Isolated cohomological representations

Transcription

1 Isolated cohomological representations p. 1 Isolated cohomological representations and some cohomological applications Nicolas Bergeron E.N.S. Paris (C.N.R.S.)

2 Isolated cohomological representations p. 2 Introduction G = real connected semi-simple Lie group.

3 Isolated cohomological representations p. 2 Introduction G = real connected semi-simple Lie group. X = G/K : associated symmetric space.

4 Isolated cohomological representations p. 2 Introduction G = real connected semi-simple Lie group. X = G/K : associated symmetric space. Ĝ = unitary dual of G equipped with the Fell topology.

5 Isolated cohomological representations p. 3 Introduction Kazhdan (1967) : non trivial global spectral and topological properties of quotients Γ\X, with Γ G lattice follow from a local property on the topological space Ĝ, namely...

6 Isolated cohomological representations p. 3 Introduction Kazhdan (1967) : non trivial global spectral and topological properties of quotients Γ\X, with Γ G lattice follow from a local property on the topological space Ĝ, namely... Property (T) : 1 G is isolated in Ĝ.

7 Isolated cohomological representations p. 4 Introduction If G has (T) and Γ is any lattice in G, then... ε = ε(g) > 0 s.t. λ 1 (Γ\X) ε,

8 Isolated cohomological representations p. 4 Introduction If G has (T) and Γ is any lattice in G, then... ε = ε(g) > 0 s.t. λ 1 (Γ\X) ε, Γ is finitely generated,

9 Isolated cohomological representations p. 4 Introduction If G has (T) and Γ is any lattice in G, then... ε = ε(g) > 0 s.t. λ 1 (Γ\X) ε, Γ is finitely generated, H 1 (Γ\X) = {0}.

10 Isolated cohomological representations p. 5 Introduction Kazhdan then shows that Property (T) is not empty...

11 Isolated cohomological representations p. 5 Introduction Kazhdan then shows that Property (T) is not empty... (Kazhdan +...) If G has no simple factor locally isomorphic to SO(n, 1) (n 2) or SU(n, 1) (n 1), then G has (T).

12 Isolated cohomological representations p. 5 Introduction Kazhdan then shows that Property (T) is not empty... (Kazhdan +...) If G has no simple factor locally isomorphic to SO(n, 1) (n 2) or SU(n, 1) (n 1), then G has (T). In this talk I will be concern with the same question with 1 G replaced by...

13 Isolated cohomological representations p. 5 Introduction Kazhdan then shows that Property (T) is not empty... (Kazhdan +...) If G has no simple factor locally isomorphic to SO(n, 1) (n 2) or SU(n, 1) (n 1), then G has (T). In this talk I will be concern with the same question with 1 G replaced by... cohomological representations.

14 Isolated cohomological representations p. 6 Cohomological representations A cohomological representation is a representation π Ĝ without any local obstruction to occur in the cohomology H (Γ\X) of a certain compact quotient of X.

15 Isolated cohomological representations p. 6 Cohomological representations A cohomological representation is a representation π Ĝ without any local obstruction to occur in the cohomology H (Γ\X) of a certain compact quotient of X. A representation π L 2 (Γ\G) corresponds to a differential form on Γ\X.

16 Isolated cohomological representations p. 6 Cohomological representations A cohomological representation is a representation π Ĝ without any local obstruction to occur in the cohomology H (Γ\X) of a certain compact quotient of X. A K-finite vector of a representation π L 2 (Γ\G) corresponds to a differential form on Γ\X, with value in a certain local system.

17 Isolated cohomological representations p. 7 Cohomological representations A representation π L 2 (Γ\G) corresponds to a differential form on Γ\X. Hom K ( k p, H π ) {0} for some k,

18 Isolated cohomological representations p. 7 Cohomological representations A representation π L 2 (Γ\G) corresponds to a differential form on Γ\X. If π is cohomological it corresponds to a non zero harmonic form on Γ\X. Hom K ( k p, H π ) {0} for some k, π(ω) = 0, where Ω is the Casimir operator of g.

19 Isolated cohomological representations p. 7 Cohomological representations A representation π L 2 (Γ\G) corresponds to a differential form on Γ\X. If π is cohomological it corresponds to a non zero harmonic form on Γ\X. In mathematical words... π Ĝ is cohomological iff Hom K ( k p, H π ) {0} for some k, π(ω) = 0, where Ω is the Casimir operator of g.

20 Isolated cohomological representations p. 7 Cohomological representations A representation π L 2 (Γ\G) corresponds to a differential form on Γ\X. If π is cohomological it corresponds to a non zero harmonic form on Γ\X. In mathematical words... π Ĝ is cohomological iff Hom K ( k p, H π ) {0} for some k, π(ω) = 0, where Ω is the Casimir operator of g. The smallest k is called the degree of the cohomological representation.

21 Isolated cohomological representations p. 8 Cohomological representations Given G, Vogan and Zuckermann classify all the cohomological representations.

22 Isolated cohomological representations p. 8 Cohomological representations Given G, Vogan and Zuckermann classify all the cohomological representations. Let s give some examples... 1 G is the unique cohomological representation of degree 0.

23 Isolated cohomological representations p. 8 Cohomological representations Given G, Vogan and Zuckermann classify all the cohomological representations. Let s give some examples... 1 G is the unique cohomological representation of degree 0. Assume G = SU(n, 1). The cohomological representations of G are classified by Borel and Wallach: π i,j i,j Z 0, i + j n.

24 Isolated cohomological representations p. 9 Cohomological representations The representation π i,j is of degree i + j, it corresponds to the Hodge-Lefschetz decomposition of the cohomology of the (Kaehlerian!) manifolds Γ\X.

25 Isolated cohomological representations p. 9 Cohomological representations The representation π i,j is of degree i + j, it corresponds to the Hodge-Lefschetz decomposition of the cohomology of the (Kaehlerian!) manifolds Γ\X. From now on, I will concentrate on the explicit and far enough general case of the group G = SU(p,q).

26 Isolated cohomological representations p. 10 Cohomological representations Cohomological representations of G = SU(p, q) are naturally parametrized by...

27 Isolated cohomological representations p. 10 Cohomological representations Cohomological representations of G = SU(p, q) are naturally parametrized by... * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *

28 Isolated cohomological representations p. 10 Cohomological representations Cohomological representations of G = SU(p, q) are naturally parametrized by... * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * couples (λ, µ) of partitions s.t...

29 Isolated cohomological representations p. 11 Cohomological representations µ p q.

30 Isolated cohomological representations p. 11 Cohomological representations µ p q. λ µ. (Here µ denotes the complementary of µ in p q.)

31 Isolated cohomological representations p. 11 Cohomological representations µ p q. λ µ. (Here µ denotes the complementary of µ in p q.) µ/λ is a disjoint union of rectangles

32 Isolated cohomological representations p. 11 Cohomological representations µ p q. λ µ. (Here µ denotes the complementary of µ in p q.) µ/λ is a disjoint union of rectangles = (a 1 b 1 )... (a k b k ).

33 Isolated cohomological representations p. 11 Cohomological representations µ p q. λ µ. (Here µ denotes the complementary of µ in p q.) µ/λ is a disjoint union of rectangles = (a 1 b 1 )... (a k b k ). Such a couple of partitions is said to be compatible.

34 Isolated cohomological representations p. 11 Cohomological representations µ p q. λ µ. (Here µ denotes the complementary of µ in p q.) µ/λ is a disjoint union of rectangles = (a 1 b 1 )... (a k b k ). Such a couple of partitions is said to be compatible. The cohomological representation is of degree λ + µ. π(λ, µ)

35 Isolated cohomological representations p. 12 Cohomological representations Note then that if Γ G is a cocompact lattice, H k (Γ\X) = {0} for k = 1,...,min(p,q) 1.

36 Isolated cohomological representations p. 12 Cohomological representations Note then that if Γ G is a cocompact lattice, H k (Γ\X) = {0} for k = 1,...,min(p,q) 1. In a paper to be published in the Park City Volume, Vogan classifies isolated cohomological representations in Ĝ.

37 Isolated cohomological representations p. 12 Cohomological representations Note then that if Γ G is a cocompact lattice, H k (Γ\X) = {0} for k = 1,...,min(p,q) 1. In a paper to be published in the Park City Volume, Vogan classifies isolated cohomological representations in Ĝ. In the SU(p,q) case, Vogan s theorem implies the following...

38 Isolated cohomological representations p. 12 Cohomological representations Note then that if Γ G is a cocompact lattice, H k (Γ\X) = {0} for k = 1,...,min(p,q) 1. In a paper to be published in the Park City Volume, Vogan classifies isolated cohomological representations in Ĝ. In the SU(p,q) case, Vogan s theorem implies the following... Theorem 4. The cohomological representation π(λ,µ) of G = SU(p,q) is isolated in Ĝ iff (λ,µ ) compatible s.t. µ /λ and µ/λ only differ by one box.

39 Isolated cohomological representations p. 12 Cohomological representations Note then that if Γ G is a cocompact lattice, H k (Γ\X) = {0} for k = 1,...,min(p,q) 1. In a paper to be published in the Park City Volume, Vogan classifies isolated cohomological representations in Ĝ. In the SU(p,q) case, Vogan s theorem implies the following... Theorem 5. The cohomological representation π(λ,µ) of G = SU(p,q) is isolated in Ĝ iff (λ,µ ) compatible s.t. µ /λ and µ/λ only differ by one box. Equivalently...

40 Isolated cohomological representations p. 13 Cohomological representations min i (a i,b i ) 2, and

41 Isolated cohomological representations p. 13 Cohomological representations min i (a i,b i ) 2, and λ and µ don t share any angle or.

42 Isolated cohomological representations p. 13 Cohomological representations min i (a i,b i ) 2, and λ and µ don t share any angle or. Corollary 3. Let G = SU(p,q) with p,q 2. There exists ε = ε(p,q) > 0 s.t. for any lattice Γ G, for i = 0,...,p + q 3. λ i 1 (Γ\X) ε

43 Isolated cohomological representations p. 14 Cohomological representations I recently obtained an elementary proof of Vogan s theorem based on the following simple idea...

44 Isolated cohomological representations p. 14 Cohomological representations I recently obtained an elementary proof of Vogan s theorem based on the following simple idea... Assume π(λ,µ) non isolated : π i π(λ,µ).

45 Isolated cohomological representations p. 14 Cohomological representations I recently obtained an elementary proof of Vogan s theorem based on the following simple idea... Assume π(λ,µ) non isolated : π i π(λ,µ). Think of (π i ) as a sequence of differential forms ω i of type (λ,µ), then...

46 Isolated cohomological representations p. 14 Cohomological representations I recently obtained an elementary proof of Vogan s theorem based on the following simple idea... Assume π(λ,µ) non isolated : π i π(λ,µ). Think of (π i ) as a sequence of differential forms ω i of type (λ,µ), then... dω i or δω i tends to a non zero harmonic form.

47 Isolated cohomological representations p. 14 Cohomological representations I recently obtained an elementary proof of Vogan s theorem based on the following simple idea... Assume π(λ,µ) non isolated : π i π(λ,µ). Think of (π i ) as a sequence of differential forms ω i of type (λ,µ), then... dω i or δω i tends to a non zero harmonic form. Take a Hodge type (λ,µ ). Because of the degree, µ /λ and µ/λ only differ by one box.

48 Isolated cohomological representations p. 15 Cohomological applications They are all based on the following principle...

49 Isolated cohomological representations p. 15 Cohomological applications They are all based on the following principle... Theorem 7 (Burger-Sarnak). Let H G be Q-semi-simple Lie groups. 1. Ind G H(ĤAut) ĜAut, and 2. Res G H(ĜAut) ĤAut.

50 Isolated cohomological representations p. 15 Cohomological applications They are all based on the following principle... Theorem 8 (Burger-Sarnak). Let H G be Q-semi-simple Lie groups. 1. Ind G H(ĤAut) ĜAut, and 2. Res G H(ĜAut) ĤAut. Recall } Ĝ Aut = Γ {π Ĝ : π L2 (Γ\G) Ĝ. So that if π ĜAut is isolated then π L 2 (Γ\G) for some congruence subgroup Γ G.

51 Isolated cohomological representations p. 16 Cohomological applications Theta series are a useful tool to construct cohomology classes of arithmetic manifolds.

52 Isolated cohomological representations p. 16 Cohomological applications Theta series are a useful tool to construct cohomology classes of arithmetic manifolds. Burger and Sarnak principle + the isolation of certain cohomological representations enable to give a non-arithmetic treatment of this method for producing cohomology classes...

53 Isolated cohomological representations p. 16 Cohomological applications Theta series are a useful tool to construct cohomology classes of arithmetic manifolds. Burger and Sarnak principle + the isolation of certain cohomological representations enable to give a non-arithmetic treatment of this method for producing cohomology classes... Consider ω the Weil representation of Sp. It is an automorphic representation.

54 Isolated cohomological representations p. 17 Cohomological applications Consider the dual reductive pair (Howe) U(r,s) U(p,q) Sp, with r + s q.

55 Isolated cohomological representations p. 17 Cohomological applications Consider the dual reductive pair (Howe) U(r,s) U(p,q) Sp, with r + s q. Li has shown that, as a representation of real Lie groups, π π((p r ), (p s )) Res Sp U(r,s) U(p,q) ω, where π belongs to the discrete series of U(r,s) and π((p r ), (p s )) is the cohomological representation associated to ((p r ), (p s ))...

56 Isolated cohomological representations p. 18 Cohomological applications * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *

57 Isolated cohomological representations p. 19 Comological applications The Weil representation is automorphic, it thus follows from the Burger and Sarnak principle that π((p r ), (p s )) Û(p,q) Aut.

58 Isolated cohomological representations p. 19 Comological applications The Weil representation is automorphic, it thus follows from the Burger and Sarnak principle that π((p r ), (p s )) Û(p,q) Aut. But Vogan s theorem implies π((p r ), (p s )) is isolated in Û(p,q) (and thus in Û(p,q) Aut and q r s 2. We get...!) as soon as p 2

59 Isolated cohomological representations p. 20 Cohomological applications Theorem 9. Let p, q, r, s integers satisfying p 2 and q r s 2. There exists a cocompact lattice Γ G = SU(p,q) s.t. π((p r ), (p s )) L 2 (Γ\G).

60 Isolated cohomological representations p. 20 Cohomological applications Theorem 10. Let p, q, r, s integers satisfying p 2 and q r s 2. There exists a cocompact lattice Γ G = SU(p,q) s.t. π((p r ), (p s )) L 2 (Γ\G). Many cases were already proved by Li using the Weil representation in a more arithmetic way (theta series, L functions...). His method extends to the rank 1 case where cohomological representation are no more isolated in the unitary dual. What about the automorphic dual?

61 Isolated cohomological representations p. 21 Automorphic isolation As first noticed by Selberg in the SL(2)-case, it may happens that 1 G is not isolated in Ĝ but isolated in Ĝ Aut.

62 Isolated cohomological representations p. 21 Automorphic isolation As first noticed by Selberg in the SL(2)-case, it may happens that 1 G is not isolated in Ĝ but isolated in Ĝ Aut. This is the Property (τ) phenomenon of Lubotzky and Zimmer. And...

63 Isolated cohomological representations p. 21 Automorphic isolation As first noticed by Selberg in the SL(2)-case, it may happens that 1 G is not isolated in Ĝ but isolated in Ĝ Aut. This is the Property (τ) phenomenon of Lubotzky and Zimmer. And... Theorem 13 (Clozel). The trivial representation 1 G is always isolated in ĜAut.

64 Isolated cohomological representations p. 21 Automorphic isolation As first noticed by Selberg in the SL(2)-case, it may happens that 1 G is not isolated in Ĝ but isolated in Ĝ Aut. This is the Property (τ) phenomenon of Lubotzky and Zimmer. And... Theorem 14 (Clozel). The trivial representation 1 G is always isolated in ĜAut. What about the cohomological representations?

65 Isolated cohomological representations p. 22 Automorphic isolation Conjecture 1. Any cohomological representation of G = SU(p,q) is isolated in ĜAut.

66 Isolated cohomological representations p. 22 Automorphic isolation Conjecture 2. Any cohomological representation of G = SU(p,q) is isolated in ĜAut. For a general group G, the (conjectural) picture is a little bit more delicate!

67 Isolated cohomological representations p. 22 Automorphic isolation Conjecture 3. Any cohomological representation of G = SU(p,q) is isolated in ĜAut. For a general group G, the (conjectural) picture is a little bit more delicate! With Clozel, we proved a first case of Conjecture 1 which does not follow from Property τ. Namely...

68 Isolated cohomological representations p. 22 Automorphic isolation Conjecture 4. Any cohomological representation of G = SU(p,q) is isolated in ĜAut. For a general group G, the (conjectural) picture is a little bit more delicate! With Clozel, we proved a first case of Conjecture 1 which does not follow from Property τ. Namely... Theorem 18. Any cohomological representation of G = SU(2, 1) is isolated in ĜAut.

69 Isolated cohomological representations p. 23 Automorphic isolation Using Burger and Sarnak principle, I recently obtained a way to reduce parts of Conjecture 1 for a given group G to some smaller subgroup H of rank one..

70 Isolated cohomological representations p. 23 Automorphic isolation Using Burger and Sarnak principle, I recently obtained a way to reduce parts of Conjecture 1 for a given group G to some smaller subgroup H of rank one. As an example, one may recover a theorem proved with Clozel.

71 Isolated cohomological representations p. 23 Automorphic isolation Using Burger and Sarnak principle, I recently obtained a way to reduce parts of Conjecture 1 for a given group G to some smaller subgroup H of rank one. As an example, one may recover a theorem proved with Clozel. Theorem 21. Any cohomological representation of G = SU(n, 1) (n 1) of degree 1 is isolated in ĜAut.

72 Isolated cohomological representations p. 23 Automorphic isolation Using Burger and Sarnak principle, I recently obtained a way to reduce parts of Conjecture 1 for a given group G to some smaller subgroup H of rank one. As an example, one may recover a theorem proved with Clozel. Theorem 22. Any cohomological representation of G = SU(n, 1) (n 1) of degree 1 is isolated in ĜAut. Or get the following new result...

73 Isolated cohomological representations p. 24 Automorphic isolation Theorem 7. The cohomological representation π((1 n ), ) of G = SU(n, 2) (n 1) (of degree n) is isolated in ĜAut.