푔 ∗ (푓 ∶ ℎ) ≤ (푓 ∶ 푔)(푔 ∶ ℎ), ℎ, 푔 ∈ 퐶푐(퐺)+, (1.5)
1 (푓 ∶ 푔) ∗ (푓 ∶ 푓), 푓, 푓, 푔 ∈ 퐶푐(퐺)+, (1.6) ≥ ≥ > 0 (푓 ∶ 푓) (푓 ∶ 푔) and that for all with , and all , there is a compact neighborhood 푓, 푓, ℎ ∈ 퐶푐(퐺)+ ℎ|Supp(푓+푓) ≥ 1 휖 > 0 퐾 ∋ 1 such that
∗ (푓 ∶ 푔) + (푓 ∶ 푔) ≤ (푓 + 푓 ∶ 푔) + 휖(ℎ ∶ 푔), 푔 ∈ 퐶푐(퐺, 퐾)+. (1.7) The properties (1.1)—(1.4) are straightforward. For (1.5), note that
푠푖 푡푗 푡푗푠푖 푓 ≤ ﾜ 푐푖푔 , 푔 ≤ ﾜ 푑푗ℎ ⟹ 푓 ≤ ﾜ 푐푖푑푗ℎ 푖 푗 푖,푗
7 ∑ ∑ .(hence (푓 ∶ ℎ) ≤ 푖 푐푖 푗 푑푗. Apply (1.5) to both 푓, 푓, 푔 and 푓, 푓, 푔 to obtain (1.6 휖ℎ To verify (1.7), let 퐹 ∶= 푓 + 푓 + . Use the condition on ℎ to define ⎧ ⎪ (푓푖(푔)/퐹(푥), 푥 ∈ Supp(푓 + 푓⎨ 휑푖(푥) = ⎪ ∈ 퐶푐(퐺)+ (푖 = 1, 2). ⎩0, otherwise.
Furthermore, given 휂 > 0 we may choose the compact neighborhood 퐾 ∋ 1 such that |휑푖(푥) − 휑푖(푦)| ≤ 휂 − whenever 푥 푦 ∈ 퐾, for 푖 = 1, 2. One readily verifies that for all 푔 ∈ 퐶푐(퐺, 퐾)+ and 푖 = 1, 2, 푠 푠 휑푖푔 ≤ (휑푖(푠) + 휂) ⋅ 푔 , 푠 ∈ 퐺. 푛 푠 푠 푠푛 ∑ 푗 Suppose that 퐹 ≤ 푐푔 + … 푐푛푔 , then 푓푖 = 휑푖퐹 ≤ 푗= 푐푗(휑푖(푠푗) + 휂)푔 by the previous step. As ∑푛 ,(휑 + 휑 ≤ 1, we infer that (푓 ∶ 푔) + (푓 ∶ 푔) ≤ (1 + 2휂) 푗= 푐푗. By (1.3) and (1.5 휖 (푓 ∶ 푔) + (푓 ∶ 푔) ≤ (1 + 2휂)(퐹 ∶ 푔) ≤ (1 + 2휂) (푓 + 푓 ∶ 푔) + (ℎ ∶ 푔) 2 휖 ≤ (푓 + 푓 ∶ 푔) + + 2휂(푓 + 푓 ∶ ℎ) + 휖휂 (ℎ ∶ 푔). 2
.(Choosing 휂 small enough relative to 푓, 푓, ℎ, 휖 yields (1.7 ∗ Proceed to the construction of Haar measure. We fix 푓 ∈ 퐶푐(퐺)+ and set
(푓 ∶ 푔) ∗ 퐼푔(푓) ∶= , 푓 ∈ 퐶푐(퐺)+, 푔 ∈ 퐶푐(퐺)+. (푓 ∶ 푔) ∗ ∗ We want to “take the limit” over 푔 ∈ 퐶푐(퐺, 퐾)+, where 퐾 shrinks to {1} and 푓, 푓 ∈ 퐶푐(퐺)+ are kept − ,fixed. By (1.6) we see 퐼푔(푓) ∈ ℑ ∶= ￮(푓 ∶ 푓) , (푓 ∶ 푓). For each compact neighborhood 퐾 ∋ 1 in 퐺 ∗ let 퐼퐾 (푓) ∶= 퐼푔(푓) ∶ 푔 ∈ 퐶푐(퐺, 퐾)+�. The family of all 퐼퐾 (푓) form a filter base in the compact space ℑ, hence can be refined into an ultrafilter which has the required limit 퐼(푓). We refer to [2, I. §6 and §9.1] for the language of filters and its relation with compactness. Alterna- tively, one can argue using the Moore–Smith theory of nets together with Tychonoff’s theorem; see [6, (15.25)]. (Then 퐼 ∶ 퐶푐(퐺)+ → ℝ≥ satisfies 퐺-invariance by (1.1). From (1.3) we infer 퐼(푓+푓) ≤ 퐼(푓)+퐼(푓 which already holds for all 퐼푔; from (1.7) we infer 퐼(푓) + 퐼(푓) ≤ 퐼(푓 + 푓) + 휖퐼(ℎ) whenever ℎ ≥ 1 on (Supp(푓 + 푓) and 휖 > 0 (true by using 푔 with sufficiently small support), so 퐼(푓 + 푓) = 퐼(푓) + 퐼(푓 follows. The behavior under dilation follows from (1.2). All in all, 퐼 is the required Haar measure on 퐺. Now turn to the uniqueness. Let us consider a left (resp. right) Haar measure 휇 (resp. 휈) on 퐺. It suffices by Lemma 1.19 to show that 휇 and 휈̌ are proportional. Fix 푓 ∈ 퐶 (퐺) such that ∫ 푓 d휇 ≠ 0. It 푐 퐺 is routine to show that the function − − 퐷푓 ∶ 푥 ⟼ ﾙ 푓 d휇 ﾙ 푓(푦 푥) d휈(푦), 푥 ∈ 퐺 퐺 퐺
is continuous. Let 푔 ∈ 퐶푐(퐺). Since (푥, 푦) ↦ 푓(푥)푔(푦푥) is in 퐶푐(퐺 × 퐺), Fubini’s theorem implies that
d휈(푦)=d휈(푦푥) ﾙ 푓 d휇 ⋅ ﾙ 푔 d휈 = ﾙ 푓(푥) ﾙ 푔(푦) d휈(푦) d휇(푥) = 퐺 퐺 퐺 퐺 d휇(푥)=d휇(푦푥) (ﾙ ﾙ 푓(푥)푔(푦푥) d휇(푥) d휈(푦) = ﾙ ﾙ 푓(푦−푥)푔(푥) d휇(푥) d휈(푦 퐺 퐺 퐺 퐺
− = ﾙ 푔(푥) ﾙ 푓(푦 푥) d휈(푦) d휇(푥) = ﾙ 푓 d휇 ⋅ ﾙ 푔퐷푓 d휇(푥). 퐺 퐺 퐺 퐺
8 Hence ∫ 푔 d휈 = ∫ 퐷 ⋅ 푔 d휇. Since 푔 is arbitrary and 퐷 is continuous, 퐷 (푥) is independent of 푓, 퐺 퐺 푓 푓 푓 hereafter written as 퐷(푥). Now 퐷(1) ﾙ 푓 d휇 = ﾙ 푓 d휈̌ 퐺 퐺 for 푓 ∈ 퐶푐(퐺) with ∫ 푓 d휇 ≠ 0. As both sides are linear functionals on the whole 퐶푐(퐺), the equality extends and yield the asserted proportionality.
Proposition 1.21. Let 휇 be a left (resp. right) Haar measure on 퐺. Then 퐺 is discrete if and only if 휇({1}) > 0, and 퐺 is compact if and only if 휇(퐺) < +∞.
Proof. The “only if” parts are easy: for discrete 퐺 we may take 휇 = Count, and for compact 퐺 we integrate the constant function 1. For the “if” part, first suppose that 휇({1}) > 0. Then every singleton has measure 휇({1}). Observe that every compact neighborhood 퐾 of 1 must satisfy 휇(퐾) > 0, therefore 퐾 is finite. As 퐺 is Hausdorff, {1} is thus open, so 퐺 is discrete. Next, suppose 휇(퐺) is finite. Fix a compact neighborhood 퐾 ∋ 1 so that 휇(퐾) > 0. If 푔, … , 푔푛 ∈ 퐺 푛 ⋃ .(are such that {푔푖퐾}푖= are disjoint, then 푛휇(퐾) = 휇( 푖 푔푖퐾) ≤ 휇(퐺) and this gives 푛 ≤ 휇(퐺)/휇(퐾 Choose a maximal collection 푔, … , 푔푛 ∈ 퐺 with the disjointness property above. Every 푔 ∈ 퐺 must − lie in 푔푖퐾 ⋅ 퐾 for some 1 ≤ 푖 ≤ 푛, since maximality implies 푔퐾 ∩ 푔푖퐾 ≠ ∅ for some 푖. Hence 푛 −⋃ .퐺 = 푖= 푔푖퐾 ⋅ 퐾 is compact by 1.3
1.5 The modulus character
Let 휃 ∶ 퐺 → 퐺 be an isomorphism of locally compact groups. A Radon measure 휇 on 퐺 transports to a Radon measure 휃∗휇 on 퐺: the corresponding positive linear functional is
휃 휃 .(퐼휃∗휇(푓) = 퐼휇(푓 ), 푓 ∶= 푓 ∘ 휃, 푓 ∈ 퐶푐(퐺
− It is justified to express this(휃 asd ∗휇)(푥) = d휇(휃 (푥)). × (Evidently, 휃∗ commutes with rescaling by ℝ>. By transport of structure, 휃∗휇 is a left (resp. right Haar measure if 휇 is. For two composable isomorphisms 휃, 휎 we have
(휃휎)∗휇 = 휃∗(휎∗휇).
−− Assume hereafter 퐺 = 퐺 = 퐺 and 휃 ∈ Aut(퐺). We consider 휃 휇 ∶= (휃 )∗휇 where 휇 is a left Haar measure on 퐺. Theorem 1.20 imlies 휃−휇 is a positive multiple of 휇, and this ratio does not depend on 휇.
Definition 1.22. For every 휃 ∈ Aut(퐺), define its modulus 훿휃 as the positive number determined by
− 휃 휇 = 훿휃휇, equivalently d휇(휃(푥)) = 훿휃 d휇(푥), where 휇 is any left Haar measure on 퐺. Remark 1.23. One can also use right Haar measures to define the modulus. Indeed, the right Haar measures are of the form form 휇̌ where 휇 is a left Haar measure, and
− −− d휇(휃(푥))̌ = d휇(휃(푥) ) = d휇(휃(푥 )) = 훿휃 ⋅ d휇(푥 ) = 훿휃 ⋅ d휇(푥).̌
− Remark 1.24. The modulus characters are sometimes defined as Δ퐺(푔) = 훿퐺(푔) in the literature, such as [1, VII §1.3] or [6]. This ambiguity is responsible for uncountably many headaches. Our convention for 훿퐺 seems to conform with most papers in representation theory.
9 × .Proposition 1.25. The map 휃 ↦ 훿휃 defines a homomorphism Aut(퐺) → ℝ> Proof. Use the fact that
− − −− (휃휎) 휇 = 휎∗ (휃∗ 휇) = 훿휃 ⋅ 휎∗ 휇 = 훿휃훿휎 ⋅ 휇 ∗ for any two automorphisms 휃, 휎 of 퐺 and any left Haar measure 휇.
Example 1.26. If 퐺 is discrete, then 훿휃 = 1 for all 휃 since the counting measure is preserved.
− Definition 1.27. For 푔 ∈ 퐺, let Ad(푔) be the automorphism 푥 ↦ 푔푥푔 of 퐺. Define 훿퐺(푔) ∶= 훿Ad(푔) ∈ − × .(ℝ>. In other words, d휇(푔푥푔 ) = 훿퐺(푔) d휇(푥
The following result characterizes 훿퐺.
× Lemma 1.28. The map 훿퐺 ∶ 퐺 → ℝ> is a continuous homomorphism. For every left (resp. right) Haar measure 휇ℓ (resp. 휇푟), we have
− d휇ℓ(푥푔) = 훿퐺(푔) d휇ℓ(푥), d휇푟(푔푥) = 훿퐺(푔) d휇푟(푥).
− Furthermore, 휇̌ℓ = 훿퐺 ⋅ 휇ℓ and 휇̌푟 = 훿퐺 휇푟.
Proof. Since Ad(⋅) ∶ 퐺 → Aut(퐺) is a homomorphism, so is 훿퐺 by Proposition 1.25. As for the continuity of 훿 , fix a left Haar measure 휇 and 푓 ∈ 퐶 (퐺) with 휇(푓) ∶= ∫ 푓 d휇 ≠ 0. Then 퐺 푐 퐺
−− 훿퐺(푔) = 휇(푓) ﾙ 푓 ∘ Ad(푔) d휇, 퐺 and it is routine to check that ∫ 푓 ∘ Ad(푔) d휇 is continuous in 푔. The second assertion results from 퐺 applying d휇ℓ(⋅) and d휇푟(⋅) to
,푥푔 = 푔(Ad(푔−)푥), 푔푥 = (Ad(푔)푥)푔 respectively. To deduce the last assertion, note that 훿퐺휇ℓ is a right Haar measure by the previous step and Remark 1.15, thus Theorem 1.20 entails × .푡 ∈ ℝ>, 휇ℓ̌ = 푡훿퐺휇ℓ∃ −− ∨∨ Hence 휇ℓ = 푡훿퐺 휇ℓ̌ = 푡 휇ℓ, and we obtain 푡 = 1. A similar reasoning for 훿퐺 휇푟 applies.
Corollary 1.29. We have 훿퐺 = 1 if and only if every left Haar measure on 퐺 is also a right Haar measure, and vice versa.
− Proposition 1.30. For every 푔 ∈ 퐺, we have 훿퐺(푔) = 훿퐺op (푔). = (Proof. Fix a left Haar measure 휇 for 퐺 and denote the multiplication in 퐺op by ⋆. Then d휇(푔푥푔− op− 훿퐺(푔) d휇(푥) is equivalent to d휇(푔 ⋆ 푥 ⋆ 푔) = 훿퐺(푔) d휇(푥) for 퐺 . Since 휇 is a right Haar measure for op − 퐺 , we infer from Remark 1.23 that 훿퐺op (푔 ) = 훿퐺(푔).
Definition 1.31. A locally compact group 퐺 with 훿퐺 = 1 is called unimodular. Example 1.32. The following groups are unimodular.
• Commutative groups.
• Discrete groups: use Example 1.26.
10 × .Compact groups: indeed, the compact subgroup 훿퐺(퐺) of ℝ> ≃ ℝ must be trivial • The normalized absolute value on a local field 퐹 is actually the modulus character for the additive group of 퐹, as shown below.
Theorem 1.33. For every local field 퐹, define ‖ ⋅ ‖퐹 to be the normalized absolute value | ⋅ |퐹 for non- × archimedean 퐹, the usual absolute value for 퐹 = ℝ, and 푥 ↦ |푥푥|̄ ℝ for 퐹 = ℂ. For 푡 ∈ 퐹 , let 푚푡 be the automorphism 푥 ↦ 푡푥 for (퐹, +). Then
훿푚푡 = ‖푡‖퐹.
Proof. Since 푚 ′ = 푚 ′ ∘ 푚 , by Proposition 1.25 we have 훿 = 훿 훿 . Fix any Haar measure 휇 on 푡푡 푡 푡 푚푡푡′ 푚푡′ 푚푡 퐹. By definition 휇(푡퐾) 훿 = 푚푡 휇(퐾) for every compact subset . The assertion is then evident for or , by taking 퐾 ⊂ 퐹 훿푚푡 = ‖푡‖퐹 퐹 = ℝ ℂ 퐾 to be the unit ball and work with the Lebesgue measure. Now suppose is non-archimedean and take . For × we have , thus . The 퐹 퐾 = 픬퐹 푡 ∈ 픬퐹 푡퐾 = 퐾 훿푚푡 = 1 same holds for and both are multiplicative. It remains to show − for any uniformizer | ⋅ |퐹 훿푚휛 = |휛|퐹 = 푞 −− 휛 of 퐹, where 푞 ∶= |픬퐹/픭퐹|. But then 휇(휛픬퐹)/휇(픬퐹) = (픬퐹 ∶ 휛픬퐹) = 푞 .
1.6 Interlude on analytic manifolds It is well known that every real Lie group admits an analytic structure. The theory of analytic manifolds generalizes to any local field 퐹. For a detailed exposition, we refer to [9, Part II]. In what follows, it suffices to assume that 퐹 is a complete topological field with respect to some 푚 ∶ (absolute value |⋅|. Let 푈 ⊂ 퐹 be some open ball. The definition of 퐹-analytic functions 푓 = (푓, … , 푓푛 푛 푈 → 퐹 is standard: we require that each 푓푖 can be expressed as convergent power series on 푈. For any open subset 푈 ⊂ 퐹푚, we say 푓 ∶ 푈 → 퐹푛 is 퐹-analytic if 푈 can be covered by open balls over which 푓 are 퐹-analytic.
Definition 1.34. An 퐹-analytic manifold of dimension 푛 is a second countable Hausdorff space 푋, en- dowed with an atlas (푈, 휙푈 ) ∶ 푈 ∈ 풰�, where • 풰 is an open covering of 푋;
푛 • for each 푈 ∈ 풰, 휙푈 is a homeomorphism from 푈 to an open subset 휙푈 (푈) of 퐹 .
The condition is that for all 푈, 푈 ∈ 풰 with 푈 ∩ 푈 ≠ ∅, the transition map
( 휙 휙− ∶ 휙 (푈 ∩ 푈 ) → 휙 (푈 ∩ 푈 푈 푈 푈 푈 is 퐹-analytic.
As in the standard theory of manifolds, the 퐹-analytic functions on 푋 are defined in terms of charts, and the same applies to morphisms between 퐹-analytic manifolds; the notion of closed submanifolds is defined in the usual manner. We can also pass to maximal atlas in the definition. The 퐹-analytic manifolds then form a category. Open subsets of 퐹-analytic manifolds inherit analytic structures. It is also possible to define 퐹-analytic manifolds intrinsically by sheaves. More importantly, the notion of tangent bundles, cotangent bundles, differential forms and their ex- terior powers still make sense on an 퐹-analytic manifold 푋. Let 푛 = dim 푋. As in the real case, locally a differential form on 푋 can be expressed as 푐(푥, … , 푥푛) d푥 ∧ ⋯ ∧ d푥푛, where 푈, 휙푈 = (푥, … , 푥푛) 푛 is any chart for 푋, and 푐 is an analytic function on 휙푈 (푈) ⊂ 퐹 .
11 Remark 1.35. When (퐹, | ⋅ |) is ultrametric, the theory of 퐹-analytic manifolds is not quite interesting since they turn out to be totally disconnected. Nevertheless, it is a convenient vehicle for talking about measures, exponential maps and so forth. Example 1.36. Let X be an 퐹-variety. The set 푋 of its 퐹-points acquires a natural Hausdorff topology (say by reducing to the case to affine varieties, then to affine sets).If X is smooth, then 푋 becomes an analytic 퐹-manifold. The assignment X ↦ 푋 is functorial. The algebraically defined regular functions, differentials, etc. on X give rise to their 퐹-analytic avatars on 푋. By convention, objects on X will often carry the adjective algebraic. Now assume that 퐹 is local, and fix a Haar measure on 퐹, denoted abusively by d푥 in conformity with the usual practice. This equips 퐹푛 with a Radon measure for each 푛 ≥ 0, and every open subset 푈 ⊂ 퐹푛 carries the induced measure. × -Let 푋 be an 퐹-analytic manifold of dimension 푛. The collection of volume forms will be an ℝ> torsor over 푋. Specifically, a volume form on 푋 is expressed in every chart (푈, 휙푈 ) as
|휔| = 푓(푥, … , 푥푛)| d푥| ⋯ | d푥푛| where 푓 ∶ 푉 ∶= 휙푈 (푈) → ℝ≥ is continuous. Define ‖ ⋅ ‖퐹 ∶ 퐹 → ℝ≥ as in Theorem 1.33. A change of local coordinates 푦푖 = 푦푖(푥, … , 푥푛) (푖 = 1, … , 푛) transports 푓 and has the effect
휕푦푖 (d푦| ⋯ | d푦푛| = det ⋅ | d푥| ⋯ | d푥푛|. (1.8 | 휕푥 푗 푖,푗 퐹 In other words, |휔| is the “absolute value” of a differential form of top degree on 푋, squared when 퐹 = ℂ. Some geometers name these objects as densities. Now comes integration. The idea is akin to the case of 퐶∞-manifolds, and we refer to [7, Chapter 16] for a complete exposition for the latter. Let 휑 ∈ 퐶푐(푋) and choose an atlas {(푈, 휙푈 ) ∶ 푈 ∈ 풰} for 푋. In fact, it suffices to take a finite subset of 풰 covering Supp(휑).
1. By taking a partitions of unity 휓푈 ∶ 푋 → [0, 1] (with ∑ 휓푈 = 1 as usual), and replacing 푈∈풰 푈 휙 by 휓 휙 for each 푈 ∈ 풰, the definition of ∫ 휑|휔| reduces to the case that Supp(휑) ⊂ 푈 for 푈 푋 some 푈 ∈ 풰. 2. In turn, we are reduced to the integration ∫ 휑푓 d푥 ⋯ d푥 on some open subset 푉 ⊂ 퐹푛; this 푉⊂퐹푛 푛 we can do by the choice of Haar measure on 퐹. Furthermore, this integral is invariant under 퐹- analytic change of coordinates. Indeed, this is a consequence of (1.8) plus the interpretation of ‖ ⋅ ‖퐹 in Theorem 1.33. 3. As in the case over ℝ, one verifies that ∫ 휑휔 does not depend on the choice of charts and partition of unity. 4. It is routine to show that 퐼 ∶ 휑 ↦ ∫ 휑|휔| is a positive linear functional, giving rise to a Radon 푋 measure on 푋. The operations of addition, rescaling and group actions on volume forms mirror those on Radon measures. Remark 1.37. In many applications, 푋 will come from a smooth 퐹-variety X or its open subsets, and |휔| will come from a nowhere-vanishing algebraic differential form 휔 thereon, say by “taking absolute values ‖ ⋅ ‖퐹”; we call such an 휔 an algebraic volume form on X. Definition 1.38. Let 푋 be an 퐹-analytic manifolds, where 퐹 is a local field. Set ⎧ ⎪ ∞ ⎨푓 ∈ 퐶푐(푋) ∶ infinitely differentiable� , archimedean 퐹 퐶푐 (푋) ∶= ⎩⎪ 푓 ∈ 퐶푐(푋) ∶ locally constant� , non-archimedean 퐹 where locally constant means that every 푥 ∈ 푋 admits an open neighborhood 푈 such that 푓|푈 is constant.
12 1.7 More examples We begin by pinning down the Haar measures on a local field 퐹, viewed as additive groups.
Archimedean case 퐹 = ℝ Use the Lebesgue measure.
Archimedean case 퐹 = ℂ Use twice of the Lebesgue measure.
Non-archimedean case To prescribe a Haar measure on 퐹, by Theorem 1.20 it suffices to assign a vol- ume to the compact subring 픬퐹 of integers. A canonical choice is to stipulate that 픬퐹 has volume 1, but this is not always the best recipe.
Example 1.39. If 퐺 is a Lie group over any local field 퐹, i.e. a group object in the category of 퐹-analytic manifolds, one can construct left (resp. right) Haar measures as follows. Choose a basis 휔, … , 휔푛 of the cotangent space of 퐺 at 1. The cotangent bundle of 퐺 is trivializable by left (resp. right) translation, therefore 휔, … , 휔푛 can be identified as globally defined, left (resp. right) invariant differentials. The nowhere-vanishing top form 휔 ∶= 휔 ∧ ⋯ ∧ 휔푛, or more precisely |휔|, gives the required measure on 퐺. The Haar measures constructed in this way are unique up to scaling, as the form 휔 (thus |휔|) is. We shall denote the tangent space at 1 ∈ 퐺 as 픤: we have dim퐹 픤 = 푛. The automorphism Ad(푔) ∶ .(푥 ↦ 푔푥푔− induces a linear map on 픤 at 1, denoted again by Ad(푔 Proposition 1.40. In the circumstance of Example 1.39, one has
훿 (푔) = ￎ det(Ad(푔) ∶ 픤 → 픤) ￎ , 푔 ∈ 퐺. 퐺 퐹 Proof. This follows from Theorem 1.33.
Example 1.41. Let 퐹 be a local field. For the multiplicative group 퐹×, the algebraic volume form
푥− d푥 defines a Haar measure: it is evidently invariant under multiplications. More generally, let 퐺 = GL(푛, 퐹). 푛 We take 휂 to be a translation-invariant nonvanishing algebraic volume form on 푀푛(퐹) ≃ 퐹 . The 퐺- invariant algebraic volume form
휔(푔) ∶= (det 푔)−푛휂(푔), 푔 ∈ 퐺 induces a left and right Haar measure on 퐺.
Example 1.42. Here is a non-unimodular group. Let 퐹 be a local field, and 푛 = 푛 + ⋯ + 푛 where 푛, 푛 ∈ ℤ≥. Consider the group of block upper-triangular matrices
푔 푥 GL GL 푃 ∶= 푔 = ∈ (푛, 퐹) ∶ 푔푖 ∈ (푛푖, 퐹), 푥 ∈ 푀푛×푛 (퐹) 푔 the unspecified matrix entries being zero. This is clearly the spaceof 퐹-points of an algebraic group P. ∗ ∗ The Lie algebra of P is simply ( ∗ ). A routine calculation of block matrices gives
−− 푔 푥 푎 푏 푔 푥 푎 푔푏푔 = 푔 푑 푔 푑 where , and . Thus Proposition 1.40 and Theorem 1.33 imply (푎 ∈ 푀푛 (퐹) 푏 ∈ 푀푛 (퐹) 푏 ∈ 푀푛×푛 (퐹
푛 −푛 . 훿푃(푔) = ‖ det 푔‖퐹 ⋅ ‖ det 푔‖퐹
13 1.8 Homogeneous spaces Let 퐺 be a locally compact group. In what follows we consider spaces with right 퐺-actions. One can switch to left 퐺-actions by passing to 퐺op. Definition 1.43. A 퐺-space in the category of locally compact spaces is a locally compact Hausdorff space 푋 equipped with a continuous right 퐺-action 푋 × 퐺 → 푋. If the 퐺-action is transitive, 푋 is called a homogeneous 퐺-space. The 퐺-spaces form a category: the morphisms are continuous 퐺-equivariant maps. Example 1.44. Let 퐻 be a closed subgroup of 퐺. The coset space 퐻\퐺 is locally compact and Hausdorff by Proposition 1.4, and 퐺 acts continuously on the right by (퐻푥, 푔) ↦ 퐻푥푔. This is a homogeneous 퐺- space.
For a general homogeneous 퐺-space 푋 and 푥 ∈ 푋, let 퐺푥 ∶= Stab퐺(푥); it is a closed subgroup, and − we have 퐺푥푔 = 푔 퐺푥푔 for 푔 ∈ 퐺. The orbit map orb푥 ∶ 푔 ↦ 푥푔 factors through
orb푥 ∶ 퐺푥\퐺 → 푋
퐺푥푔 ⟼ 푥푔, which is a continuous 퐺-equivariant bijection. If orb푥 is actually an isomorphism, then 푋 ≃ 퐺푥\퐺 in the category of 퐺-spaces. This indeed holds under mild conditions on 퐺, thereby showing that Example 1.44 is the typical sort of homogeneous spaces.
Proposition 1.45. Suppose that 퐺 is second countable as a topological space. Then for any homogeneous 퐺-space 푋 and any 푥 ∈ 푋, the map orb푥 ∶ 퐺푥\퐺 → 푋 is an isomorphism of 퐺-spaces.
Proof. It suffices to show that orb푥 is an open map. Let 푈 ∋ 1 be any compact neighborhood in 퐺. There ⋃ ⋃ ,is a sequence 푔, 푔, … ∈ 퐺 such that 퐺 = 푖≥ 푈푔푖, therefore 푋 = 푖≥ 푥푈푔푖. Each 푥푈푔푖 is compact hence closed in 푋. Baire’s theorem implies that some 푥푈푔푖 has nonempty interior. Since 푥푈푔푖 ≃ 푥푈 by right translation, ,we obtain an interior point 푥ℎ of the subset 푥푈 of 푋, with ℎ ∈ 푈. Then 푥 is an interior point of 푥푈ℎ− .hence of the bigger subset 푥푈푈− Now consider any open subset 푉 ⊂ 퐺 and 푔 ∈ 푉. Take a compact neighborhood 푈 ∋ 1 such that 푈푈−푔 ⊂ 푉. Then .푥푔 ∈ 푥푈푈−푔 ⊂ 푥푉 ,As 푥 has been shown to an interior point of 푥푈푈−, so is 푥푔 in 푥푈푈−푔 ⊂ 푥푉. Since 푔, 푉 are arbitrary we conclude that 푔 ↦ 푥푔 is an open map. The same holds for orb푥 by the definition of quotient topologies.
Corollary 1.46. If 퐺 is an 퐹-analytic Lie group, where 퐹 is a local field, then the condition in Proposition 1.45 holds, thus every homogeneous 퐺-space takes the form 퐻\퐺 for some 퐻. Proof. The second countability is built into the definitions.
-Example 1.47. Take 푋 to be the (푛 − 1)-sphere 핊푛−, on which the orthogonal group O(푛, ℝ) acts con tinuously (in fact analytically, or even algebraically) and transitively. The action is given by (푣, 푔) ↦ 푣푔 where we regard 푣 as a row vector. The subgroup SO(푛, ℝ) also acts transitively on 핊푛−, since every vector 푣 ∈ 핊푛− is fixed by reflections relative to any hyperplane containing 푣, which have determinant −1. This gives .(핊푛− ≃ O(푛 − 1, ℝ)\O(푛, ℝ Indeed, the stabilizer group of 푣 = (1, 0, … , 0) can be identified with O(푛 − 1, ℝ).
14 Example 1.48. Another important example of homogeneous space is given by the Siegel upper half plane. Here it is customary to work with left actions by setting
−1푛×푛 Sp(2푛, ℝ) ∶= 푔 ∈ GL(2푛, ℝ) ∶ 푔 ⋅ 퐽 ⋅ 푡푔 = 퐽� , 퐽 ∶= . 1푛×푛
In a coordinate-free language, Sp(2푛, ℝ) is the isometry group of the symplectic form prescribed by 퐽. Take 푋 to be
ℋ푛 ∶= {푍 = 푅 + 푖푆 ∈ 푀푛×푛(ℂ) ∶ 푅, 푆 ∈ 푀푛×푛(ℝ) symmetric, 푆 > 0} .
Let Sp(2푛, ℝ) act on the left of ℋ푛 by
− 푔푍 ∶= (퐴푍 + 퐵)(퐶푍 + 퐷) , 푍 ∈ ℋ푛, 퐴 퐵 푔 = ∈ Sp(2푛, ℝ), 퐴, 퐵, 퐶, 퐷 ∈ 푀 (ℝ). 퐶 퐷 푛×푛
This is manifestly an analytic map in (푔, 푍). One can show that • this is indeed a transitive group action, and
• the stabilizer group of 푍 = 푖 ⋅ 1푛×푛 is the unitary group 푈(푛).
Therefore ℋ푛 ≃ Sp(2푛, ℝ)/푈(푛). When 푛 = 1, it reduces to the usual ℋ ∶= {휏 ∈ ℂ ∶ ℑ(휏) > 0} with SL(2, ℝ) acting by linear fractional transformations.
Example 1.49. Let 퐹 be a non-archimedean local field, with ring of integers 픬퐹. By convention, we identify 퐹푛 with the space of row vectors, and let GL(푛, 퐹) act from the right. A lattice in an 푛-dimensional 퐹-vector space 푉 is a free 픬퐹-submodule 퐿 ⊂ 푉 of rank 푛 such that 푛 푛 퐿 ⋅ 퐹 = 푉; the standard example is 픬퐹 inside 퐹 . Define the discrete space
푋 ∶= {lattices 퐿 ⊂ 푉} with right GL(푉)-action. Exercise: check that the GL(푉)-action is continuous and transitive: in fact StabGL(푉)(퐿) is closed and open in GL(푉). 푛 푛 Now identify 푉 with 퐹 by choosing a basis. The stabilizer of 퐿 = 픬퐹 is GL(푛, 픬퐹), hence
푋 ≃ GL(푛, 픬퐹)\ GL(푛, 퐹).
Examples 1.47—1.48 are well-known instances of Riemannian symmetric spaces. Loosely speaking, the space 푋 of lattices is in contrast some non-archimedean counterpart of symmetric spaces of the same type as Example 1.48: it consists of the vertices of the enlarged Bruhat–Tits building of GL(푉). Next, we turn to the measures on the 퐺-spaces 퐻\퐺, where 퐻 is a closed subgroup of 퐺.
Lemma 1.50. Fix a right Haar measure 휇퐻 on 퐻. The map
퐶푐(퐺) ⟶ 퐶푐(퐻\퐺)
♭ 푓 ⟼ 푓 ∶= 퐻푔 ↦ ﾙ 푓(ℎ푔) d휇퐻 (ℎ) 퐻 is well-defined, surjective and 퐺-equivariant with respect to the left 퐺-actions on 퐶푐(퐺) and 퐶푐(퐻\퐺) 푔 given by 푓 ↦ ￮ 푓 ∶ 푥 ↦ 푓(푥푔). It maps 퐶푐(퐺)+ onto 퐶푐(퐻\퐺)+.
15 ♭ Proof. The only nontrivial part is the surjectivity. Given 푓 ∈ 퐶푐(퐻\퐺), by Lemma 1.5 there exists a compact 퐾 ⊂ 퐺 such that 푔 ↦ 푓♭(퐻푔) is supported on 퐻퐾. We want to define ⎧ ⎪ 휙(푔) ⎨⎪푓♭(퐻푔) ⋅ , 푔 ∈ 퐻퐾 푓(푔) ∶= ⎪ 휙♭(퐻푔) ⎪ ⎩0, 푔 ∉ 퐻퐾 for some 휙 ∈ 퐶푐(퐺)+ depending solely on 퐾. Specifically, take any 휙 ∈ 퐶푐(퐺)+ such that 휙|퐾 > 0. For any 푔 ∈ 퐻퐾 we have 휙♭(푔) ≥ inf 휙♭(퐻푥) > 0 푥∈퐾
♭ by assumption, hence 푓 is a well-defined element in 퐶푐(퐺). If 푓 ∈ 퐶푐(퐻\퐺)+ then 푓 ∈ 퐶푐(퐺)+. It is also readily verified that ∫ 푓(ℎ푔) d휇 (ℎ) = 푓♭(퐻푔) for 푔 ∈ 퐻퐾, and zero for 푔 ∉ 퐻퐾, this 퐻 퐻 affords the required preimage of 푓♭.
♭ Fix a right Haar measure 휇퐻 on 퐻 to define 푓 ↦ 푓 as in Lemma 1.50. Every positive Radon measure 휇 on 푋 induces 휇♮ on 퐺, characterized by ∫ 푓 d휇♮ = ∫ 푓♭ d휇 . 퐻\퐺 퐻\퐺 퐺 퐻\퐺 퐻\퐺 퐻\퐺 Also recall the notion of quasi-invariance from Definition 1.14.
Lemma 1.51. Fix a right Haar measure 휇퐻 on 퐻 and let 휇퐻\퐺 be a quasi-invariant positive measure × ♮ on 푋 with eigencharacter 휒 ∶ 퐺 → ℝ>. Then 휇퐻\퐺 is of eigencharacter 휒 (resp. 훿퐻 (ℎ)) under right 퐺-action (resp. left 퐻-action).
♭ ♮ Proof. Since 푓 ↦ 푓 respects right 퐺-translation, the eigencharacter of 휇퐻\퐺 is the same 휒. Now let ℎ ℎ ♭ 푓 ∈ 퐶푐(퐺); recall that 푓 (푔) = 푓(ℎ푔). By Lemma 1.28, (푓 ) (퐻푔) equals
′ ′ ℎ − ﾙ 푓 (푘푔) d휇퐻 (푘) = ﾙ 푓(ℎ푘푔) d휇퐻 (푘) = 훿퐻 (ℎ) ﾙ 푓(ℎ 푔) d휇퐻 (ℎ ) 퐻 퐻 퐻
.which is 훿 (ℎ)−푓♭(퐻푔). This implies that ∫ 푓ℎ d휇♮ = 훿 (ℎ)− ∫ 푓 d휇♮ as asserted 퐻 퐺 퐻\퐺 퐻 퐺 퐻\퐺 × Theorem 1.52. Let 휒 ∶ 퐺 → ℝ> be a continuous homomorphism. Then there exists a quasi-invariant positive measure on 퐻\퐺 of eigencharacter 휒 if and only if
− 휒|퐻 = 훿퐻 (훿퐺|퐻 ) .
× Moreover, such a quasi-invariant measure is unique up to ℝ> when 휒|퐻 = 훿퐻 훿퐺|퐻 , and every choice of right Haar measures 휇퐺 (resp. 휇퐻 ) on 퐺 (resp. 퐻) gives rise to such a measure 휇퐻\퐺, characterized by
ﾙ 푓(푔)휒(푔) d휇퐺(푔) = ﾙ ﾙ 푓(ℎ푔) d휇퐻 (ℎ) d휇퐻\퐺(퐻푔) (1.9) 퐺 퐻\퐺 퐻 where 푓 ∈ 퐶푐(퐺). Consequently, 퐺-invariant positive measures exists on 퐻\퐺 if and only if 훿퐻 = 훿퐺|퐻 , in which case it is unique up to rescaling.
− Proof. First, suppose 휇퐻\퐺 with eigencharacter 휒 is given on 퐻\퐺. Let us verify 휒|퐻 = 훿퐻 (훿퐺|퐻 ) and ♮ (1.9). Fix a right Haar measure 휇퐻 on 퐻 to apply Lemma 1.51 to obtain 휇퐻\퐺. Lemma 1.51 implies ♮ 휇퐻\퐺 = 휒 ⋅ 휇퐺 for a right Haar measure 휇퐺 on 퐺. Both sides are quasi-invariant under left 퐻-action: ♮ the eigencharacter of 휇퐻\퐺 is 훿퐻 , whilst that of 휒 ⋅ 휇퐺 is (휒훿퐺)|퐻 by Remark 1.15 and Lemma 1.28. The formula (1.9) is built into our construction.
16 − Conversely, given 휒 with 휒|퐻 = 훿퐻 (훿퐺|퐻 ) , fix 휇퐺 and 휇퐻 . We claim that
퐼 ∶ 퐶푐(퐻\퐺) ⟶ ℂ ♭ ♭ 푓 ⟼ ﾙ 푓휒 d휇퐺, 푓 ↦ 푓 퐺 is well-defined. Granting this, one readily sees that 퐼 defines a Radon measure 휇퐻\퐺 by Lemma 1.50. ♭ Furthermore, since 푓 ↦ 푓 respects right 퐺-action, 휇퐻\퐺 will have the same eigencharacter as 휒휇퐺, namely 휒. What remains to show is that 푓♭ = 0 implies ∫ 푓휒 d휇 = 0. For any 휑 ∈ 퐶 (퐺), Fubini’s 퐺 퐺 푐 theorem entails
♭ 0 = ﾙ 휑(푔)푓 (퐻푔)휒(푔) d휇퐺(푔) = ﾙ ﾙ 휑(푔)푓(ℎ푔)휒(푔) d휇퐻 (ℎ) d휇퐺(푔) 퐺 퐺 퐻 ′ ′ ′ − −− ′ = ﾙ ﾙ 휒(푔 )휑(ℎ 푔 )푓(푔 )휒(ℎ) 훿퐺(ℎ) d휇퐺(푔 ) d휇퐻 (ℎ) 퐻 퐺 ′ − ′ by first swapping the integrals and then substituting 푔 = ℎ푔, using d휇퐺(푔) = 훿퐺(ℎ) d휇퐺(푔 ) from ′ ′ −− ′ Lemma 1.28. Next, substitute ℎ = ℎ into the last integral and use d휇퐻 (ℎ) = 훿퐻 (ℎ ) d휇퐻 (ℎ ) from Lemma 1.28 to arrive at
′ ′ − ′ ′ ′ ′ ′ ′ ′ 0 = ﾙ ﾙ 휒(푔 )휑(ℎ 푔 )푓(푔 ) 휒(ℎ )훿퐺(ℎ )훿퐻 (ℎ ) d휇퐻 (ℎ ) d휇퐺(푔 ) 퐻 퐺 =
♭ ′ ′ ′ ′ = ﾙ 휑 (퐻푔 )푓(푔 )휒(푔 ) d휇퐺(푔 ). 퐺 Select 휑 so that 휑♭ = 1 on the image of Supp(푓) in 퐻\퐺 to deduce ∫ 푓휒 d휇 = 0. 퐺 퐺 When 퐺 is an 퐹-analytic Lie group and 퐻 is a Lie subgroup, the results can also be deduced by considerations of volume forms. We leave this to the curious reader.
Definition 1.53. Denote by 휇퐺/휇퐻 the quasi-invariant measure on 퐻\퐺 of eigencharacter 휒 determined × .by (1.9). To see the benefits of this notation, note that (푠휇퐺)/(푡휇퐻 ) = (푠/푡) ⋅ 휇퐺/휇퐻 for all 푠, 푡 ∈ ℝ> Remark 1.54. For homogeneous spaces under left 퐺-actions, one uses left Haar measures on 퐻 to define 푓 ↦ 푓♭, and the condition in Theorem 1.52 becomes
− 휒|퐻 = (훿퐻 ) ⋅ 훿퐺|퐻 . Indeed, one switches to right 퐺op-actions and applies the previous result; the eigencharacter 휒 is unal- tered, whereas the modulus characters are replaced by inverses by Proposition 1.30.
2 Convolution
Let 퐺 be a locally compact group.
2.1 Convolution of functions (Fix a right Haar measure 휇 on 퐺, and define 퐿푝(퐺) ∶= 퐿푝(퐺, 휇) for all 1 ≤ 푝 ≤ ∞. Write 푓(푥)̌ ∶= 푓(푥− (sometimes as 푓∨(푥)) for all functions 푓 on 퐺. Given two functions 푓, 푓 on 퐺, their convolution is defined as the function
− (푓 ⋆ 푓)(푥) = ﾙ 푓(푥푦 )푓(푦) d휇(푦) 퐺 (2.1) − ﾙ 푓(푦 )푓(푦푥) d휇(푦), 푥 ∈ 퐺 = 퐺
17 whenever it converges. We will also consider (2.1) as an element in various 퐿푝-spaces on 퐺, by estimating .(푓 ⋆ 푓‖퐿푝 in terms of the norms of 푓 and 푓 (see below‖ Remark 2.1. If we use left Haar measures instead, or equivalently work in 퐺op, then (2.1) becomes .(푓 (푥푦)푓 (푦−) d휇(푦) = ∫ 푓 (푦)푓 (푦−푥) d휇(푦 ∫ 퐺 퐺 ,(Definition–Proposition 2.2. Let 푓, 푓 ∈ 퐿 (퐺). Then 푓 ⋆ 푓 is a well-defined element in 퐿 (퐺 satisfying
. 푓 ⋆ 푓‖퐿 ≤ ‖푓‖퐿 ⋅ ‖푓‖퐿‖ -The convolution product makes 퐿(퐺) into a Banach algebra, which is in general non-commutative, non unital.
Proof. To show that 푓 ⋆푓 ∈ 퐿(퐺) is well-defined, evaluate ∬ |푓 (푥푦−)|⋅|푓 (푦)| d휇(푦) d휇(푥) using 퐺×퐺 Fubini’s theorem to get ‖푓‖퐿 ⋅ ‖푓‖퐿 . It is straightforward to verify that
,푓 ⋆ (푓 ⋆ 푓) = (푓 ⋆ 푓) ⋆ 푓
,푓 ⋆ (푓 + 푓) = 푓 ⋆ 푓 + 푓 ⋆ 푓
.푓 + 푓) ⋆ 푓 = 푓 ⋆ 푓 + 푓 ⋆ 푓)
.Hence (퐿(퐺), ⋆) is a Banach algebra It is important to extend the convolution to more general functions and deduce the corresponding estimates. Following analysts’ convention, define a bijection 푝 ↦ 푝′ from [1, +∞] to itself by requiring
1 1 + = 1. 푝 푝′
푝 ,(Theorem 2.3 (Minkowski’s inequality). Suppose that 1 ≤ 푝 ≤ ∞. Then for 푓 ∈ 퐿 (퐺) and 푓 ∈ 퐿 (퐺 (2.1) is well-defined in 퐿푝(퐺) and we have
. ￎ푓 ⋆ 푓 ￎ ≤ ‖푓 ‖ 푝 ⋅ ‖푓 ‖ 퐿푝 퐿 퐿 Proof. The case 푝 = 1 has just been addressed, and the case 푝 = ∞ is easy. Assume 1 < 푝 < ∞. To (ease notations, we may and do assume 푓, 푓 ≥ 0. Apply Hölder’s inequality to the measure 푓(푦) d휇(푦 to obtain
′/푝 /푝 − 푝− . (ﾙ 푓(푥푦 )푓(푦) d휇(푦) ≤ ﾙ 푓(푥푦 ) 푓(푦) d휇(푦) ﾙ 푓(푦) d휇(푦 퐺 퐺 퐺 Hence
/푝 푝− − 푝 푝 d d (푓 ⋆ 푓‖퐿 ≤ ‖푓‖퐿 ⋅ ｪ 푓(푥푦 ) 푓(푦) 휇(푦) 휇(푥‖ 퐺×퐺 /푝 푝− 푝 d d (‖푓‖퐿 ⋅ ｪ 푓(푥) 푓(푦) 휇(푥) 휇(푦 = 퐺×퐺 푝− 푝 /푝 푝 푝 푓‖퐿 ⋅ ‖푓‖퐿 ⋅ ‖푓‖퐿 = ‖푓‖퐿 ⋅ ‖푓‖퐿‖ = where the right invariance of 휇 is used.
18 Theorem 2.4 (Young’s inequality). Suppose that 1 ≤ 푝, 푞, 푟 ≤ ∞ satisfy 1 1 1 + 1 = + . 푞 푝 푟
̌ 푝 푟 푞 (Then for all 푓, 푓 such that 푓 ∈ 퐿 (퐺) and 푓 ∈ 퐿 (퐺), the convolution (2.1) is well-defined in 퐿 (퐺 and satisfies ̌ . 푓 ⋆ 푓‖퐿푞 ≤ ‖푓‖퐿푝 ⋅ ‖푓‖퐿푟‖ Proof. The premises imply 1 1 1 푝 푝 푟 푟 + + = 1, + = 1, + = 1. 푟′ 푞 푝′ 푞 푟′ 푞 푝′
′ ′ First, assume 푓, 푓 ≥ 0. Apply Hölder’s inequality with three exponents (푟 , 푞, 푝 ) and use the right invariance of 휇 to deduce that
− (푓 ⋆ 푓)(푥) = ﾙ 푓(푦 )푓(푦푥) d휇(푦) 퐺 푝 푝 푟 푟 ′ ′ − 푞 푞 푝− (ﾙ 푓(푦 ) 푟 푓(푦 ) 푓(푦푥) 푓(푦푥) d휇(푦 = 퐺 ′/푟′ /푞 /푝 푝 − 푝 푟 푟 ̌ (ﾙ (푓) d휇 ﾙ 푓(푦 ) ⋅ 푓(푦푥) d휇(푦) ﾙ 푓(푦푥) d휇(푦 ≥ 퐺 퐺 퐺 /푞 푝/푟′ ′ 푝 푟 푟/푝− ̌ . ￎ푓ￎ 푝 ﾙ 푓(푦 ) 푓(푦푥) d휇(푦) ‖푓‖퐿푟 = 퐿 퐺 It follows from Fubini’s theorem that
/푞 ′ 푝/푟 푟/푝′ 푞 ̌ − 푝 푟 (푓 ⋆ 푓‖퐿 ≤ ￎ푓ￎ 푝 ‖푓‖퐿푟 ｪ 푓(푦 ) 푓(푦푥) d휇(푥) d휇(푦‖ 퐿 퐺×퐺 ′ 푝/푟 푟/푝′ 푝/푞 푟/푞 = ￎ푓̌ ￎ ⋅ ‖푓 ‖ 푟 ⋅ ‖푓̌ ‖ 푝 ⋅ ‖푓 ‖ 푟 = ‖푓̌ ‖ 푝 ⋅ ‖푓 ‖ 푟 . 퐿푝 퐿 퐿 퐿 퐿 퐿
.In general, we can work with |푓|, |푓| to reduce to the previous setting 푟 ̌ 푟 Proposition 2.5. If 휑 ∈ 퐶푐(퐺) and 푓 ∈ 퐿 (퐺) (resp. 푓 ∈ 퐿 (퐺)) with 1 ≤ 푟 ≤ ∞, then 휑 ⋆ 푓 (resp. 푓 ⋆ 휑) is defined by a convergent integral and is a continuous function on 퐺. Proof. Recall 휑푔(푥) ∶= 휑(푔푥) and 푔휑(푥) = 휑(푥푔) for 푔, 푥 ∈ 퐺. Apply Theorem 2.4 with 푞 = ∞ and + = 1 to see 푝 푟
|(휑 ⋆ 푓)(푔푥) − (휑 ⋆ 푓)(푥)| = |((휑 − 휑푔) ⋆ 푓)(푥)| ≤ ￎ휑∨ − (휑푔)∨ￎ ￎ푓ￎ . 퐿푝 퐿푟 We conclude by noting that
푔→ ∨ 푔 ∨ ∨ 푔− ∨ ￎ휑 − (휑 ) ￎ 푝 = ￏ휑 − (휑 )ￏ −−−−→0. 퐿 퐿푝 The case with 푓̌ ∈ 퐿푟(퐺) is completely analogous.
2.2 Convolution of measures [ UNDER CONSTRUCTION ]
19 3 Continuous representations
3.1 General representations We will work exclusively with topological vector spaces over ℂ. Some words on Hausdorff property: a topological vector space 푉 is also a topological group under +, therefore 푉 being Hausdorff is the same as {0} = {0}. As easily observed, {0} ⊂ 푉 is a closed vector subspace, thus 푉/{0} is a Hausdorff topological vector space. The assignment 푉 ↦ 푉/{0} is functorial and is the left adjoint of the inclusion functor {Hausdorff ones} ↪ {topological vector spaces}. The main topological vector spaces under consideration are Hausdorff and locally convex; the latter means that there is a basis of neighborhoods consisting of convex subsets. Alternatively, locally convex Hausdorff spaces are exactly the topological vector spaces whose topology is defined by afamilyof separating semi-norms. Definition 3.1. Henceforth we denote by TopVect the category of locally convex Hausdorff topological vector spaces over ℂ. In contrast, Vect will stand for the category of ℂ-vector spaces. Remark 3.2. The property of being locally convex and Hausdorff is preserved under passing to subspaces and to quotients by a closed subspace. The direct sums of locally convex Hausdorff spaces also carry a locally convex Hausdorff topology, characterized in categorical terms. More generally, one can form the inductive limits lim inside the category of locally convex spaces; −−→ these limits are not necessarily Hausdorff, but one can take quotient by {0} to land in TopVect. Example 3.3. The Hilbert, Banach, Fréchet and LF-spaces are all in TopVect. Every finite-dimensional vector space 푉 admits a unique structure of Hausdorff topological vector space, namely the usual one coming from 푉 ≃ ℂdim 푉 , which is also locally convex.
Denote by AutTopVect(푉) (resp. AutVect(푉)) the group of automorphisms of 푉 taken in TopVect (resp. in the category of ℂ-vector spaces); AutVect(푉) is much larger than AutTopVect(푉) for infinite- dimensional 푉. We will consider certain linear left actions of a locally compact group 퐺 on a topological vector space 푉. This is the same as prescribing a homomorphism 휋 ∶ 퐺 → AutVect(푉) of groups; it can also be recorded by the corresponding “action map” 푎 ∶ 퐺 × 푉 → 푉 with 푎(푔, 푣) = 푔푣 = 휋(푔)푣. The issue of continuity is more delicate. Definition 3.4. Let 퐺 be a locally compact group and let 푉 in TopVect.A continuous representation of 퐺 on the left of 푉 is a homomorphism 휋 ∶ 퐺 → AutVect(푉) such that the corresponding action map 푎 ∶ 퐺 × 푉 ⟶ 푉 (푔, 푣) ⟼ 푔푣 ∶= 휋(푔)푣 is continuous. The case of right actions is defined analogously, with the action written as (푣, 푔) ↦ 푣푔 with the property 푣(푔푔′) = (푣푔)푔′, etc. We denote a continuous representation of 퐺 as (휋, 푉), 푉 or 휋 interchangeably, depending on the context; call 푉 = 푉휋 the underlying vector space of 휋. The shorthand 퐺-representation will be used extensively.
-Definition 3.5. A morphism of 퐺-representations 휑 ∶ 푉 → 푉 is understood to be continuous homo morphism of topological vector spaces such that 휑(푔푣) = 푔휑(푣) for all 푔 ∈ 퐺 and 푣 ∈ 푉. This turns the collection of all 퐺-representations into a category 퐺-Rep, so one can talk about isomorphisms, automorphisms, etc. in the usual manner. The morphisms between 퐺-representations are also called intertwining operators.
20 Definition 3.6. A subrepresentation of a 퐺-representation (푉, 휋) is a closed subspace 푉 ⊂ 푉 that is stable under 퐺-action. A quotient representation is a quotient space 푉̄ = 푉/푉 by a subrepresentation .푉, endowed with the quotient topology. In both cases, 푉 and 푉̄ are naturally 퐺-representations The 퐺-action on the direct sum ⨁ 푉 of 퐺-representations {푉 } makes it into a 퐺-representation. 푖∈퐼 푖 푖 푖∈퐼 The same holds for more general lim of 퐺-representations. −−→ Call a 퐺-representation 푉 ≠ {0} simple or irreducible if the only subrepresentations (equivalently, quotient representations) are {0} and 푉. Call it indecomposable if 푉 = 푉 ⊕ 푉 with subrepresentations .{푉, 푉 ⊂ 푉 implies that either 푉 = {0} or 푉 = {0 One of the basic goals of representation theory is to classify the irreducible 퐺-representations under various constraints. Remark 3.7. The category 퐺-Rep is additive and ℂ-linear. In particular, the endomorphisms of a 퐺- representation 푉 form a ℂ-algebra Endℂ(푉). We caution the reader that, unlike the case of finite- dimensional representations, 퐺-Rep is not an abelian category: already in the case of 퐺 = {1} so that 퐺-Rep = TopVect, every homomorphism 휑 ∶ 푉 → 푊 of topological vector spaces admits kernel 휑−(0) and cokernel 푊/im(휑); by taking 휑 with dense image, one can show that
휑 ker(휑) = 0 = coker(휑), coim(휑) ≃ 푉 −→푊 ≃ im(휑) so that the canonical morphism coim(휑) → im(휑) is not an isomorphism if 휑 is not surjective, contra- dicting the axioms for abelian categories. Such a 휑 ∶ 푉 → 푊 already exists for Banach spaces.
Let 푉 be in TopVect with a given homomorphism 휋 ∶ 퐺 → AutVect(푉), so that 퐺 acts linearly on the left of 푉. Consider the orbit map for every 푣 ∈ 푉
orb푣 ∶ 퐺 ⟶ 푉 푔 ⟼ 푔푣.
The following result gives a convenient means to check the continuity of the action map 푎 ∶ 퐺 × 푉 → 푉 corresponding to 휋. Proposition 3.8. For any 푉 in TopVect and a left linear action of 퐺 on 푉 (no continuity condition so far), denote by 휋(푔) the map 푣 ↦ 푔푣. The following are equivalent. 1. the action map 푎 ∶ 퐺 × 푉 → 푉 is continuous, i.e. (푉, 휋) is a 퐺-representation; the conditions below hold: (a) there is a dense subspace 푉 ⊂ 푉 such that orb푣 is continuous for .2 all 푣 ∈ 푉, and (b) for every compact subset 퐾 ⊂ 퐺 the family 휋(푔) ∶ 푔 ∈ 퐾� of operators on 푉 is equicontinuous.
Proof. Suppose that 푎 ∶ 퐺 × 푉 → 푉 is continuous, then orb푣 is continuous for all 푣 ∈ 푉 ∶= 푉. Let 퐾 ⊂ 퐺 be compact. Given a neighborhood 푊 ∋ 0 in 푉, we have to find another neighborhood 푈 ∋ 0 so that 휋(푔)(푈) ⊂ 푊 for each 푔 ∈ 퐾. By the definition of product topology, for each 푔 ∈ 퐾 there is a − ′ neighborhood 푈푔 × 푈푔 ∋ (푔, 0) in 퐺 × 푉 that is contained in 푎 (푊). There is a finite subset 푇 ⊂ 퐾 such ⋃ ′ ⋂ that 퐾 = 푡∈푇 푈푡 ; it remains to take 푈 ∶= 푡∈푇 푈푡. Conversely, assume (a) and (b), consider (푔, 푣) ∈ 퐺 × 푉 and let 푊 ∋ 0 be a neighborhood in 푉. We seek a neighborhood 푈′ × 푈 of (푔, 푣) such that for all (푔′, 푣′) ∈ 푈′ × 푈 we have
푎(푔′, 푣′) − 푎(푔, 푣) = 휋(푔′)푣′ − 휋(푔)푣 ∈ 푊.
′ ′ ′ For any such neighborhoods 푈 , 푈, use density to pick 푣 ∈ 푉 ∩ 푈 ≠ ∅; then write 휋(푔)푣 − 휋(푔 )푣 as
′ ′ ′ ′ . 휋(푔)푣 − 휋(푔)푣 + 휋(푔)푣 − 휋(푔 )푣 + 휋(푔 )푣 − 휋(푔 )푣
21 ′ ′ ′ ′ Assume 푈 has compact closure, then 휋(푔)푣 − 휋(푔)푣 and 휋(푔 )푣 − 휋(푔 )푣 approaches zero when 푈 ′ shrinks, by applying (b) to 퐾 ∶= 푈̄ . Once 푣 ∈ 푉 ∩ 푈 is chosen, we can further shrink 푈 to make ′ .휋(푔)푣 − 휋(푔 )푣 arbitrarily small. This concludes the proof Corollary 3.9. Let (푉, ‖ ⋅ ‖) be a Banach space and a left action of 퐺 on 푉 (no continuity conditions so far). Then 푉 is a 퐺-representation if and only if 휋(푔) is continuous for every 푔 and the orbit map 푔 ↦ 푔푣 is continuous for every 푣 ∈ 푉.
∋ Proof. It suffices to prove the “if” part. Take 푉 = 푉 in 3.8 and check the equicontinuity of {휋(푔) ∶ 푔 퐾} ⊂ AutTopVect(푉) for every compact 퐾 ⊂ 퐺. More precisely, by the uniform boundedness principle of Banach–Steinhaus, it suffices to notice that the continuity of orbit maps implies
sup ￎ휋(푔)푣ￎ < +∞ 푔∈퐾 for every 푣 ∈ 푉, which in turn implies the asserted equicontinuity.
Example 3.10. Suppose that 푋 is a locally compact Hausdorff space with a continuous right 퐺-action. Equip continuous with the topology defined by semi-norms sup 퐶(푋) ∶= {푓 ∶ 푋 → ℂ, } ‖ ⋅ ‖∞,︵ ∶= ︵ ‖ ⋅ ‖ where Ω ranges over compact subsets of 푋. This makes 퐶(푋) into a member of TopVect. Let 퐺 act on the left of 퐶(푋) by 푔푓 ∶= 푔푓 = [푥 ↦ 푓(푥푔)], 푓 ∈ 퐶(푋), 푔 ∈ 퐺. Let us verify that 퐶(푋) is a 퐺-representation. It has been observed that 푔(푔′푓) = (푔푔′)푓. Apply 3.8 with .푉 = 푉 as follows • For every chosen 푓 ∈ 퐶(푋), we contend that 푔 ↦ 푔푓 is a continuous map from 퐺 to 퐶(푋). Continuity can be checked just at 푔 = 1. This amounts to showing that for every compact Ω ⊂ 푋 and every , there exists a neighborhood in such that sup 휖 > 0 푈 ∋ 1 퐺 푥∈︵ |푓(푥푔) − 푓(푥)| < 휖 whenever 푔 ∈ 푈. Indeed, for every 푥 ∈ 푋 there exist open 푈푥 ∋ 1 and 푊푥 ∋ 푥 in 퐺 and 푋, such that (푔, 푦) ∈ 푈푥 × 푊푥 ⟹ |푓(푦푔) − 푓(푦)| < 휖.
Now take a finite subcover of the open covering {푊푥}푥∈︵ of Ω, and take 푈 to be the finite inter- section of the corresponding 푈푥. • Fix a compact 퐾 ⊂ 퐺. For every 푔 ∈ 퐾 and every compact Ω ⊂ 푋 we know Ω퐾 is compact, and
.∞,휋(푔)푓 − 휋(푔)푓‖︵,∞ = sup |푓(푥푔) − 푓(푥푔)| ≤ ‖푓 − 푓‖︵퐾‖ 푥∈︵
This shows the equicontinuity of the operators {휋(푔) ∶ 푔 ∈ 퐾}.
Remark 3.11. When 푋 is second countable, an increasing countable sequence of Ω exhausts 푋, and it suffices to treat the corresponding semi-norms. In this case 퐶(푋) turns out to be Fréchet. If 푋 is compact, it suffices to take Ω = 푋 and 푋 is a Banach space.
Example 3.12. The same result in 3.10 holds for 퐺 acting on 퐶푐(푋), where
퐶 (푋) = lim 퐶 (푋, Ω) 푐 −−→ 푐 ︵ is equipped with the inductive limit topology, where
• Ω ⊂ 푋 ranges over the compact subsets, and
22 • 퐶푐(푋, Ω) ∶= 푓 ∈ 퐶푐(푋) ∶ Supp(푓) ⊂ Ω�, topologized by the sup-norm ‖ ⋅ ‖∞.
Then 퐶푐(푋) with 푔푓(푥) = 푓(푥푔) is also a 퐺-representation on an LF-space, and this coincides with 퐶(푋) when 푋 is compact. We check the continuity of 퐺 × 퐶푐(푋) → 퐶푐(푋) directly. Since 퐺 is locally compact, it suffices to show the continuity of 퐾 × 퐶푐(푋) → 퐶푐(푋) a given compact subset 퐾 ⊂ 퐺. By the definition of the topology of lim, we further reduce to the composite −−→
퐾 × 퐶푐(푋, Ω) → 퐶푐(푋, Ω ⋅ 퐾) ↪ 퐶푐(푋)
where Ω ⊂ 푋 is any compact. The continuity of ↪ is evident, whilst that of 퐾×퐶푐(푋, Ω) → 퐶푐(푋, Ω⋅퐾) is straightforward to check, as everything is compact.
Example 3.13. Let 푋 be a locally compact Hausdorff space, endowed with a continuous right 퐺-action. For simplicity, we assume that 푋 admits a 퐺-invariant Radon positive measure 휇; this is indeed the case when 푋 is a homogeneous 퐺-space isomorphic to 퐻\퐺 satisfying 훿퐺|퐻 = 훿퐻 , by 1.52. For every 1 ≤ 푝 ≤ ∞, we have 퐺 acting on the Banach space 푉 ∶= 퐿푝(퐺) by 푔푓 = 푔푓 ∶ 푥 ↦ 푓(푥푔) as 푔 usual. This makes sense if 푓 ∈ 퐶푐(푋), and ‖ 푓‖푝 = ‖푓‖푝 for all 푔 ∈ 퐺. Therefore the action extends to all 푓 ∈ 퐿푝(푋) by an approximation argument. (Claim: 퐿 (푋) is a 퐺-representation. We apply 3.8 with 푉 = 퐶푐(푋) as follows. The condition (b is satisfied since 휋(푔) preserves ‖ ⋅ ‖푝, hence equicontinuous. It remains to check (a) that 푔 ↦ 푔푓 is continuous for every given 푓 ∈ 퐶푐(푋), i.e. for every 휖 > 0, there exists a neighborhood 푈 ∋ 1 in 퐺 such that 푔 ∈ 푈 ⟹ ‖푔푓 − 푓‖푝 < 휖. Since Supp(푓) is compact, there exists a neighborhood 푈 ∋ 1 such that
.푓(푥푔) − 푓(푥)| < 휖휇(Supp(푓))−/푝 for all 푥|
Then ‖푔푓 − 푓‖푝 ≤ 휖 as required; note that the argument also works for 푝 = ∞, and in that case we do not need measures.
Representations on function spaces arising from 퐺-actions on 푋, such as the examples 3.10, 3.12 and 3.13 are called regular representations. The most important case is 푋 = 퐺. When 푋 = 퐺 is finite, we revert to the usual regular representation on the finite-dimensional space ℂ[퐺] = Maps(퐺, ℂ). We end these discussions by more terminologies.
Definition 3.14. Let (휋, 푉) be a 퐺-representation. If 푉 is a Hilbert (resp. Banach, Fréchet) space, we say that (휋, 푉) is a Hilbert (resp. Banach, Fréchet) representation. In each case, such representations form a full subcategory of 퐺-Rep.
3.2 Matrix coefficients
∗ Definition 3.15. For (휋, 푉) be in 퐺-Rep. Denote by 푉 ∶= HomTopVect(푉, ℂ) the topological dual of 푉. It is customary to write ⟨휆, 푣⟩ ∶= 휆(푣) for 휆 ∈ 푉∗, 푣 ∈ 푉. Given 휆, 푣, the corresponding matrix coefficient is the continuous function
푐푣⊗휆 ∶ 퐺 ⟶ ℂ 푔 ⟼ ﾉ휆, 휋(푔)푣ｼ .
Matrix coefficients is bilinear in 휆 and 푣, which justifies the tensor notation.
Remark 3.16. Suppose 푉 ≠ {0}. The Hahn–Banach theorem implies that for every 푣 ≠ 0 there exists 휆 such that 푐푣⊗휆(1) = ⟨휆, 푣⟩ ≠ 0.
23 We can let 퐺 act on the left of 푉∗ by the contragredient: 휆 ↦ ̌휋(푔)∶= 휆∘휋(푔−). However, the choice of topological structure on 푉∗ is a subtle issue, and in practice it is necessary to pass to some subspace of 푉∗ to obtain interesting continuous representations. At present, we do not put any extra structure on 푉∗. A Hilbert representation (휋, 푉) is called unitary if 휋(푔) is a unitary operator for all 푔 ∈ 퐺. We will say more about this in 4.1.
Remark 3.17. By Riesz’s theorem, the topological dual of a Hilbert space (푉, (⋅|⋅)푉 ) is its Hermitian conjugate 푉̄ : it is the same space with the new scalar multiplication (푧, 푣) ↦ 푧푣̄ for 푧 ∈ ℂ, and becomes ̄ a Hilbert space with (푣|푤)푉̄ = (푤|푣)푉 = (푣|푤)푉 . Every 푤̄ ∈ 푉 corresponds to the linear functional 푣 ↦ (푣|푤)푉 . The contragredient representation is therefore given by
휋(푔)∶= ∗휋(푔)− ∶ 푉̄ → 푉̄̌ where ∗(⋯) denotes the Hermitian adjoint. It satisfies ( ̌휋(푔)푤|휋(푔)푣)= (푤|푣). When 퐺 is unitary, the contragredient reduces ̌휋(푔)= 휋(푔); to emphasize the complex conjugation, we will also write ( ̄휋, 푉)̄ = ( ̌휋, 푉)̄ in the unitary case, and it is again unitary. Therefore 3.15 specializes as follows.
Definition 3.18. Let (휋, 푉) be a Hilbert representation of 퐺. For 푣 ∈ 푉 and 푤 ∈ 푉̄ , the corresponding matrix coefficient is the continuous map
푐푣⊗푤 ∶ 퐺 ⟶ ℂ
푔 ⟼ (휋(푔)푣|푤)푉 where (⋅|⋅) is the Hermitian form on 푉. We record some basic properties of matrix coefficients.
Lemma 3.19. Let (휋, 푉) be a Hilbert representation of 퐺. The matrix coefficients map 푤 ⊗ 푣 ↦ 푐푣⊗푤 is bilinear in 푉 × 푉̄ . Moreover:
− 푐푤⊗푣(푏 푥푎) = 푐휋(푎)푣⊗ ̌휋(푏)푤(푥), (푎, 푏) ∈ 퐺 × 퐺, 푥 ∈ 퐺, (3.1) − 푐푣⊗푤(푔) = 푐푤⊗푣(푔 ), 푣, 푤 ∈ 푉, 푔 ∈ 퐺, if 휋 is unitary; (3.2)
Here we write ̄휋 for the representation 휋 of 퐺 on 푉̄ , emphasizing that 푐푣⊗푤 is conjugate-linear in 푤. It is also additive under orthogonal direct sum 푉 ⊕ 푉 of Hilbert representations, namely
푐(푣,푣′)⊗(푤,푤′)(푔) = 푐푣⊗푤(푔) + 푐푣′⊗푤′ (푔) (3.3)
′ ′ .with 푔 ∈ 퐺, 푣, 푤 ∈ 푉 and 푣 , 푤 ∈ 푉
,Proof. When 휋 is unitary, the identity (휋(푔)푤|푣) = (푤|휋(푔−)푣) = (휋(푔−)푣|푤) proves (3.2). Next
(휋(ℎ−)휋(푥)휋(푔)푣|푤 = (휋(푥)휋(푔)푣| ̌휋(ℎ)푤 translates into (3.1).
24 3.3 Vector-valued integrals Vector-valued integrals are immensely useful for studying continuous representations. We shall consider a weak version thereof, called weak integrals or Gelfand–Pettis integrals. The canon is surely Bourbaki [3]; shorter surveys include [8, 3.26 — 3.29] and [4].
Definition 3.20. Let 푋 be a locally compact Hausdorff space and 휇 a positive Radon measure thereon. Let 푉 be in TopVect, and 푓 ∶ 푋 → 푉 is a function such that 휆 ∘ 푓 ∶ 푋 → ℂ is 휇-measurable for all 휆 in the continuous dual 푉∗. Suppose that 휆 ∘ 푓 is 휇-integrable for all 휆, define the integral ∫ 푓 d휇 as the 푋 element 휆 ⟼ ﾙ ﾉ휆, 푓(푥)ｼ d휇(푥), 휆 ∈ 푉∗ 푋 ∗ ∗ in HomVect(푉 , ℂ), i.e. the algebraic dual of 푉 . Since we are working with locally convex Hausdorff spaces, 푉∗ separates points on 푉 by the Hahn– ∗ Banach theorem, and the natural map 푉 → HomVect(푉 , ℂ) is injective. Therefore it makes sense to ask if ∫ 푓 d휇 exists inside 푉, which will then take a definite value, called the weak integral of 푓 d휇. 푋 When 푉 is a Banach space, the Bochner or the “strong” integrals are instances of weak integrals. Due to its “weak” definition, Gelfand–Pettis integrals satisfy agreeable functorial properties, as illustrated below.
Proposition 3.21. Let 푇 ∶ 푉 → 푊 be a morphism in TopVect. If ∫ 푓 d휇 in the situation of 3.20 exists, 푋 then so does ∫ 푇푓 d휇 and we have ∫ 푇푓 d휇 = 푇 ∫ 푓 d휇. Here we identify 푇 with the natural linear 푋 푋 푋 ∗ ∗ map HomVect(푉 , ℂ) → HomVect(푊 , ℂ) it induces.
Proof. By definition, 푇 ∫ 푓 d휇 is the linear functional on 푊 ∗ mapping each 휆 ∈ 푊 ∗ to ﾊ휆 ∘ 푇, ∫ 푓 d휇ｽ, 푋 푋 which is nothing but ∫ ﾉ휆, 푇푓ｼ d휇. Also, ﾉ휆, 푇푓ｼ = ﾉ휆 ∘ 푇, 푓ｼ is 휇-integrable by assumptions. 푋
A Radon measure on 푋 is called bounded if the integration functional 퐼 ∶ 퐶푐(푋) → ℂ is continuous for the 퐿∞-norm. A similarly recipe defines the locally bounded measures on 푋. Denote the support of a measure 휇 by Supp(휇).
Theorem 3.22. Suppose that 휇 is bounded in the situation of 3.20. If 푓(Supp(휇)) is contained in a convex, balanced, bounded and complete subset 퐵 ⊂ 푉, then 휆 ∘ 푓 is 휇-integrable for all 휆 ∈ 푉∗ and
ﾙ 푓 d휇 ∈ 휇(푋) ⋅ 퐵 ⊂ 푉. 푋 Proof. This is in [3, §1.2, Proposition 8].
Remark 3.23. Most often we encounter the case of
• 푓 ∶ 푋 → 푉 is continuous,
• 휇 has compact support.
In particular, 푓(Supp(휇)) is compact. The premises in 3.22 are met whenever compact subsets in 푉 have compact convex hulls. The latter property holds for quasi-complete spaces. A topological vector space is called quasi-complete if every bounded closed subset is complete. Completeness implies quasi- completeness as expected. In particular, 3.22 applies to Fréchet spaces. Another example of quasi-complete spaces are the continuous duals of barreled spaces, endowed with the topology of pointwise convergence. For instance, Fréchet spaces and LF spaces are all barreled.
25 Proposition 3.24. In the circumstance of 3.22, suppose that ‖ ⋅ ‖ ∶ 푉 → ℝ≥ is a continuous semi-norm on 푉. Then ﾙ 푓 d휇 ≤ ﾙ ‖푓‖ d휇. 푋 푋 Proof. Take 푣 = ∫ 푓 d휇 ∈ 푉. There exists 휆 ∈ 푉∗ with ⟨휆, 푣⟩ = ‖푣‖ and | ⟨휆, 푤⟩ | ≤ ‖푤‖ for all 푤 ∈ 푉 푋 by Hahn–Banach theorem. Insert this 휆 into the characterization of ∫ 푓 d휇. 푋
3.4 Action of convolution algebra In what follows, we will often write the integration of a measure 푓 on a locally compact Hausdorff space 푋 as ∫ 푓(푥), omitting the useless d푥. It is convenient to adopt the notations for functions: given an 푥∈푋 isomorphism Φ ∶ 푌 → 푋, write 푓 ∘ Φ for the measure transported to 푌, so that ∫ 푓 ∘ Φ = ∫ 푓 holds 푌 푋 for tautological reasons. Let 퐺 be a locally compact group. Define the following space of Radon measures on 퐺
ℳ (퐺) ∶= {bounded Radon measures on 퐺} ,
ℳ푐(퐺) ∶= 푓 ∈ ℳ (퐺) ∶ Supp(푓) is compact� , ℳ̃ (퐺) ∶= 푓 ∈ ℳ (퐺) ∶ 푓 ≪ any right Haar measure� , ̃ ̃ ℳ푐(퐺) ∶= ℳ (퐺) ∩ ℳ푐(퐺).
.(If a right Haar measure 휇 on 퐺 is chosen, then 퐿(퐺) d휇 = ℳ̃ (퐺 Recall that a subset of 퐺 is called symmetric if it is invariant under 푔 ↦ 푔−. Likewise, we say a function or a measure on 퐺 is symmetric if it has the same invariance. A topological group always has a .symmetric neighborhood basis at 1: replacing each neighborhood 푈 ∋ 1 by 푈 ∩ 푈− ∋ 1 suffices
Definition 3.25. An approximate identity is a symmetric neighborhood basis 픑 at 1 ∈ 퐺 together with a ̃ family 휑푈 ∈ ℳ푐(퐺), where 푈 ranges over 픑, such that for all 푈 ∈ 픑,
• 휑푈 is positive of the form 휙푈 d휇 where 휙푈 ∈ 퐶푐(퐺) and 휇 is a right Haar measure; • ∫ 휑 = 1; 퐺 푈 − • Supp(휑푈 ) ⊂ 푈, 휑푈 (푔) = 휑푈 (푔 ).
When 픑 is countable, an approximate identity can be conveniently described as a sequence 휑, 휑,… in ̃ ℳ푐(퐺) with each 휑푖 symmetric and positive, and Supp(휑푖) shrinks to {1} as 픑 does. Proposition 3.26. For every symmetric neighborhood basis 픑 at 1 ∈ 퐺, there exists approximate iden- tities supported on 픑.
− Proof. Given 푈 ∈ 픑, use Urysohn’s lemma to construct 휑푈 . We can force 휑푈 (푔) = 휑푈 (푔 ) by aver- aging. ̃ Definition 3.27. Let (휋, 푉) be a continuous representation of 퐺 on a quasi-complete space. Let ℳ푐(퐺) act on the left of 푉 by 푉-valued integrals
̃ 휋(휑)푣 = ﾙ 휑(푔)휋(푔)푣, 푣 ∈ 푉, 휑 ∈ ℳ푐(퐺), 퐺 whose existence is guaranteed by 3.22.
26 One should imagine that 휋(푔) = 휋(훿푔) where 훿푔 stands for the Dirac measure at 푔. In general,
(휋(ℎ−)휋(푓)휋(푔−) = 휋 ℎ푓푔 , ℎ푓푔(푥) = 푓(ℎ푥푔), (3.4 where the left/right translation should be understood in terms of measures. It preserves the spaces ℳ (퐺), etc. ̃ Lemma 3.28. This makes 푉 into an ℳ푐(퐺)-algebra, namely
휋(휑) ∈ Endℂ(푉),
,휋(푎휑 + 푏휑)푣 = 푎휋(휑)푣 + 푏휋(휑)푣
휋(휑 ⋆ 휑) = 휋(휑)휋(휑)푣 ̃ for all 휑, 휑 ∈ ℳ푐(퐺) and 푎, 푏 ∈ ℂ. Furthermore, if (휑푈 )푈∈픑 is an approximate identity, then
lim 휋(휑푈 )푣 = 푣, 푣 ∈ 푉 푈∈픑 in the sense of filter bases.
Proof. The linearity is clear. As for convolutions, take any 휆 ∈ 푉∗ and verify that
− ﾉ휆, 휋(휑 ⋆ 휑)푣ｼ = ﾙ ﾙ 휑(ℎ )휑(ℎ푔) ﾉ휆, 휋(푔)푣ｼ 푔∈퐺 ℎ∈퐻
−− Fubini’s theorem) = ﾙ 휑(ℎ ) ﾋ휆, ﾙ 휋(ℎ )휑(ℎ푔)휋(ℎ푔)푣ｾ ∵) ℎ∈퐻 푔∈퐺
−− ﾙ 휑(ℎ ) ﾊ휆, 휋(ℎ ) 휋(휑)푣ｽ = (3.21 ∵) ℎ∈퐻 . ﾉ휆, 휋(휑)휋(휑)푣ｼ =
For the convergence, it suffices to show for every continuous semi-norm ‖ ⋅ ‖ that
lim ‖휋(휑푈 )푣 − 푣‖ = 0 푈 since 푉 is locally convex. By 3.24, we have
‖휋(휑푈 )푣 − 푣‖ = ﾙ 휑푈 (푔)(휋(푔)푣 − 푣) 퐺
≤ ﾙ 휑푈 (푔)‖휋(푔)푣 − 푣‖. 퐺 Given 휖 > 0, for sufficiently small 푈 we have ‖휋(푔)푣 − 푣‖ < 휖 when 푔 ∈ 푈. Hence the last expression is bounded by 휖 ∫ 휑 = 휖. 퐺 푈 ̃ Lemma 3.29. The subspace 휋 ℳ푐(퐺) 푉 of 푉 is 퐺-stable and dense. In fact, the subspace spanned by 휋(휑푈 )푉 is already dense, when 휑푈 runs over an approximate identity (recall 3.25). ̃ ̃ Proof. Apply 3.28. Note that ℳ푐(퐺) is stable under (3.4), thus 휋 ℳ푐(퐺) 푉 is 퐺-stable as well.
27 3.5 Smooth vectors: archimedean case
Let 퐺 be a locally compact group and (휋, 푉) be a 퐺-representation. The orbit map orb푣 ∶ 푔 ↦ 푔푣 can also be used to define representation-theoretic properties of vectors 푣 ∈ 푉.
Definition 3.30. Let P stand for a property of continuous functions 퐺 → 푉. For a 퐺-representation 푉 and 푣 ∈ 퐺, we say that 푣 has the property P if orb푣 ∶ 퐺 → 푉 has. Since 푣 ↦ orb푣 is a ℂ-linear map from 푉 to Maps(퐺, 푉), if the functions with property P form a vector space, so are the vectors with property P.
In particular, when 퐺 is an analytic Lie group over a local field 퐹, we can talk about 퐶∞ (infinitely differentiable) and 퐶휔 (analytic) vectors in a 퐺-representation, thereby bringing the smooth structure into the picture.
Definition 3.31. The 퐶∞-vectors in 푉 are called smooth vectors. They form a subspace
푉∞ ∶= {푣 ∈ 푉 ∶ smooth vector} .
In the next few sections we will mainly be interested in the case of archimedean 퐹. The same definition pertains to non-archimedean case as well (see 1.38), but it has a quite different flavor. Note that for archimedean 퐹, one can actually talk about 퐶푘-vectors where 푘 ∈ {0, 1, … , ∞, 휔}. In what follows we consider only 퐹 = ℝ. The case of ℂ follows by “Weil restriction”, namely one can view a ℂ-group as an ℝ-group by dropping its complex structure. For a Lie group 퐺 over 퐹 = ℝ, let 픤 ∶= Lie(퐺) ⊗ℝ ℂ denote its complexified Lie algebra, and let 푈(픤) designate the universal enveloping algebra of 픤.
Definition 3.32. Suppose that 퐺 is a Lie group over ℝ. Let (휋, 푉) be in 퐺-Rep. Define the eponymous action 휋 of the Lie algebra 픤 by
d 휋(푔(ℎ))푣 − 휋(푔(0))푣 휋(푋)푣 ∶= | 휋(푔(푡))푣 = lim , 푋 ∈ 픤 d푡 푡= ℎ→ ℎ for every 푣 ∈ 푉 ∞, where 푔 is any differentiable curve in 퐺 with 푔(0) = 1 and 푔′(0) = 푣; it is customary to take 푔(푡) = exp(푡푋).
Lemma 3.33. The operators 휋(푋) above yield a homomorphism of ℂ-algebras
∞ 푈(픤) → Endℂ(푉 ).
Proof. In other words (푋, 푣) ↦ 휋(푋)푣 is a representation of the Lie algebra 픤. It boils down to check that 휋(푋)(휋(푌)푣) − 휋(푌)(휋(푋)푣) = 휋([푋, 푌])푣 for all 푣 ∈ 푉∞ and 푋, 푌 ∈ 픤. This reduces the to basic fact that 푠푡 exp(푡푋) exp(푠푌) = exp 푡푋 + 푠푌 + [푋, 푌] + higher . 2
[ TO BE CONTINUED... ]
4 Unitary representation theory
Fix a locally compact group 퐺.
28 4.1 Generalities Recall that a pre-Hilbert space is a ℂ-vector space 푉 equipped with a positive-definite Hermitian form / (⋅|⋅) = (⋅|⋅)푉 ; if 푉 is complete with respect to the norm ‖푣‖ ∶= (푣|푣) , we say 푉 is a Hilbert space. Definition 4.1. A unitary representation (resp. pre-unitary representation) of 퐺 is a Hilbert (resp. pre- Hilbert) space (푉, (⋅|⋅)) together with a linear left 퐺-action on 푉 such that
• for every 푣 ∈ 푉, the map 푔 ↦ 푔푣 from 퐺 to 푉 is continuous,
• for each 푔 ∈ 퐺, the operator 휋(푔) ∶ 푉 → 푉 from the 퐺-action is unitary, i.e. ‖휋(푔)(푣)‖ = ‖푣‖ for all 푣 ∈ 푉.
Remark 4.2. In view of 3.9 with 푉 = 푉, a unitary representation is automatically continuous, and is the same as a Hilbert representation (푉, 휋) of 퐺 such that all 휋(푔) are unitary operators. On the other hand, pre-unitary representations can be completed to unitary ones.
Lemma 4.3. If 푉 is a pre-unitary representation, then the Hilbert space completion 푉̂ of 푉 with respect to ‖ ⋅ ‖ can be endowed with an induced 퐺-action, and becomes a unitary representation.
Proof. For every 푔 ∈ 퐺, the operator 휋(푔) extends uniquely to a unitary operator on 푉̂ . The relation 휋(푔)휋(푔′) = 휋(푔푔′) also extends to 푉̂ . It remains to check the continuity of 푔 ↦ 휋(푔)푣 where 푣 ∈ 푉̂ is fixed. To see this, take 푣 ∈ 푉 and note that
′ ′ ′ ′ ‖휋(푔)푣 − 휋(푔 )푣‖ ≤ ‖휋(푔)푣 − 휋(푔)푣‖ + ‖휋(푔)푣 − 휋(푔 )푣‖ + ‖휋(푔 )푣 − 휋(푔 )푣‖ ′ .‖푣 − 푣‖ + ‖휋(푔)푣 − 휋(푔 )푣‖ + ‖푣 − 푣‖ =
.We conclude from the continuity of 푔 ↦ 휋(푔)푣, by taking ‖푣 − 푣‖ sufficiently small
Definition 4.4. A morphism or an intertwining operator 휑 ∶ 푉 → 푉 between unitary representations is a ℂ-linear homomorphism 휑 such that • is unitary, i.e. preserves inner products: for all , 휑 (푣|푤)푉 = (휑(푣)|휑(푤))푉 푣, 푤 ∈ 푉
.휑 respects 퐺-actions: 푔휑(푣) = 휑(푔푣) for all 푣 ∈ 푉 and 푔 ∈ 퐺 • This turns the collection of all unitary representations into a category, which is a non-full subcate- gory of Hilbert representations. Isomorphisms between unitary representations are also called unitary equivalences.
Example 4.5. Let 퐺 acts continuously on a locally compact Hausdorff space; for simplicity assume that 푋 carries a 퐺-invariant positive Radon measure. It is shown in 3.13 with 푝 = 2 that 퐿(퐺) is a 퐺-representation. It is actually unitary, since we have pointed out in 3.13 that 푓 ↦ 푔푓 preserves the .퐿 -norm ‖ ⋅ ‖ 푛 The finite direct sum ⨁ 푉 of unitary representations 푉 , … , 푉 is defined so that the underlying 푖= 푖 푛 …,Hilbert space is the orthogonal direct sum of 푉, … , 푉푛. The case of countably infinite sum of 푉 requires more care: we have to use the completed direct sum ⨁̂ 푉 , which is the completion of the 푖≥ 푖 pre-Hilbert space ⨁ 푉 . 푖≥ 푖 The notion of subrepresentations and irreducibility of unitary 퐺-representations are the same as the case of continuous representations in general. On the other hand, quotients are unnecessary as shown by the following complete reducibility.
29 Lemma 4.6. Let 푊 be a subrepresentation of a unitary 퐺-representation 푉. Then
푉 = 푊 ⊕ 푊 ⟂
as unitary representations, where 푊 ⟂ ∶= {푣 ∈ 푉 ∶ ∀푤 ∈ 푊, (푣|푤) = 0}. Proof. The orthogonal decomposition surely works on the level of Hilbert spaces. It remains to notice that 푊 ⟂ is stable under 퐺-action.
In contrast to the case of modules, 4.6 does not imply that 푉 can be written as a direct sum of irreducibles. In fact, in many cases 푉 has no simple submodules, as we shall see in 4.11. The following is a unitary variant of the familiar Schur’s Lemma for irreducible representations of finite groups.
Theorem 4.7. Let (휋, 푉) be an irreducible unitary representation. Then
End퐺-Rep(푉) = ℂ ⋅ id푉 .
Proof. Let 푇 ∶ 푉 → 푉 be continuous and linear. Suppose that 푇 commutes with 퐺-action, then so is its adjoint ∗푇 since (푤|푔∗푇푣) = (푇푔−푤|푣) = (푔−푇푤|푣) = (푇푤|푔푣) = (푤|∗푇푔푣) for all 푤 ∈ 푉 and 푔 ∈ 퐺. Write 푇 + ∗푇 푇 − ∗푇 푇 = + √−1 ⋅ . 2 2√−1 Therefore, to show that 푇 is a scalar, we may suppose ∗푇 = 푇. Now let 휎(푇) ⊂ ℝ be the spectrum of 푇. Apply the spectral decomposition for self-adjoint bounded operators (e.g. [8, 12.23 Theorem]) to express 푇 via its spectral measure 퐸:
푇 = ﾙ 휆 d퐸(휆). 휎(푇) Here 퐸 can be thought as a projection-valued measure on the Borel subsets of 휎(푇); it is canonically associated to 푇. The irreducibility of 푉 together with the 퐺-equivariance of 푇 imply that 퐸(퐴 ∩ 휎(푇)) is either 0 or id푉 for all Borel subsets 퐴 ⊂ ℝ; i.e. 휎(푇) is an atom in measure-theoretic terms. It follows ; that there exists a point 휆 with 퐸({휆}) = id푉 . Indeed, take any interval 퐼 with spectral measure id푉 bisect it and pass to the one with spectral measure id푉 , named 퐼, and so forth to obtain nested intervals ⋂ 퐼 ⊃ 퐼 ⊃ ⋯ such that 푖≥ 퐼푖 = {휆} for some 휆. The zero-one dichotomy then implies that 퐸 is supported at {휆}. Therefore 푇 = 휆 ⋅ id푉 .
Corollary 4.8. A unitary representation (휋, 푉) is irreducible if and only if End퐺-Rep(푉) = ℂ ⋅ id푉 . Proof. The “only if” part follows from 4.7. Conversely, if 푊 ⊂ 푉 is a proper subrepresentation then 푉 = 푊 ⊕ 푊 ⟂ by 4.6, hence
⟂ End퐺-Rep(푉) ⊃ End퐺-Rep(푊) ⊕ End퐺-Rep(푊 )
has dimension ≥ 2. Another important consequence of Schur’s Lemma is the existence of central characters in the unitary case.
Theorem 4.9. Let (휋, 푉) be an irreducible unitary representation. There is a continuous homomorphism 휔휋 ∶ 푍퐺 → 핊 , where 푍퐺 denotes the center of 퐺, such that 휋(푧) = 휔휋(푧)id푉 for every 푧 ∈ 푍퐺. We call 휔휋 the central character of 휋.
30 Proof. Central elements act on 푉 by scalar multiplication since they commute with 퐺. This yields the homomorphism 휔휋 whose continuity and unitarity follow from that of (휋, 푉).
In practice, it is convenient to allow any closed subgroup 푍 of 푍퐺. By abuse of language, we also .say that (휋, 푉) has central character 휔 on 푍, if 푍 acts via the continuous homomorphism 휔 ∶ 푍 → 핊
Corollary 4.10. If 퐺 is commutative, the irreducible unitary representations are precisely the one- dimensional unitary representations.
Proof. For commutative 퐺 we have 퐺 = 푍퐺 acts via 휔휋 on irreducible (휋, 푉). Conversely, one- dimensional representations are clearly irreducible.
.Example 4.11. Consider the unitary ℝ-representation 퐿(ℝ). It has no irreducible subrepresentations ,(Indeed, an irreducible subrepresentation must be generated by an 퐿-function 푓 with 푓(푥 + 푡) = 휒(푡)푓(푥 .where 휒 is a continuous homomorphism ℝ → 핊. The only such 퐿-function is 0 The next enhancement of 4.7 will be needed.
Theorem 4.12. Let (휋, 푉) and (휎, 푊) be unitary representations of 퐺 and assume that (휋, 푉) is irre- ducible. Let 푇 be a 퐺-equivariant linear map from a dense 퐺-stable vector subspace 푉 ⊂ 푉 to 푊. If the graph of 푇 {Γ푇 ∶= {(푣, 푇푣) ∈ 푉 × 푊 ∶ 푣 ∈ 푉
-is closed in 푉 ⊕ 푊, then 푉 = 푉 and 푇 is a scalar multiple of a morphism between unitary representa tions.
Proof. We shall make use of basic facts on unbounded operators from Hilbert spaces 푉 to 푊 in general; see for example [8, Chapter 13]. Such an operator 푇 is only defined on some subspace 풟 (푇) of 푉, and continuity is not presumed. When taking their composites, etc., domains must shrink suitably. In our .case 풟 (푇) = 푉 is dense The first result is the existence of an adjoint ∗푇 defined on
∗ , �(풟 ( 푇) ∶= 푤 ∈ 푊 ∶ (푇(⋅)|푤)푊 ∈ HomTopVect(푉, ℂ
∗ ∗ uniquely determined by (푇푣|푤)푊 = (푣| 푇푤)푉 . Secondly, assume that Γ푇 is closed, then 풟 ( 푇) is also dense; then consider the unbounded operator 푄 ∶= 1 + ∗푇푇 from 푉 to itself. According to [8, Theorem 13.13],
• 풟 (푄) = {푣 ∈ 풟 (푇) ∶ 푇푣 ∈ 풟 (∗푇)} maps onto 푉 under 푄;
• there is a continuous endomorphism 퐵 of 푉 such that im(퐵) ⊂ 풟 (푄) and 푄퐵 = id푉 . In fact 푏 ∶= 퐵푣 (for all 푣 ∈ 푉) is characterized in 푊 ⊕ 푉 by
(0, 푣) = (푐, ∗푇푐) + (−푇푏, 푏), 푏 ∈ 풟 (푇), 푐 ∈ 풟 (∗푇). (4.1)
∗ Now apply these results to our setting. The domain 풟 ( 푇) is a 퐺-stable subspace as 푉 = 풟 (푇) is. One readily checks that ∗푇 ∶ 풟 (∗푇) → 푉 is also 퐺-equivariant. From the characterization (4.1), we conclude × that 퐵 is 퐺-equivariant as well, hence 퐵 = 휆 ⋅ id푉 for some 휆 ∈ ℂ by 4.7, as 푄퐵 = id푉 . ∗ − Consequently 풟 (푄) = 풟 (푇) = 푉 and 푄 = 휆 id풟 (푄). This also implies 풟 ( 푇) ⊃ im(푇). Now
′ ′ − ′ ∗ ′ 푇푣|푇푣 )푉 = ( 푇푇푣|푣 )푉 = (휆 − 1)(푣|푣 )푉 , 푣, 푣 ∈ 푉) hence 푇 is a scalar multiple of an isometry 푉 → 푊.
31 4.2 External tensor products
The theory of completed tensor products for general locally convex spaces 푉, 푉 is delicate. For our present purposes, the case of Hilbert spaces suffices.
Definition 4.13. Let (푉, (⋅|⋅)) and (푉, (⋅|⋅)) be Hilbert spaces. Write 푉 ⊗ 푉 ∶= 푉 ⊗ℂ 푉 and equip it with an Hermitian form by requiring
푣 ⊗ 푣 | 푤 ⊗ 푤) = (푣|푤) ⋅ (푣|푤), 푣, 푤 ∈ 푉, 푣, 푤 ∈ 푉) and extend by sesquilinearity to all 푉 ⊗푉. This is well-defined and positive-definite. Denote by 푉⊗푉̂ .the corresponding completion of 푉 ⊗ 푉
The operation ⊗̂ is functorial in 푉 and 푉, turning the collection of Hilbert spaces into a symmetric monoidal category. Now suppose that 푉푖 is the underlying space of a unitary representations 휋푖 of a locally compact group 퐺푖, for 푖 = 1, 2. Let 퐺 × 퐺 act on 푉 ⊗ 푉 via 휋 ⊗ 휋; this extends to an action on 푉⊗푉̂ by unitary operators. In order to stress that it is acted upon by 퐺 × 퐺, it is customary to designate the resulting representation by 휋 ⊠ 휋. Summing up, we obtain
.푉⊗푉̂ , 휋 ⊠ 휋 ∶ unitary representation of 퐺 × 퐺
-Proposition 4.14. If 휋 and 휋 are both irreducible, then 휋 ⊠ 휋 is irreducible as a unitary represen .tation of 퐺 × 퐺
Proof. In view of 4.8, it suffices to show that every continuous 퐺 × 퐺-equivariant endomorphism 푇 of 휋 ⊠ 휋 is 푐 ⋅ id for some 푐 ∈ ℂ. Since this property can be tested by taking inner product with all elements from 푉 ⊗ 푉, which are dense in 푉⊗푉̂ , it is equivalent to that
푇(푣 ⊗ 푣) | 푤 ⊗ 푤 = 푐(푣|푤)푉 ⋅ (푣|푤)푉
.for all 푣, 푣 and 푤, 푤 To achieve this, first fix 푣, 푤 ∈ 푉 and define the endomorphism
휑푤,푣 ∶ 푉 ⟶ 푉 ⊗ ℂ = 푉 id ( ;(푣 ⟼ 푉 ⊗ (⋅|푤)푉 (푇(푣 ⊗ 푣 here we view as . It is continuous and -equivariant, hence 4.7 implies = 푤)푉 푉 → ℂ 퐺 휑푤,푣|⋅) id for some scalar . Now we have a continuous linear map 푎푤,푣 ⋅ 푉 푎푤,푣
∗ (퐴 ∶ 푉 ⟶ (푉
. 푤 ⟼ ￮푣 ↦ 푎푤,푣
∗ ,Recall that (푉) ≃ 푉 via the Hermitian form. A careful verification reveals that 퐴 is 퐺-equivariant hence id for some . In other words, for all . This implies 퐴 = 푐 ⋅ 푉 푐 ∈ ℂ 푎푤,푣 = 푐(푣|푤)푉 푣, 푤 ∈ 푉 . 푇(푣 ⊗ 푣) | 푤 ⊗ 푤 = 푐(푣|푤)푉 ⋅ (푣|푤)푉
Proposition 4.15. Let (휋푖, 푉푖) and (휎푖, 푊푖) be irreducible unitary representations of 퐺푖 for 푖 = 1, 2. If .휋 ⊠ 휎 ≃ 휋 ⊠ 휎, then 휋푖 ≃ 휎푖 for 푖 = 1, 2
32 Proof. Suppose that 푇 ∶ 휋 ⊠ 휎 ≃ 휋 ⊠ 휎 is nonzero. Then there must exist 푣 ∈ 푉 and 푤 ∈ 푊 such that
휑푤,푣 ∶ 푉 ⟶ 푊 ⊗ ℂ = 푊 id ((푣 ⟼ 푊 ⊗ (⋅|푤)푊 (푇(푣 ⊗ 푣 is nonzero. As in the proof of 4.14, it is 퐺-equivariant and continuous, hence 휋 ≃ 휎 by 4.12. Switching .the roles of 퐺, 퐺 yields 휋 ≃ 휋
Of particular importance is the case 퐺 = 퐺 = 퐺 and (푉, 푉) = (푉, 푉)̄ , where (휋, 푉) is a unitary ̄ representation of 퐺 and ( ̄휋, 푉) is its Hermitian conjugate; see 3.17. Recall that (푥|푦)푉̄ ∶= (푦|푥)푉 . By linear algebra, there is a ℂ-linear isomorphism
∼ /ℂ 푉 ⊗ 푉̄ ⟶ Endf.r.(푉) ∶= 푇 ∶ 푉 → 푉 ∶ finite rank, continuous (4.2)
푣 ⊗ 푤 ⟼ 퐴푣⊗푤 ∶= (⋅|푤)푣.
Equip Endf.r.(푉) with the Hilbert–Schmidt Hermitian inner product, which is positive-definite:
∗ (퐴|퐵)HS ∶= Tr( 퐵 ⋅ 퐴).
∗ It is routine yet amusing to verify that 퐴푤⊗푣 = 퐴푣⊗푤. Hence ′ ′ ′ ′ (퐴 |퐴 ′ ′ ) = (푣 |푣) ⋅ (푤|푤 ) = (푣|푣 ) ⋅ (푤|푤 ) ̄ 푣⊗푤 푣 ⊗푤 HS 푉 푉 푉 푉 which equals the Hermitian inner product in 푉 ⊗ 푉̄ . Also recall that the space of Hilbert–Schmidt operators 푉 → 푉 is defined as
HS(푉) ∶= completion of Endf.r.(푉) relative to (⋅|⋅)HS.
The group 퐺 × 퐺 acts on Endf.r.(푉) and HS(푉) by .푎, 푏)푇 = 휋(푎)푇휋(푏)−) This is visibly unitary and corresponds to the action on 푉 ⊗ 푉̄ under (4.2). Theorem 4.16. The identification (4.2) extends to an isomorphism
푉 ⊠ 푉̄ ≃ HS(푉) between unitary 퐺 × 퐺-representations.
Proof. The identification is a 퐺 × 퐺-equivariant isometry between 푉 ⊗ 푉̄ and Endf.r.(푉). Now pass to completion.
4.3 Square-integrable representations
Fix a closed subgroup 푍 ⊂ 푍퐺. Central characters of irreducible unitary representations will be defined × relative to 푍. In view of 1.52, up to ℝ> there exists a unique right Haar measure on 퐺/푍, since the condition 훿퐺|푍 = 훿푍 is trivially satisfied: both are trivial from the very definition of 1.27. As unitary representations are a special kind of Hilbert representations, it makes sense to consider the matrix coefficients 푐푣⊗푤(푔) = (휋(푔)푣|푤)푉 for 푣 ⊗ 푤 ∈ 푉 ⊗ 푉̄ as in 3.18. If (휋, 푉) admits a central character 휔휋 as in 4.9, for example when (휋, 푉) is irreducible, then
푐푣⊗푤(푧푔) = 휔휋(푧)푐푣⊗푤(푔), 푧 ∈ 푍, 푔 ∈ 퐺.
In this case |푐푤⊗푣| factors through 퐺/푍.
33 .Definition 4.17. Denote by 퐿(퐺/푍) the 퐿-space defined relative to any right Haar measure on 퐺/푍 More precisely, for every continuous homomorphism 휔 ∶ 푍 → 핊}, set
푓(푧푥) = 휔(푧)푓(푥), 푧 ∈ 푍 . ,퐿(퐺/푍, 휔) ∶= 푓 ∶ 퐺 → ℂ 푓| ∈ 퐿(퐺/푍) | 
It is a Hilbert space under ‖푓‖ = ∫ |푓| d휇 once a right Haar measure 휇 is chosen. Specifically, one 퐺/푍 starts from measurable 푓, and take the quotient by functions with ‖⋅ ‖ = 0 and so on. Likewise, we define the other function spaces on 퐺 with (휔, 푍)-equivariance such as
푓(푧푥) = 휔(푧)푓(푥), 푧 ∈ 푍 퐶 (퐺/푍, 휔) ∶= 푓 ∶ 퐺 → ℂ continuous, . 푐  Supp(푓)/푍 is compact
.Notice that 퐿(퐺/푍, 휔) is also a unitary representation of 퐺 under 푔 ∶ 푓(푥) ↦ 푓(푥푔) as in 4.5 Definition 4.18. Suppose 퐺 is unimodular. An irreducible unitary representation (휋, 푉) of 퐺 is called square-integrable or a discrete series representation if the matrix coefficient 푐푣⊗푤 is nonzero and lies in 퐿 (퐺/푍, 휔휋) for some 푤, 푣. Such representations are also called essentially square-integrable in the literature, while the term square-integrable is reserved to those with 푐푣⊗푤 ∈ 퐿 (퐺), i.e. for 푍 = {1}. Square-integrable represen- tations in the latter strict sense exists only when 푍퐺 is compact, cf. 1.21 and the integration formula in 1.52; when 푍퐺 is compact, both definitions coincide. In 4.20 we will extend this definition to non-unimodular groups. Lemma 4.19. Let (휋, 푉) be an irreducible unitary representation (휋, 푉) of a unimodular group 퐺. The following are equivalent. 1. (휋, 푉) is square-integrable;
′ 2. 푐푣⊗푤 ∈ 퐿 (퐺/푍, 휔휋) for all 푣, 푤 ≠ 0, and if 푇 ∶ 푣 ↦ 푐푣′⊗푤 is not identically zero, 푇 is a 퐺- equivariant linear map from 푉 into 퐿 (퐺/푍, 휔휋) with closed image, which is a scalar multiple of an isometry.
3. there exists an embedding 푉 ↪ 퐿 (퐺/푍, 휔휋) as unitary 퐺-representations. Proof. We begin with (1) ⟹ (2). Fix 푤, 푣 ∈ 푉 with 푐푣⊗푤 ∈ 퐿 (퐺/푍, 휔휋) nonzero. We begin by ′ showing that 푐푣′⊗푤 ∈ 퐿 (퐺/푍, 휔휋) for all 푣 . Set
′ . �(푉 ∶= 푣 ∈ 푉 ∶ 푐푣′⊗푤 ∈ 퐿 (퐺/푍, 휔휋 This is a 퐺-stable vector subspace by (3.1); it contains 푣, hence we infer from the irreducibility of 푉 that 푉 is dense. Define a linear map
′ .푇 ∶ 푉 → 퐿 (퐺/푍, 휔휋), 푣 ↦ 푐푣′⊗푤
By (3.1) we see 푇 is 퐺-equivariant. Equip 푉 with the Hermitian pairing
′ ′ ′ ′ ′ ′ . 푣|푣)) = (푣|푣)푉 + (푇푣|푇푣)퐿)) .(We claim that 푉 is a Hilbert space under ((⋅|⋅)), and the graph Γ푇 is closed in 푉 × 퐿 (퐺/푍, 휔휋 Indeed, ((⋅|⋅)) is evidently Hermitian and positive definite; we set out to show its completeness. The ′ ∞ ′ (choice of ((⋅|⋅)) implies that for every Cauchy sequence (푣푖 )푖= in 푉, there exists (푣 , 푐) ∈ 푉 ×퐿 (퐺/푍, 휔휋 ′ ′ ′ ′ ′ such that 푣푖 → 푣 in 푉 and 푇푣푖 → 푐 in 퐿 (퐺/푍, 휔휋). Observe that 푣푖 → 푣 implies that
′ ′ ′ ′ |푐 ′ (푥) − 푐 ′ (푥)| = |(휋(푥)(푣 − 푣 )|푤)| ≤ ‖푣 − 푣 ‖ ⋅ ‖푤‖ 푣푖 ⊗푤 푣 ⊗푤 푖 푖
34 ′ ′ so that 푇푣푖 → 푐푣′⊗푤 uniformly on 퐺. On the other hand, one can pass to a subsequence to ensure 푇푣푖 → 푐 ′ pointwise and a.e. Therefore 푐푣′⊗푤 = 푐 so that 푣 ∈ 푉. This proves both the completeness and closedness parts of our claim. By 4.12, the claim implies that 푉 = 푉 and 푇 is a scalar multiple of an isometry. Such maps between Hilbert spaces always have closed images. ′ To conclude (2), we must also show that 푐푣⊗푤′ ∈ 퐿 (퐺/푍, 휔휋) for all 푤 ∈ 푉. Via (3.2) and the unimodularity of 퐺/푍, this is subsumed into the previous case. It is immediate that (2) ⟹ (3): 푇 furnishes the required embedding. (3) ⟹ (1): Assume that (휋, 푉) is a unitary subrepresentation of 퐿 (퐺/푍, 휔휋) and 푣 ∈ 푉 is nonzero. ( There exists 푤̃ ∈ 퐿 (퐺/푍, 휔 ) such that (푣|푤)̃ ≠ 0. We may even assume that 푤̃ ∈ 퐶 (퐺/푍, 휔 휋 퐿 (퐺/푍,휔휋) 푐 휋 ′ ′ by density. Then 푣 ↦ (푣 |푤)̃ restricts to a continuous linear functional on 푉, of the form 퐿 (퐺/푍,휔휋) ′ ′ 푣 ↦ (푣 |푤)푉 for a unique 푤 ∈ 푉. Thus 푐푣⊗푤(1) ≠ 0 and it remains to show 푐푣⊗푤 ∈ 퐿 (퐺/푍, 휔휋). − Denote by 휇 the chosen right Haar measure on 퐺/푍. Put 푢(푥) ∶= 푤(푥̃ ) to express 푐푣⊗푤(푔) = 휋(푔)푣|푤)̃ as) 퐿 (퐺/푍,휔휋) .(ﾙ 푢(푥−)푣(푥푔) d휇(푥 퐺/푍 Its absolute value is bounded by (|푢| ⋆ |푣|)(푔) (convolution on 퐺/푍). As |푢| ∈ 퐶푐(퐺/푍) and |푣| ∈ 퐿 (퐺/푍), it follows either from a direct analysis or Young’s inequality 2.4 with (푝, 푞, 푟) = (1, 2, 2) that |푢| ⋆ |푣| ∈ .퐿(퐺/푍), as required
The third condition in 4.19 makes sense for any locally compact 퐺, which leads to the following general definition.
Definition 4.20. Let 퐺 be a locally compact group. An irreducible unitary representation (휋, 푉) of 퐺 is called square-integrable or a discrete series representation if it is isomorphic to a unitary subrepresen- tation of 퐿 (퐺/푍, 휔휋). Theorem 4.21 (Schur orthogonality relation). Let (휋, 푉) and (휎, 푊) be square-integrable representa- tions of a unimodular group 퐺 with 휔휋 = 휔휎, and fix a right Haar measure 휇 on 퐺/푍. 1. If (휋, 푉) and (휎, 푊) are not isomorphic as unitary representations, then
ﾙ 푐푣⊗푣′ 푐푤⊗푤′ d휇 = 0 퐺/푍
for all 푣, 푣′ ∈ 푉 and 푤, 푤′ ∈ 푊.
In the case 휋 = 휎, there exists 푑(휋) ∈ ℝ>, inverse-proportional to the choice of 휇, such that .2
′ ′ − ﾙ 푐푣⊗푣′ 푐푤⊗푤′ d휇 = 푑(휋) (푣|푤)푉 (푤 |푣 )푉 퐺/푍
for all 푣, 푣′, 푤, 푤′ ∈ 푉.
Proof. Write 푇 ∶ 푣 ↦ 푐푣⊗푣′ and 푆 ∶ 푤 ↦ 푐푤⊗푤′ . They are scalar multiples of isometries and are 퐺-equivariant. Now
∗ . (푐 ′ 푐 ′ d휇 = (푇푣|푆푤) = ( 푆푇푣|푤 ﾙ 푣⊗푣 푤⊗푤 퐿 (퐺/푍,휔휋) 푊 퐺/푍
But ∗푆 is also continuous and 퐺-equivariant, thus 퐴 ∶= ∗푆푇 ∶ 푉 → 푊 is a morphism in 퐺-Rep. Then 4.12 implies that 퐴 is a scalar multiple of a morphism between unitary representations. If (휋, 푉) and (휎, 푊) are non-isomorphic, the only possibility is 퐴 = 0 and the first item is proved.
35 Next, assume 휋 = 휎. Then 퐴 = 휆푣′,푤′ ⋅ id푉 for some 휆푣′,푤′ ∈ ℂ. This amounts to
ﾙ 푐푣⊗푣′ 푐푤⊗푤′ d휇 = 휆푣′,푤′ (푣|푤)푉 . 퐺/푍 Switching 푣 ↔ 푣′ and 푤 ↔ 푤′ replaces the matrix coefficients by their complex conjugates. As 퐺/푍 is unimodular, we deduce ′ ′ 휆푣′,푤′ (푤|푣)푉 = 휆푣,푤(푣 |푤 )푉 . ′ ′ Fix 푣 = 푤 ≠ 0 in the equation above to see that 휆푣′,푤′ = 푐(푤 |푣 )푉 for some constant 푐 depending solely ′ ′ . (on (휋, 푉) and 휇; it is proportional to 휇. Setting 푣 = 푤 = 푤 = 푣 to deduce ‖푐푣⊗푣‖ = 푐(푣|푣 퐿 (퐺/푍,휔휋) 푉 − As 푐푣⊗푣(1) ≠ 0 whenever 푣 ≠ 0, we see 푐 > 0. Now set 푑(휋) = 푐 to conclude. Corollary 4.22. The convolutions of square-integrable matrix coefficients on 퐺/푍 are well-defined. In fact, for irreducible unitary representations (휋, 푉), (휎, 푊) we have
− 푐푣⊗푣′ ⋆ 푐푤⊗푤′ (푥) ∶= ﾙ 푐푣⊗푣′ (푔 )푐푤⊗푤′ (푔푥) d휇(푔) 퐺/푍 ⎧ ′ − ⎪ ⎨푑(휋) (푣|푤 )푐푣′⊗푤(푥), 휋 = 휎 = ⎪ ⎩0, 휋 ≄ 휎, for all 푣′, 푣 ∈ 푉 and 푤, 푤′ ∈ 푊. Proof. Use 3.2, 3.1 and apply 4.21 to deduce
− ﾙ 푐푣⊗푣′ (푔 )푐푤⊗푤′ (푔푥) d휇(푔) = ﾙ 푐휋′(푥)푤⊗푤′ (푔)푐푣′⊗푣(푔) d휇(푔) 퐺/푍 퐺/푍 ⎧ 푑(휋)−(휋′(푥)푤|푣′)(푣|푤′), 휋 = 휎⎪⎨ = ⎪ ⎩0, 휋 ≄ 휎.
′ − In the first case, it equals 푑(휋) (푣|푤 )푐푤⊗푣′ (푥). Now enters the bilateral translation on 퐺. Observe that 퐺 × 퐺 acts on the right of 퐺 by (푎, 푏) ∶ 푥 ↦ 푏−푥푎. It acts on the left of functions 퐺 → ℂ by .푎, 푏)푓 ∶ 푥 ↦ 푓(푏−푥푎), 푥 ∈ 퐺) Proposition 4.23. The action ((푎, 푏)푓)(푥) = 푓(푏−푥푎) makes 퐿(퐺/푍, 휔) into a unitary representation of 퐺 × 퐺. Proof. Since 퐺/푍 is assumed to be unimodular, this action is unitary; in other words 퐺 carries a 퐺 × 퐺- invariant positive Radon measure. The rest of the verification is akin to 4.5.
Corollary 4.24. Let 휋 be a square-integrable unitary representation of 퐺 with central character 휔 on 푍. − Then 푣⊗푤 ↦ 푑(휋) 푐푣⊗푤 yields an isomorphism from 휋⊠ ̄휋 onto the irreducible 퐺×퐺-subrepresentation of 퐿 (퐺/푍퐺, 휔) generated by all matrix coefficients of 휋. Proof. The map is 퐺 × 퐺-equivariant by (3.1) on 푉 ⊗ 푉̄ . It is an isometry on 푉 ⊗ 푉̄ by 4.21 since
′ ′ ′ ′ ′ ′ (푣 ⊗ 푣 |푤 ⊗ 푤 )푉⊗푉̄ = (푣|푤)푉 (푣 |푤 )푉̄ = (푣|푤)푉 (푤 |푣 )푉 . These properties extends to 푉⊗̂ 푉̄ by density. The image of any isometry is closed. Irreducibility of 휋 ⊠ ̄휋 follows from 4.14.
Definition 4.25. The positive constant 푑(휋) is called the formal degree of 휋. It depends only on the isomorphism class of 휋 and 퐺, 휇. It is inverse-proportional to the choice of 휇 and generalizes the notion of dimension, cf. 4.40.
36 4.4 Spectral decomposition: the discrete part
Fix a closed subgroup 푍 ⊂ 푍퐺. We still assume 퐺/푍 unimodular, and fix a Haar measure 휇 on it. Definition 4.26. Let 휔 ∶ 푍 → 핊 be a continuous character. The discrete spectrum with central character 휔 is the subrepresentation
퐿disc(퐺/푍, 휔) ∶= ﾜ 휋 ⊂ 퐿 (퐺/푍, 휔) ∶ irred. unitary subrep.�.
By 4.19, every 휋 in the sum is square-integrable and every square-integrable unitary representation 휋 with 휔휋|푍 = 휔 intertwines into 퐿disc(퐺/푍, 휔) via matrix coefficients. Lemma 4.27. Every irreducible unitary subrepresentation of 퐿(퐺/푍, 휔) lies in the closed subspace generated by matrix coefficients of square-integrable representations of central character 휔 on 푍.
Proof. Let 퐶 stand for the closed subspace generated by all square-integrable matrix coefficients of cen- tral character 휔 on 푍. If the assertion does not hold, some irreducible subrepresentation (휋, 푉) would have non-trivial orthogonal projection to 퐶⟂. The projection being 퐺-equivariant, 휋 can be embedded inside 퐶⟂. This is contradictory as explained below. ⟂ Assume that 푉 ⊂ 퐶 and 푓 ∈ 푉 is nonzero. Express 휇 as 휇퐺/휇푍 according to 1.53. There exists a real-valued 휑 ∈ 퐶푐(퐺) such that the function
Φ(푔) ∶= (휑̌ ⋆ 푓)(푔) (continuous by 2.5)
= ﾙ 푓(ℎ푔)휑(ℎ) d휇퐺(ℎ) = ﾙ ﾙ 푓(ℎ푔)휑(푧ℎ)휔(푧) d휇푍(푧) d휇(ℎ) 퐺 퐺/푍 푍
= ﾙ 푓(ℎ푔)휑휔(ℎ) d휇(ℎ) 퐺/푍 is not identically zero, where
휑휔(ℎ) ∶= ﾙ 휑(푧ℎ)휔(푧) d휇푍(ℎ), 휑휔 ∈ 퐶푐(퐺/푍, 휔). 푍 This is indeed possible by 3.29 since
− Φ(푔) = ﾙ 휑(ℎ)푓(ℎ̌ 푔) d휇퐺(ℎ) = 퐿(휑)푓(푔)̌ 퐺
.(where 퐿 denotes the continuous 퐺-representation 퐿(ℎ)푓(푥) = 푓(ℎ−푥) on 퐿(퐺/푍, 휔 = (Note that Φ(푔) = 푐푓⊗푤(푔) where 푤 ∈ 푉 is such that (⋅|휑휔)퐿(퐺/푍,휔) = (⋅|푤)푉 . We infer that (Φ|푐푓⊗푤 , . On the other hand, since is unimodular Φ‖퐿(퐺/푍,휔) ≠ 0 퐺‖
(Φ|푐푓⊗푤) = ｪ 푓(ℎ푔)휑휔(ℎ) ⋅ 푐푓⊗푤(푔) d휇(ℎ) d휇(푔) 퐺/푍×퐺/푍
− = ﾙ ﾙ 푓(푔)푐푓⊗푤(ℎ 푔) d휇(푔) 휑휔(ℎ) d휇(ℎ) 퐺/푍 퐺/푍 (3.1) d .(ﾙ 푓 | 푐푓⊗휋(ℎ)푤 휑휔(ℎ) 휇(ℎ = ∵ 퐺/푍 퐿 (퐺/푍,휔) The inner integral vanishes by assumption. Contradiction.
In general, it may happen that 퐿disc(퐺/푍, 휔) = {0} for all 휔. See 4.11.
37 Theorem 4.28. For each square-integrable unitary representation 휋 with central character 휔 on 푍, there is an isomorphism of unitary 퐺 × 퐺-representations
∽ ̂ ̂ Coeff ∶ ﾘ HS(푉휋) = ﾘ 휋 ⊠ ̄휋 ⟶ 퐿disc(퐺/푍, 휔) 휋 휋
(푣휋 ⊗ 푤휋)휋 ⟼ ﾜ 푑(휋)푐푣휋⊗푤휋 . 휋
• (휋, 푉휋) ranges over the square-integrable representations with central character 휔 on 푍, taken up to isomorphism,
퐺 × 퐺 acts on 퐿(퐺/푍, 휔) by ((푎, 푏)푓)(푥) = 푓(푏−푥푎), and •
• the ⨁̂ is a completed orthogonal direct sum of unitary 퐺 × 퐺-representations.
Moreover, 휋 ⊠ ̄휋 are pairwise non-isomorphic; i.e. the decomposition above of unitary representations is multiplicity-free.
Proof. Combine 4.24 with 4.27. Matrix coefficients from different 휋 are orthogonal, thus the ∑ ⋯ in 4.26 becomes an orthogonal ⨁̂ . The representations 휋 ⊠ ̄휋 are pairwise distinct by 4.15. The relation to HS(푉휋) is explicated in 4.16.
Proposition 4.29. Equip each 푉휋⊗̂ 푉휋 (resp. HS(푉휋)) in 4.28 with the ℂ-algebra structure with multi- plication (푣 ⊗ 푣′) ⋆ (푤 ⊗ 푤′) ∶= (푣|푤′)푣′ ⊗ 푤, (resp. 퐴 ⋆ 퐵 ∶= 퐵 ∘ 퐴). Then Coeff restricted to ⨁ ̄휋⊠ 휋 respects the ⋆-multiplications on both sides. 휋 Proof. This reduces to 4.22 and the rule
(⋅|푣′)푣 ⋆ (⋅|푤′)푤 = (푣|푤′)(⋅|푣′)푤
′ ′ inside each HS(푉휋), which mirrors (푣 ⊗ 푣 ) ⋆ (푤 ⊗ 푤 ). The inverse of Coeff can be described on the linear span of matrix coefficients as follows.
Lemma 4.30. Let (휋, 푉) be a square-integrable unitary representation with central character 휔 on 푍. Consider 휑 ∶= Coeff(푣 ⊗ 푤) ∈ 퐿(퐺/푍, 휔) ∩ 퐶(퐺/푍, 휔), where 푣 ⊗ 푤 ∈ 푉 ⊗ 푉̄ . Then the vector-valued integrals 휋(휑)푣̌ ′ = ﾙ 휑(푔)휋(푔)푣̌ d휇(푔), 푣′ ∈ 푉 퐺/푍 defines an operator 푉 → 푉, which equals the (⋅|푤)푣 ∈ HS(푉) determined by 푣 ⊗ 푤.
Proof. We determine 휋(휑)푣̌ ′ by applying (⋅|푤′) ∈ 푉∗, where 푤′ ∈ 푉 is arbitrary. By (3.2) and 4.21, this euqlas
′ ′ − ′ ′ ﾙ 휑(푔)(휋(푔)푣̌ |푤 ) d휇(푔) = 푑(휋) ﾙ 푐푣⊗푤(푔 )(휋(푔)푣 |푤 ) d휇(푔) 퐺/푍 퐺/푍 ′ ′ = ﾙ 푐푣′⊗푤′ 푐푤⊗푣 d휇 = (푣 |푤)(푣|푤 ). 퐺/푍
Hence 휋(휑)푣̌ ′ = (푣′|푤)푣 and defines a linear endomorphism on 푉. The same description pertains to (⋅|푤)푣.
38 4.5 Usage of compact operators We make systematic use of the formalism in 3.27. − ̃ Lemma 4.31. Let 휑 ∈ ℳ푐(퐺) and let (휋, 푉) be a unitary representation of 퐺. Define 휑(푔)̌ = 휑(푔 ) in the sense of measures. Then ∗휋(휑) = 휋 휑̌ . .(In particular, if 휑 is real-valued and 휑(푔) = 휑(푔−) for all 푔, then 휋(휑) = ∗휋(휑
Proof. For all 푣, 푤 ∈ 푉,
(휋(휑)푣|푤) = ﾙ 휑(푔)(휋(푔)푣|푤) = ﾙ 휑(푔)(푣|휋(푔−)푤) 퐺 퐺
= ﾙ 휑(푔)̌ (휋(푔)푤|푣) = ﾙ 휑(푔)(휋(푔)푤|푣)̌ 퐺 퐺 = 휋 휑̌ 푤|푣 = 푣|휋 휑̌ 푤 .
Example 4.32. Now consider 퐺 acting on 퐿 (퐺/푍, 휔) where 푍 ⊂ 푍퐺 is closed and 휔 ∶ 푍 → 핊 is a chosen continuous homomorphism; call this representation 푅. Choose a right Haar measure 휇 on 퐺/푍, with 휇 = 휇퐺/휇푍 as in 1.53. Assume that 휑 ∈ 퐶푐(퐺) d휇, then we have
푅(휑)푓 (푥) = ﾙ 휑(푔)푓(푥푔) d휇퐺(푔) 퐺
= ﾙ 퐾휑(푥, 푦)푓(푦) d휇퐺/푍(푔) 퐺/푍 with − 퐾휑(푥, 푦) ∶= ﾙ 휑(푥 푧푦)휔(푧) d휇푍(푧), 푥, 푦 ∈ 퐺/푍. (4.3) 푍
Clearly 퐾휑(푥, 푦) ∶ 퐺 × 퐺 → ℂ is continuous and satisfies
″ ′ − ″ ′ ″ ′ 퐾휑(푥푧 , 푦푧 ) = 휔(푧 )휔(푧 ) 퐾휑(푥, 푦), 푧 , 푧 ∈ 푍.
It expresses 푅(휑) as an integral transform on 퐿 (퐺/푍, 휔) with kernel 퐾휑. This is slightly different from the usual picture, since a character 휔 intervenes.
Lemma 4.33 (Gelfand–Graev–Piatetski-Shapiro). Let (휋, 푉) be a unitary representation of 퐺. Suppose that there exists an approximate identity (휑푈 )푈∈픑 (recall 3.25) such that each 휋(휑푈 ) ∶ 푉 → 푉 is a compact operator. Then (휋, 푉) is a completed orthogonal sum of its irreducible subrepresentations. Moreover, each irreducible constituent appears in this decomposition with finite multiplicity.
Proof. The following arguments are due to Langlands. Set
풮 ∶= {sets of mutually orthogonal irreducible subrepresentations ⊂ 휋} .
Note that 풮 is nonempty since ∅ ∈ Σ, and it is partially ordered by set inclusion. Zorn’s lemma affords a maximal element 푆 ∈ 풮 : indeed, every chain in 풮 has the upper bound furnished by taking union. Consider the subrepresentation 휋 ∶= ⨁̂ 휎 of 휋. We claim that 휋 = 휋. 휎∈푆 -Let the subrepresentation 휋 of 휋 be the orthogonal complement of 휋. We will derive a contradic tion from 휋 ≠ {0} by exhibiting an irreducible subrepresentation inside 휋, which would violate the maximality of 푆.
39 Denote by 푉 ⊂ 푉 the underlying space of 휋. By 3.28, there exists 휑푈 from the approximate identity such that . Note that is a self-adjoint compact operator since is, ( 푇 ∶= 휋(휑푈 )|푉 ≠ 0 푇 휋(휑푈 by 3.25 and 4.31. The spectral theorem for such operators (eg. [8, Theorems 4.25 + 12.23]) implies that 푇=휆 푇 has an eigenvalue 휆 ∈ ℝ ∖ {0}, and the eigenspace 푉 is nonzero and finite-dimensional. Take a ♭ 푇=휆 ♭ 푇=휆 푇=휆 subspace 푉 ⊂ 푉 of minimal dimension subject to the condition that 푉 = 푊 = 푊 ∩ 푉 for 푇=휆 = some 퐺-stable closed subspace 푊 ⊂ 푉. Among these 푊, their intersection 푊min ∶= ⋂{푊 ∶ 푊 ♭ 푉} is the smallest one. We contend that 푊min is irreducible. Otherwise there would be an orthogonal decomposition 푊min = 퐴 ⊕ 퐵 into nonzero 퐺-stable closed subspaces, and
♭ 푇=휆 푇=휆 푇=휆 . 푉 = 푊min = 퐴 ⊕ 퐵 ♭ ♭ 푇=휆 푇=휆 -The minimality assumption on 푉 implies that 푉 equals 퐴 or 퐵 , but this contradicts the mini mality of 푊min. We conclude that 휋 = ⨁̂ 휎. Suppose that some irreducible unitary representation 휎 (considered 휎∈푆 up to isomorphism) intervenes with multiplicity 푚. Then for every 휑푈 in the approximate identity, 휋(휑푈 ) will restrict to 푚-copies of 휎(휑푈 ). By assumption we may choose 휑푈 such that 휎(휑푈 ) ≠ 0, thus has a nonzero eigenvalue 휆. As the 휆-eigenspace of 휋(휑푈 ) is finite-dimensional, 푚 must be finite as well. Proposition 4.34. In the setting of 4.32, suppose that 퐺/푍 is compact, then 퐿(퐺/푍, 휔) decomposes into a completed orthogonal direct sum of irreducible unitary representations, each constituent appearing with finite multiplicity. In particular, 퐿 (퐺/푍, 휔) = 퐿disc(퐺/푍, 휔) when 퐺 is unimodular. Proof. In view of 4.33, it suffices to show that each 휑 ∈ 퐶푐(퐺) acts on 퐿 (퐺/푍, 휔) via compact operators. (In fact they act as Hilbert–Schmidt operators: a standard result asserts that integral operators on 퐿(푋 prescribed by a kernel 퐾 ∈ 퐿(푋 × 푋) are Hilbert–Schmidt. If 푋 is compact then every continuous 퐾 ∶ 푋 × 푋 → ℂ is 퐿 . One can take 푋 = 퐺/푍 if 휔 = 1, but the arguments with 휔 ∶ 푍 → 핊 and 퐾휑 satisfying (4.3) are entirely analogous.
4.6 The case of compact groups Suppose that 퐺 is a unimodular group and 퐺/푍 is compact. Therefore all irreducible unitary representa- tions of 퐺 are square-integrable relative to 푍. Fix a Haar measure 휇 on 퐺/푍. Proposition 4.35. Every irreducible unitary representation (휋, 푉) of 퐺 is finite-dimensional. Proof. The argument below is credited to L. Nachbin. Take a nonzero 푣 ∈ 푉. Define for all 푤 ∈ 푉 the vector-valued integral
퐴푤 ∶= ﾙ (푤|휋(푔)푣)푉 ⋅ 휋(푔)푣 d휇(푔). 퐺/푍 By 3.24 we know ‖퐴푤‖푉 ≤ ‖푤‖푉 ⋅ ‖푣‖푉 ⋅ 휇(퐺/푍), and 퐴 is easily seen to be 퐺-equivariant as 퐺 acts unitarily. Hence 퐴 = 휆 ⋅ id푉 for some 휆 ∈ ℂ by 4.7; moreover (퐴푣|푣)푉 = ∫ |(푣|휋(푔)푣)| d휇(푔) implies 휆 > 0. Now let 푉 ⊂ 푉 be any finite-dimensional subspace with the corresponding orthogonal projection 퐸 ∶ 푉 → 푉 ⊂ 푉. Consider 퐸퐴 restricted to 푉 to obtain
d id .ﾙ ⋅ |퐸휋(푔)푣 ⋅ 퐸휋(푔)푣 휇(푔) = 휆 ⋅ 푉 ∶ 푉 → 푉 퐺/푍
Inside Endℂ(푉), we take traces on both sides and pass it under the integral sign to deduce
.휇(퐺/푍)‖푣‖푉 ≥ ﾙ ‖퐸휋(푔)푣‖ d휇(푔) = 휆 dim 푉 퐺/푍 This bound shows that dim 푉 is finite.
40 Theorem 4.36 (Peter–Weyl: the 퐿 version). Let 휔 ∶ 푍 → 핊 be a continuous homomorphism. There is an isomorphism of unitary 퐺 × 퐺-representations
∽ ̂ ̂ Coeff ∶ ﾘHS(푉휋) = ﾘ휋 ⊠ ̄휋 ⟶ 퐿 (퐺/푍, 휔) 휋∶irred 휋∶irred
(푣휋 ⊗ 푤휋)휋 ⟼ ﾜ 푑(휋)푐푣휋⊗푤휋 휋
and the decomposition is multiplicity-free. Consequently, the matrix coefficients of irreducible unitary .(representations with central character 휔 span a dense subspace in 퐿(퐺/푍, 휔 Furthermore, the inverse of Coeff is determined as follows. Let 휑 be a matrix coefficient of 휋, then
− Coeff (휑) = 휋(휑)̌ ∶= ﾙ 휑(푔)휋(푔)̌ d휇(푔) ∈ Endℂ(푉휋). 퐺/푍 Proof. To obtain the decomposition, simply combine 4.28 with 4.34. The inverse map follows from 4.30.
Remark 4.37. In deriving 4.36, we did not use the fact that each irreducible 휋 has finite dimension. In fact, we can deduce 4.35 from 4.36 as follows. Restrict 퐺 × 퐺-representations to 퐺 × {1} ≃ 퐺, so that the spectral decomposition degenerates into an isomorphism of unitary 퐺-representations
̂ ﾘ 휋⊗푉̂ 휋 ≃ 퐿 (퐺/푍, 휔) 휋
where 푉휋 still carries its Hilbert space structure, but now with trivial 퐺-action. By 4.34 we know 휋 must occur in 퐿 (퐺/푍, 휔) with finite multiplicity, which is exactly dim 푉휋. Corollary 4.38. The space 퐿 (퐺/푍, 휔) is closed under convolution ⋆. Specifically, if each 푉휋⊗̂ 푉휋 (resp. HS(푉휋)) is endowed with the ℂ-algebra structure from 4.29, then Coeff is an isomorphism between ℂ- algebras with multiplication given by ⋆ on both sides.
Proof. Since 퐿(퐺/푍, 휔) ⊂ 퐿(퐺/푍, 휔) by compactness, 퐿-property is preserved under convolution by 2.3. It suffices to check the compatibility with convolution on a dense subspace of ⨁̂ 휋 ⊠ ̄휋, and 4.29 휋 will do the job.
Finer properties of the spectral decomposition will be studied in the section on direct integrals.
4.7 Basic character theory
We keep the assumptions from §4.6: 퐺 is a locally compact group, 푍 ⊂ 푍퐺 is closed and 퐺/푍 comes with a chosen Haar measure 휇. Fix a continuous homomorphism 휔 ∶ 푍퐺 → 핊 . Throughout this subsection, we choose the Haar measure normalized by
휇(퐺/푍) = 1.
For endomorphisms of finite-dimensional vector spaces, it is safe to take trace. The following isthus justified.
Definition 4.39. The character of a finite-dimensional unitary representation (휋, 푉) of 퐺 is the contin- uous function
Θ휋 ∶ 퐺 ⟶ ℂ 푔 ⟼ Tr 휋(푔) ∶ 푉 → 푉 .
41 We clearly have Θ휋⊕휎 = Θ휋 +Θ휎 and Θ ̄휋 = Θ휋. Note that Θ휋 does not involve choices of measures. Proposition 4.40. The formal degree of an irreducible unitary representation (휋, 푉) equals dim 푉.
Proof. Choose an orthonormal basis 푣, … , 푣푑 of 푉. Then 휋(푔)푣, … , 휋(푔)푣푑 form an orthonormal basis for every 푔 ∈ 퐺. Then 푐푣 ⊗푣 (푔) is a transition matrix between orthonormal bases, hence unitary. 푖 푗 ≤푖,푗≤푑 , It follows that ∑ . Integrate both sides over and apply 4.21 to see ∑ − 푖,푗 |푐푣푖⊗푣푗 (푔)| = 푑 퐺/푍 푖,푗 푑(휋) = 푑 that is, 푑(휋) = 푑.
Theorem 4.41. If 푣, … , 푣푑 is an orthonormal basis of an irreducible unitary representation (휋, 푉) of 퐺, then 푑 ﾜ Θ휋(푔) = 푐푣푖⊗푣푖 (푔) ∈ 퐶(퐺/푍퐺, 휔휋). 푖= Furthermore, for any two irreducible unitary representations 휋, 휎 of 퐺 with central character 휔 on 푍, ⎧ ⎨⎪1, 휋 ≃ 휎 (Θ휋|Θ휎) = ⎪ .퐿(퐺/푍,휔) ⎩0, 휋 ≄ 휎
∑푑 Proof. For any linear endomorphism 퐴 ∶ 푉 → 푉 we have Tr(퐴) = 푖=(퐴푣푖|푣푖). This proves the first assertion. The second assertion stems from Schur’s orthogonality relations 4.21 and 4.40.
푔 Theorem 4.42. For the adjoint action 푓(푥) ↦ 푓(푔−푥푔) of 퐺 on 퐿(퐺/푍, 휔), the invariant-subspace is
̂ 퐺 퐿 (퐺/푍, 휔) = ﾘ ℂΘ휋. 휋∶irred 휔휋|푍=휔
Proof. We know that 휋⊠ ̄휋= HS(푉휋) = Endℂ(푉휋) as 퐺×퐺-representations by 4.16. The adjoint action 푓(푥) ↦ 푓(푔−푥푔) mirrors the diagonal 퐺-action on 휋⊠ ̄휋, which in turn corresponds to 퐴 ↦ 휋(푔)퐴휋(푔)− on Endℂ(푉휋); topology does not matter here since dim 푉휋 < +∞. Hence the space of 퐺-invariants matches End퐺(휋) = ℂ ⋅ id by 4.12. In fact, id corresponds to ∑ ∋ 푖 푣푖 ⊗ 푣푖 ∈ 푉휋 ⊗ 푉휋 where 푣, 푣,… is any orthonormal basis; it has norm 푑(휋) and maps to 푑(휋)Θ휋 .퐿(퐺/푍, 휔) under Coeff As in the case of finite groups, characters distinguish representations.
Proposition 4.43. Let 휋, 휎 be finite-dimensional unitary representations. Then 휋 ≃ 휎 if and only if Θ휋 = Θ휎. Proof. Decompose 휋 and 휎 into orthogonal sums of irreducibles, which is possible by 4.6, and then apply 4.41.
5 Unitary representation of compact Lie groups (a sketch)
Throughout this section, Lie groups are always taken over ℝ. For a Lie group 퐺 we denote by 픤 its real .Lie algebra, and 픤 ∶= 픤 ⊗ℝ ℂ. Denote the center of 픤 as 픷. Below we write 푖 = √−1
42 5.1 Review of basic notions Let 퐺 be a connected compact Lie group. Below is an incomplete collection of basic results on the structure of such groups. The reader may consult any available text on compact Lie groups for details, such as [5].
.The exponential map exp ∶ 픤 → 퐺 is surjective •
.The Lie algebra is reductive, namely 픤 = 픷 ⊕ [픤, 픤] and [픤, 픤] is semisimple • ∘ [The Lie subalgebra 픷 corresponds to the identity connected component (푍퐺) of 푍퐺, and [픤, 픤 • ∘ corresponds to a closed Lie subgroup 퐺푠푠. We have 퐺 = (푍퐺) ⋅ 퐺푠푠. In particular, 퐺푠푠 is a compact semisimple Lie group.
∘ • A torus means a connected commutative compact Lie group. Therefore the group 푍퐺 above is a torus.
• A theorem of Weyl asserts that every compact semisimple Lie group admits a finite covering which is a connected, compact and simply connected Lie group. Applied to 퐺푠푠, there is a finite covering of 퐺 of the form 푍 × 퐺sc ↠ 퐺, (5.1)
where 푍 is a torus and 퐺sc is a connected and simply connected compact Lie group.
Let 푇 be a torus. Since exp ∶ 픱 → 푇 is an open surjective homomorphism between Lie groups, we dim 푇 .see 픱/ℒ ≃ 푇 where ℒ ∶= ker(exp) is a lattice in 픱. Consequently, 푇 ≃ (ℝ/ℤ) as Lie groups Define the character lattice as ∗ ∗ ∗ 푋 (푇) ∶= Hom(푇, 핊 ) ↪ 푖픱 ⊂ 픱 Here Hom is taken in the category of Lie groups, and the inclusion is given by mapping 휒 ∶ 푇 → 핊 to ∗ the 휆 ∈ 푖픱 such that ;휒(exp(푋)) = exp(⟨휆, 푋⟩), 푋 ∈ 픱 ∗ ∗ so 휆 is essentially the derivative of 휒 at 1. This identifies 푋 (푇) with a lattice in 푖픱, namely
∗ , �휆 ∈ 푖픱 ∶ 휆(ℒ ) ⊂ 2휋푖ℤ and 휒̄ corresponds to −휆 whenever 휒 corresponds to 휆. It is sometimes beneficial to read 휒 as “푒휆”. Now we consider maximal tori in 퐺: they are closed tori 푇 ⊂ 퐺, maximal with respect to inclusion.
.The Lie algebra 픱 of a maximal torus 푇 is an abelian subalgebra in 픤, and vice versa • • The maximal tori in 퐺 are all conjugate.
• Given a maximal torus 푇, every element is conjugate to some element of 푇. If 푡, 푡′ ∈ 푇 are conjugate in 퐺, then they are actually conjugate in the normalizer 푁퐺(푇).
• 푍퐺 is contained in the intersection of all maximal tori. Fix a maximal torus 푇 ⊂ 퐺, there is a decomposition
픤 = 픱 ⊕ ﾘ 픤훼 훼∈︾(픤,픱)
∗ into 푇-eigenspaces under the adjoint action Ad ∶ 푇 → Endℂ(픤). Here Φ = Φ(픤, 픱) ⊂ 푖픱 is the set of roots for (픤, 픱); all the subspaces 픤훼 are 1-dimensional. This yields a reduced root system on 푖픱. By choosing a system of positive roots, we decompose Φ = Φ + ⊔ (−Φ +).
43 .Given any root 훼, the associated coroot is denoted by 훼̌ ∈ 푖픱 The positive coroots 훼̌ cut out the “acute Weyl chamber”
∗ ∗ ;풞+ ∶= ﾎ {훼̌ ≥ 0} ⊂ 푋 (푇) ⊗ ℝ = 푖픱 훼∈︾+
∘ ⋂ the elements of 풞+ are called dominant. Denote 풞+ ∶= 훼∈︾+ {훼̌ > 0}. The algebraically defined Weyl group Ω(픤, 픱) is generated by root reflections
푠훼 ∶ 휆 ↦ 휆 − ⟨휆, 훼̌⟩ 훼, 훼 ∈ Φ
− ∗ on 푖픱. It is a finite group. It also acts on 푖픱 by duality: we require that ⟨푤휆, 푋⟩ = ﾊ휆, 푤 푋ｽ for all 푤 ∈ Ω(픤, 픱) It turns out that Ω(픤, 픱) together with its action on 푖픱∗ is canonically isomorphic to the analytically de- ,fined Ω(퐺, 푇) = 푁퐺(푇)/푍퐺(푇). This is essentially done by reducing to the case 픤 = 픰픲(2). Furthermore 풞+ is a fundamental domain for the Ω(퐺, 푇)-action: we have
∗ ,+푖픱 = ﾌ 푤풞 푤∈︵(퐺,푇) ′ ∘ ′ ∘ 푤 ≠ 푤 ⟹ 푤풞+ ∩ 푤 풞+ = ∅,
(휆 ∈ 풞+ ∧ 푤휆 ∈ 풞+) ⟹ 푤휆 = 휆.
The reflections relative to simple roots in Φ(픤, 픱) generate Ω(퐺, 푇). We write ℓ ∶ 푊 → ℤ≥ for the length function for Ω(퐺, 푇) with respect to these generators. Recall that (−1)ℓ(⋅) ∶ Ω(퐺, 푇) → {±1} is a homomorphism of groups. When 퐺 is semisimple, we define
푃 ∶= {휆 ∈ 푋∗(푇) ⊗ ℝ ∶ ∀훼 ∈ Φ, ⟨휆, 훼̌⟩ ∈ ℤ} , 푄 ∶= ﾜ ℤ훼. 훼∈︾+
It is a general fact about root systems that they are both lattices in 푋∗(푇) ⊗ ℝ, and
푃 ⊃ 푋∗(푇) ⊃ 푄
∗ ∗ If 퐺 is simply connected (resp. 푍퐺 = {1}, i.e. adjoint), then 푃 = 푋 (푇) (resp. 푄 = 푋 (푇)). For ∗ semisimple 퐺 in general, 푃/푋 (푇) ≃ 휋(퐺, 1). In any case, 푃 contains the half-sum of positive roots 1 휌 ∶= ﾜ 훼 ∈ 푃. 2 훼∈︾+
5.2 Algebraic preparations Definition 5.1. For any Ω-stable lattice Λ ⊂ 푋∗(푇) ⊗ ℝ, and any commutative ring 퐴 (usually 퐴 = ℤ), the elements of the group 퐴-algebra 퐴[Λ] are expressed uniquely in the exponential notation
휆 ﾜ 푐휆푒 , 푐휆 ∈ 퐴 ∶ finite sum 휆∈︹ subject to the relation 푒휆푒휇 = 푒휆+휇 and the usual laws of algebra. The Weyl group Ω(퐺, 푇) acts 퐴-linearly by mapping 푒휆 to 푒푤휆.
44 푟 ± ±⊕ ,Once an isomorphism Λ ≃ ℤ is chosen, ℤ[Λ] is identifiable with ℤ[푋 , … , 푋푟 ]. In particular ∗ .[(ℤ[Λ] is a unique factorization domain since it is a localization of ℤ[푋, … , 푋푟]. Same for ℂ[푋 (푇 Fix a maximal torus 푇 ⊂ 퐺 together with a system of positive roots Φ + ⊂ Φ = Φ(픤, 픱). We will mainly work inside ℤ[푋∗(푇)]. Keep in mind that 휆 ∈ 푋∗(푇) (thus the symbol 푒휆) also signifies the character 푇 → 핊 mapping exp(푋) to exp(⟨휆, 푋⟩), for all 푋 ∈ 픱. By linearity, we obtain a map ℂ[푋∗(푇)] ⟶ 퐶(푇).
∗ ∗ ∗ As 푋 (푇) ⊂ 푖픱, it is reasonable to define the complex conjugation in ℂ[푋 (푇)] by
휆 −휆 ﾜ 푐휆푒 = ﾜ 푐휆푒 . 휆 휆
.[(which extends the complex conjugation on Hom(푇, 핊) ⊂ ℤ[푋∗(푇 Lemma 5.2. The map ℂ[푋∗(푇)] → 퐶(푇) is an embedding of ℂ-algebras, where 퐶(푇) is equipped with pointwise addition and multiplication. It is Ω(퐺, 푇)-equivariant and respects complex conjugation on both sides. Proof. For injectivity, apply the linear independence of characters to 푇. The other assertions are inherent in the construction. It is convenient to enlarge 푋∗(푇) to the commensurable lattice Λ ∶= 푋∗(푇) to accommodate for .[elements like 푒훼/, for 훼 ∈ Φ. Notice that Λ′ ⊂ Λ implies ℤ[Λ′] ⊂ ℤ[Λ Definition 5.3. The Weyl denominator is the element
Δ ∶= 푒휌 ﾟ (1 − 푒−훼) = ﾟ 푒훼/ − 푒−훼/ 훼∈︾+ 훼∈︾+
which lives in ℤ 푋∗(푇) , or ℤ[푃] if 퐺 is semisimple. Several algebraic lemmas are in order. Lemma 5.4. For every 푤 ∈ Ω(퐺, 푇) we have 푤Δ = (−1)ℓ(푤)Δ. 훼/ −훼/ ∏ + Proof. It suffices to check this for 푤 = 푠훽 where 훽 ∈ Φ is a simple root. Use Δ = 훼∈︾+ (푒 − 푒 ) + + and the fact that 푠훽(훼) ∈ Ψ for all 훼 ∈ Φ ∖ {훽}, and 푠훽(훽) = −훽. 훼 − ∑ −푘훼− In what follows, it would be convenient to be able to expand (1 − 푒 ) in a formal series 푘≥ 푒 for every 훼 ∈ Φ +. For every lattice Λ ⊂ 푋∗(푇) ⊗ ℝ, we can ”complete” ℤ[Λ] in the direction of the ∗ + closed cone −풞 ⊂ 푋 (푇)⊗ℝ (the negative “obtuse Weyl chamber”) generated by −(Φ ) to accommodate such formal series. Specifically, we start with the monoid algebra ℤ[−풞 ∩ Λ], take its adic completion ∘ 휆 with respect to the interior ideal generated by −풞 ∩ Λ; finally, invert {푒 ∶ 휆 ∈ Λ} to reach the desired algebra. Lemma 5.5. Suppose that Λ is a lattice in 푋∗(푇)⊗ℝ. If 휇, 휈 ∈ Λ∖{0} are not proportional, then 1−푒휇, 1 − 푒휈 do not generate the same ideal in ℤ[Λ]. Proof. By working in some “completion” of ℤ[Λ] as explained earlier, we expand 1 − 푒휇 휇 ﾜ 푘휈 ﾜ 푘휈 ﾜ 훼+푘훽 휈 = (1 − 푒 ) 푒 = 푒 − 푒 . 푒 푘≥ 푘≥ 푘≥ − 1
If 훼, 훽 are linearly independent in 푋∗(푇)⊗ℝ, there will be infinitely many nonzero terms, so 1−푒휈 cannot −푒휈 divide 1 − 푒휇 in ℤ[Λ]. The same reasoning applies to . −푒휇
45 Lemma 5.6. Suppose that Λ is an Ω(퐺, 푇)-stable lattice in 푋∗(푇)⊗ℝ such that ⟨Λ, 훼̌⟩ ⊂ ℤ for all 훼 ∈ Φ. An element 휒 ∈ ℤ[Λ] satisfies 푤휒 = (−1)ℓ(푤)휒 for all 푤 ∈ Ω(퐺, 푇) if and only if 휒 ∈ Δ ⋅ ℤ[Λ]︵(퐺,푇). Proof. Given 5.4, it suffices to prove the “only if” direction. Using
• the unique factorization property in ℤ[Λ],
• 1 − 푒−훼 and 1 − 푒−훽 do not generate the same ideal since Φ is a reduced root system,
• and the fact that 푒휇 ∈ ℤ[Λ]× for all 휇, −훼 + ∑ 휆 it suffices to show that (1 − 푒 ) ∣ 휒 for all 훼 ∈ Φ . Write 휒 = 휆 푐휆푒 . By assumption we have
푐푠훼휆 = −푐휆, 휆 ∈ Λ and 푠훼(휆) = 휆 − ⟨휆, 훼̌⟩ 훼. Collecting {1, 푠훼}-orbits, we see that 휒 can be expressed as a ℤ-linear sum of expressions 푒휆 − 푒휆−⟨휆,훼̌⟩훼 = 푒휆 1 − 푒−⟨휆,훼̌⟩훼 . The last term is divisible by 1 − 푒−훼 as 푘 ∶= ⟨휆, 훼̌⟩ ∈ ℤ; handle the cases 푘 ≥ 0 and 푘 < 0 separately.
Remark 5.7. There exists a lattice Λ ⊂ 푋∗(푇) ⊗ ℝ such that Λ ⊃ 푋∗(푇) ∪ {휌} and ⟨Λ, 훼̌⟩ ⊂ ℤ for all 훼 ∈ Φ. For example, in the finite covering 휋 ∶ 푍 × 퐺sc ↠ 퐺 of (5.1), take a maximal torus in 푍 × 퐺sc ∗ ∗ of the form 푍 × 푇sc that surjects onto 푇, and consider Λ ∶= 푋 (푍 × 푇sc). Pull-back induces 푋 (푇) ↪ Λ whereas 푋∗(푇) ⊗ ℚ = Λ ⊗ ℚ. On the other hand, the roots remain unaltered under pull-back since 휋 induces an isomorphism on Lie algebras.
Definition 5.8. Take Λ as in 5.7. For 휆 ∈ ℤ[푋∗(푇)] ∩ 풞 +, define
( ℓ(푤) 푤(휆+휌− 휒휆 ∶= Δ ﾜ (−1) 푒 푤∈︵(퐺,푇) .ﾟ (1 − 푒−훼)− ﾜ (−1)ℓ(푤)푒푤(휆+휌)−휌 = 훼∈︾+ 푤∈︵(퐺,푇)
It is a priori an element in the ring of fractions of ℤ[Λ].
+ We will show in 5.10 that 휒휆 is independent of the choice of Φ .
∗ ∗ ︵(퐺,푇) Lemma 5.9. For every 휆 ∈ ℤ[푋 (푇)], we have 휒휆 ∈ ℤ[푋 (푇)] . ∑ ℓ(푤) 푤(휆+휌) ℓ(⋅) Proof. Fix 휆. Clearly 푤∈︵(퐺,푇)(−1) 푒 varies by (−1) under the Ω(퐺, 푇)-action on ℤ[Λ], ︵(퐺,푇) hence lies in Δℤ[Λ] by 5.6. On the other hand, the second expression of 휒휆 can be expanded into a formal series by completing ℤ[Λ] (thus ℤ[푋∗(푇)]) in the direction of the closed cone generated by Φ +. Observe that 푤(휆 + 휌) = 푤휆 + (푤휌 − 휌) ∈ 푋∗(푇) as
푤휌 − 휌 = − ﾜ 훼 ∈ 푄. 훼∈︾+ 푤훼∉︾+
휇 ∗ So the expansion of 휒휆 in formal series involves only 푒 with 휇 ∈ 푋 (푇). A comparison with the previous ∗ ︵(퐺,푇) step shows that 휒휆 ∈ ℤ[푋 (푇)] . Remark 5.10. It is a standard fact that the choices of Φ + in Φ form a Ω(퐺, 푇)-torsor under conjugation. + + If we pass from Φ to 푤Φ , then 휒휆 is also transported by 푤. The result above shows that 휒휆 is actually independent of Φ +.
46 ∑ 휇 ∗ Write 휒휆 = 휇 푐휇푒 and let Supp(휒휆) ∶= 휇 ∈ 푋 (푇) ∶ 푐휇 ≠ 0�. As observed above, Supp(휒휆) is Ω(퐺, 푇)-stable. For 휆, 휆′ ∈ 푋∗(푇) ⊗ ℝ, we write
′ ′ 휆 ≺ 휆 ⟺ 휆 − 휆 = ﾜ 푛훼 훼. (5.2) + 훼∈︾ ≥
∗ Lemma 5.11. Let 휆 ∈ 푋 (푇). We have 휆 ∈ Supp(휒휆) and it appears with coefficient 1. If 휇 ∈ Supp(휒휆) ∑ .then 휇 = 휆 − 훼∈︾+ 푛훼훼 for some coefficients 푛훼 ∈ ℤ≥; in particular 휇 ≺ 휆 Proof. Work in a suitably completed group algebra to write
(휌 −훼 −훼 ℓ(푤) 푤(휆+휌− 휒휆 = 푒 ﾟ 1 + 푒 + 푒 + ⋯ ﾜ (−1) 푒 훼∈︾+ 푤∈︵(퐺,푇) .(푒휆 ﾟ 1 + 푒−훼 + 푒−훼 + ⋯ ﾜ (−1)ℓ(푤)푒푤(휆+휌)−(휆+휌 = 훼∈︾+ 푤∈︵(퐺,푇)
∑ The basic theory of root systems says 푤(휆 + 휌) − (휆 + 휌) is always of the form − 훼∈︾+ 푛훼훼, with .푛훼 ∈ ℤ≥, and is trivial only when 푤 = 1
∗ Lemma 5.12. Let 휆, 휇 ∈ 푋 (푇) ∩ 풞+. The coefficient of 1 = 푒 in 휒휆Δ휒휇Δ equals |Ω(퐺, 푇)| (resp. 0) if 휆 = 휇 (resp. 휆 ≠ 휇).
Proof. Look at the coefficient of 푒 in
(Ω(퐺, 푇)|− ﾜ(−1)ℓ(푤)푒푤(휆+휌) ⋅ ﾜ(−1)ℓ(푣)푒−푣(휇+휌| 푤 푣 .(Ω(퐺, 푇)|− ﾜ(−1)ℓ(푤)+ℓ(푣)푒푤(휆+휌)−푣(휇+휌| = 푤,푣
∘ Suppose that 푤(휌 + 휆) = 푣(휌 + 휇). Since 휌 + 휆, 휌 + 휇 ∈ 풞+, this implies 푤 = 푣 and 휆 = 휇. Therefore ( ∑ ℓ(푤)+ℓ(푣) 푤(휆+휌)−푣(휇+휌 푒 appears with multiplicity |Ω(퐺, 푇)| in 푤,푣(−1) 푒 .
5.3 Weyl character formula: semisimple case Still assume 퐺 to be a connected compact Lie group with maximal torus 푇, and choose a system of + positive roots Φ ⊂ Φ = Φ(픤, 픱). Fix the Haar measures 휇퐺 on 퐺 and 휇푇 on 푇, both of total mass 1. Notice that since Φ = Φ + ⊔ (−Φ +),
ΔΔ̄ = ﾟ(1 − 푒훼) = ﾟ(푒훼 − 1) ∈ ℤ[푋∗(푇)], 훼∈︾ 훼∈︾
.thus defines a continuous function on 푇. We denote this function as |Δ|
Theorem 5.13 (Weyl’s integration formula). If 푓 ∶ 퐺 → ℂ is measurable, then
− 1 ﾙ 푓 d휇퐺 = ｪ 푓(푔 푡푔)|Δ| (푡) d(휇 × 휇푇 )(푇푔, 푡) 퐺 |Ω(퐺, 푇)| (푇\퐺)×푇 where 휇 = 휇퐺/휇푇 is the measure on 푇\퐺 prescribed in 1.53.
47 Proof. Note that 휇(퐺/푇) = 1 by plugging 푓 = 1 into (1.9). We shall transform the integral by pulling it back through the submersion
퐴 ∶ (푇\퐺) × 푇 ⟶ 퐺 .푇푔, 푥) ⟼ 푔−푥푔)
In doing integration we can work on a dense open subset of 퐺 (resp. 푇). Here we consider the regular lo- − ⋃ cus 퐺reg (resp. 푇reg); 푡 ∈ 푇 lies in 푇reg if and only if 푍퐺(푡) = 푇; set 퐺reg = 푔∈퐺 푔 푇reg푔. Once restricted − to 퐺reg, the map 퐴 becomes a Ω(퐺, 푇)-torsor with the left Ω(퐺, 푇)-action by (푇푔, 푥) ↦ (푇푤푔, 푤푥푤 ). −− Indeed, if 푔푥푔 = ℎ푦ℎ ∈ 퐺reg, then 푥 is conjugate to 푦 through a unique 푤 ∈ Ω(퐺, 푇), with represen- − − −− tative 푤̃ ∈ 퐺. Thus 푤̃ ℎ 푔 centralizes 푥, so that 푤̃ ℎ 푔 ∈ 푁퐺(푇) and has trivial image in Ω(퐺, 푇), .(showing that 푔푥푔− and ℎ푦ℎ− differ by a unique 푤 ∈ Ω(퐺, 푇 It remains to pinpoint the Jacobian. Choose the invariant volume forms on 픤, 픱 and 픤/픱 which match the volume forms for 휇퐺, 휇푇 and 휇 at 1 = exp(0) via the exponential. Decompose 픤 = 픱 ⊕ 픭 with 픭 = ⨁ 픤 ; the latter is actually defined over ℝ, namely 픭 = 픭 ⊗ ℂ, and 픭 carries the corresponding 훼 훼 volume form. Scrutinize the local behavior of 퐴 around a chosen (푇푔, 푥):
− .픭 × 픱 ∋ (푋, 푌) ↦ 푔 exp(−푋) exp(푌)푥 exp(푋)푔 Using the invariance of Haar measures under bilateral translations, it suffices to consider
푋, 푌) ↦ exp(−푋) exp(푌)푥 exp(푋)푥−) ;(exp(−푋) exp(푌) exp(Ad(푥−)푋 = by writing Ad(푥−)푋 = 푥−푋푥, its differential at (0, 0) is seen to be
.푋, 푌) ↦ Ad(푥−) − 1 푋 + 푌)
Relative to our compatible choice of volume forms, the Jacobian factor at (푇푔, 푥) is precisely the absolute value of det Ad(푥−) − 1|픭. To conclude the computation, note that
; det Ad(푥−) − 1|픭 = ﾟ(훼(푥)− − 1) = ﾟ |훼(푥) − 1| 훼∈︾ 훼∈︾+ here 훼 is the character corresponding to what we denoted by 푒훼.
Remark 5.14. By the basic properties of conjugacy classes on 퐺, the inclusion 푇 ↪ 퐺 induces a bijection 푇/푊 → 퐺/conj. This induces a matching between invariant functions. A conjugation-invariant function ♭ ♭ 푓 ∶ 퐺 → ℂ is continuous if and only if its Ω(퐺, 푇)-invariant avatar 푓 = 푓|푇 ∶ 푇 → ℂ is. We supply a low-tech proof of this fact as follows. ♭ The “only if” part is obvious. Conversely, assume 푓 is continuous and consider any sequence 푥푗 → 푥 − in 퐺; write 푥푗 = 푔푗푡푗푔푗 with 푡푗 ∈ 푇. Thanks to compactness, we may pass to subsequences to assume ♭ − that the limits 푔푗 → 푔 and 푡푗 → 푡 exist, thus 푥 = 푔푡푔 . We deduce that 푓(푥푗) = 푓 (푡푗) → 푓(푡) = 푓(푥), whence the continuity of 푓. .Denote the spaces of conjugation-invariant functions on 퐺 as 퐶(퐺)퐺−inv, 퐿(퐺)퐺−inv and so on Corollary 5.15. For 휉, 휂 ∈ ℂ[푋∗(푇)]︵(퐺,푇), we identify them as Ω(퐺, 푇)-invariant continuous functions on 푇, then viewed as a conjugation-invariant continuous functions on 퐺 by 5.14. Let 푐(휉, 휂) be the coefficient of 푒 = 1 in 휉Δ휂Δ = 휉휂|Δ|̄ , then
(푐(휉, 휂 . = 휉|휂) ) 퐿 (퐺) |Ω(퐺, 푇)|
48 Proof. Plug these objects into 5.13 to see
1 . 휉|휂)퐿(퐺) = ﾙ 휉휂|Δ| d휇푇) |Ω(퐺, 푇)| 푇
Recall that the integral over 푇 of a character 휒 ∶ 푇 → 핊 equals zero (resp. 1 = 휇푇 (푇)) when 휒 ≠ 1 (resp. 휒 = 1). The assertion becomes visible after expanding 휉Δ휂Δ in ℤ[푋∗(푇)].
∘ ∗ Lemma 5.16. Assume 퐺 is semisimple. The elements 휒휆 for 휆 ∈ 풞+ ∩ 푋 (푇) form a ℤ-basis for ℤ[푋∗(푇)]︵(퐺,푇). Moreover, in terms of the embedding ℤ[푋∗(푇)]︵(퐺,푇) ↪ 퐶(퐺)퐺−inv of 5.2, they are .(orthonormal with respect to (⋅|⋅)퐿(퐺 ∗ ︵(퐺,푇) ∑ 휇 ∘ ∗ Proof. Observe that ℤ[푋 (푇)] has a ℤ-basis of the form 휉휆 ∶= 휇∈푊⋅휆 푒 where 휆 ∈ 풞+ ∩ 푋 (푇). Write 휒휆 = ﾜ 푐휆,휇휉휇, 푐휆,휇 ∈ ℤ. ∘ ∗ 휇∈풞+∩푋 (푇)
Then 5.11 entails that 푐휆,휆 = 1 and 푐휆,휇 ≠ 0 ⟹ 휇 ≺ 휆. Note that the ≺ defined in (5.2) induces a total ∗ order on 푋 (푇) ⊗ ℝ since 픤 is semisimple. This shows that (푐휆,휇)휆,휇 is an upper triangular matrix over ℤ when 휆, 휇 are enumerated along ≺, with diagonal entries equal to 1. Hence it is invertible, and this ∗ ︵(퐺,푇) implies that {휒휆}휆 form a ℤ-basis of ℤ[푋 (푇)] . The orthonormal property follows from 5.12 and 5.15.
Now apply the results in §4.6 with 푍 = {1}. Given a unitary representation 휋 of 퐺 with dim 푉휋 < ∞, its restriction to 푇 decomposes into irreducibles. By 4.10 this is
⊕mult(휋∶휇) 휋|푇 = ﾘ 휇 , 휇∈푋∗(푇) where mult(휋 ∶ 휇) ∈ ℤ≥ stands for the multiplicity of 휇 in 휋|푇 . On the other hand, its character Θ휋 restricted to 푇 gives a finite sum
Θ휋(푡) = ﾜ mult(휋 ∶ 휇)휇(푡), 푡 ∈ 푇. 휇∈푋∗(푇)
∗ ︵(퐺,푇) This function is Ω(퐺, 푇)-invariant. Therefore Θ휋|푇 actually yields an element of ℤ[푋 (푇)] , cor- ∑ 휇 responding to 휇 mult(휋 ∶ 휇)푒 . We call those 휇 with nonzero multiplicities as the weights of 퐺. The weight-multiplicities mult ∗ determine and then , which in turn deter- { (휋 ∶ 휇)}휇∈푋 (푇)∩풞+ Θ휋|푇 Θ휋 mines 휋 up to isomorphism, by 4.43.
∗ Theorem 5.17 (Weyl character formula). Assume 퐺 is semisimple. Then the set of 휆 ∈ 푋 (푇) ∩ 풞+ is in bijection with the set of isomorphism classes of irreducible unitary representations 휋 of 퐺. It is determined by 휆 ↔ 휋 ⟺ Θ휋 = 휒휆 ∗ ︵(퐺,푇) where we identify Θ휋 with an element of ℤ[푋 (푇)] as above. In fact, 휆 is the highest weight of 휋 relative to Φ +, namely all the other weights are ≺ 휆, see (5.2); it occurs with multiplicity one in 휋. ∗ ︵(퐺,푇) ∑ Proof. Given 휋, we regard Θ휋 as an element of ℤ[푋 (푇)] , and expand it into 휆 푛휆휒휆 with 푛휆 ∈ ℤ using 5.16. Since those 휒휆 have been shown to be orthonormal, 4.41 yields
.Θ휋|Θ휋)퐿(퐺) = ﾜ 푛휆) = 1 휆
49 Hence there is exact one nonzero 푛휆, with 푛휆 ∈ {−1, +1}. We must have 푛휆 = +1 so that Θ휋 = 휒휆, otherwise by 5.11, mult(휋 ∶ 휆) = 푛휆 = −1 would be absurd. That result also implies the assertion about highest weight. All in all, we have an injection 휋 ↦ 휆 by matching Θ휋 = 휒휆. Since all the Θ휋 form an orthonormal 퐺−inv basis of the Hilbert space 퐿 (퐺) , whilst the 휒휆 are also orthonormal by 5.16, this map must be surjective as well.
Proposition 5.18 (Weyl denominator formula). Assume 퐺 is semisimple. Then
Δ = ﾜ (−1)ℓ(푤)푒푤휌. 푤∈︵(퐺,푇)
Proof. By considerations of the highest weight, the trivial representation of 퐺 must correspond to 휆 = 0. .The assertion then follows from 휒 = 1 Theorem 5.19 (Weyl dimension formula). Assume 퐺 is semisimple. Let 휋 be an irreducible unitary representation with highest weight 휆. Then
∏ + ﾉ휆 + 휌, 훼̌ｼ dim 푉 = 훼∈︾ . 휋 ∏ ﾉ ｼ 훼∈︾+ 휌, 훼̌ ∑ 푤휌 푤(휆+휌) Proof. The dimension equals Θ휋(1). Write ΔΘ휋 = 푤(−1) 푒 . Take appropriate derivatives......
[ NOT FINISHED YET ]
5.4 Extension to non-semisimple groups [ UNDER CONSTRUCTION ]
References
[1] N. Bourbaki. Éléments de mathématique. Fascicule XXIX. Livre VI: Intégration. Chapitre 7: Mesure de Haar. Chapitre 8: Convolution et représentations. Actualités Scientifiques et Industrielles, No. 1306. Hermann, Paris, 1963, 222 pp. (2 inserts) (cit. on pp. 7, 9). [2] N. Bourbaki. Éléments de mathématique. Topologie générale. Chapitres 1 à 4. Hermann, Paris, 1971, xv+357 pp. (not consecutively paged) (cit. on p. 8). [3] N. Bourbaki. Éléments de mathématique. XXV. Première partie. Livre VI: Intégration. Chapitre 6: Intégration vectorielle. Actualités Sci. Ind. No. 1281. Hermann, Paris, 1959, 106 pp. (1 insert) (cit. on p. 25). [4] Paul Garrett. “Vector-valued integrals”. http://www.math.umn.edu/∼garrett/m/fun. Apr. 21, 2014 (cit. on p. 25). [5] Brian Hall. Lie groups, Lie algebras, and representations. Second. Vol. 222. Graduate Texts in Mathematics. An elementary introduction. Springer, Cham, 2015, pp. xiv+449. isbn: 978-3-319- 13466-6; 978-3-319-13467-3. doi: 10.1007/978-3-319-13467-3. url: http://dx.doi. org/10.1007/978-3-319-13467-3 (cit. on p. 43). [6] Edwin Hewitt and Kenneth A. Ross. Abstract harmonic analysis. Vol. I. Second. Vol. 115. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Structure of topological groups, integration theory, group representations. Springer-Verlag, Berlin-New York, 1979, pp. ix+519. isbn: 3-540-09434-2 (cit. on pp. 8, 9).
50 [7] John M. Lee. Introduction to smooth manifolds. Second. Vol. 218. Graduate Texts in Mathematics. Springer, New York, 2013, pp. xvi+708. isbn: 978-1-4419-9981-8 (cit. on p. 12). [8] Walter Rudin. Functional analysis. Second. International Series in Pure and Applied Mathematics. McGraw-Hill, Inc., New York, 1991, pp. xviii+424. isbn: 0-07-054236-8 (cit. on pp. 25, 30, 31, 40). [9] Jean-Pierre Serre. Lie algebras and Lie groups. Vol. 1500. Lecture Notes in Mathematics. 1964 lectures given at Harvard University, Corrected fifth printing of the second (1992) edition. Springer- Verlag, Berlin, 2006, pp. viii+168. isbn: 978-3-540-55008-2; 3-540-55008-9 (cit. on p. 11).
51