On the unitary dual of classical and exceptional real split groups. Alessandra Pantano, University of California, Irvine MIT, March, 2010

Transcription

1 On the unitary dual of classical and exceptional real split groups. Alessandra Pantano, University of California, Irvine MIT, March,

2 PLAN OF THE TALK unitarizability of Langlands quotients for real split groups petite K-types? unitarizability of Langlands quotients for Hecke algebras Part 1. Spherical unitary dual of affine graded Hecke algebras Part 2. Unitary dual of real split groups Part 3. Petite K-types Part 4. Embedding of unitary duals 2

3 PART 1 Spherical unitary dual of affine graded Hecke algebras 3

4 Affine graded Hecke algebras (X,, ˇX, ˇ ) : root datum, with simple roots Π and Weyl group W h := X Z C and A := Sym(h). The affine graded Hecke algebra for is H := C[W] A. generators: {t sα : α Π} {ω: ω h} relations: t 2 s α = 1; ωt sα = t sα s α (ω) + ω, ˇα α Π, ω h Example: type A1. C[W] = C1 + Ct sα (t 2 s α = 1), A = Sym(Cα) αt sα = t sα α + 2. Elements of H := C[W] A are of the form: p(α) + q(α)t sα. 4

5 Spherical Unitary H-modules The Hecke algebra H := C[W] A has a -operation. Let V be an H-module. V is spherical if it contains the trivial representation of W. V is Hermitian if it has a non-degenerate invariant Hermitian form: x v 1, v 2 + v 1, x v 2 = 0 x H, v 1, v 2 V. V is unitary if, is positive definite. 5

6 Spherical Langlands quotients of H ν ȟ h : real and dominant C ν : the corresponding character of A = Sym(h) Spherical Principal Series X(ν) := H A C ν X(ν) is spherical, dimhom W (triv, X(ν)) = 1. (H acts on the left) Langlands Quotient L(ν) := unique irreducible quotient of X(ν) Every irreducible spherical H-module (with real central character) is isomorphic to L(ν) for some ν dominant. 6

7 An Hermitian form on L(ν) w 0 : longest Weyl group element L(ν) is Hermitian if and only if w 0 ν = ν. In this case, Barbasch and Moy define a (normalized) intertwining operator such that A(w 0, ν) has no poles A(w 0, ν): X(ν) X(w 0 ν) The image of A(w 0, ν) is L(ν), hence L(ν) X(ν) Ker(A(w 0,ν)) A(w 0, ν) gives a non-deg. invariant Hermitian form on L(ν). ψ Ŵ, we find an operator A ψ (w 0, ν) on V ψ. (A triv(w 0, ν) = 1.) 7

8 The operator A ψ (w 0, ν) on the W-type ψ A ψ (w 0, ν) decomposes a product of operators corresponding to simple reflections. If w 0 = s α1 s α2 s αn is a reduced decomposition of w 0 in W, then A ψ (w 0, ν) = n i=1 with ν i = s αn+1 i s αn 1 s αn ν. Each factor acts on Vψ, and has the form A ψ (s αi, ν i 1 ) A ψ (s α, γ) := Id+ α,γ ψ(s α) 1+ α,γ. (+1)-eigensp. of ψ(sα) 1 (+1)-eigensp. of ψ(sα) ( 1)-eigensp. of ψ(sα) 1 γ, α 1 + γ, α ( 1)-eigensp. of ψ(sα) 8

9 Untarity of a spherical Langlands Quotient L(ν) is unitary A ψ (w 0, ν) is positive semi-definite, ψ Ŵ Barbasch and Ciubotaru found a small subset of Ŵ which is sufficient to detect unitarity. These are the relevant W-types. A n 1 Bn, Cn Dn {(n m, m): 0 m [n/2]} {(n m, m) (0): 0 m [n/2]} {(n m) (m): 0 m n} {(n m, m) (0): 0 m [n/2]} {(n m) (m): 0 m [n/2]} G 2 {1 1, 2 1, 2 2 } F 4 {1 1, 4 2, 2 3, 8 1, 9 1 } E 6 {1p, 6p, 20p, 30p, 15q } E 7 {1a, 7 a, 27 a, 56 a, 21 b, 35 b, 105 b } E 8 {1x, 8z, 35x, 50x, 84x, 112z, 400z, 300x, 210x} 9

10 Spherical Unitary dual of a Hecke algebra of type A1 In type A1, there are two W-types (triv and sgn). L(ν) is unitary if and only if A ψ (w 0, ν) is positive semidefinite for ψ = triv and sgn. Because w 0 has length one, the intertwining operator has a single factor: A ψ (w 0, ν) = A ψ (s α, ν) := Id+ α,ν ψ(s α) 1+ α,ν. ψ ψ(s α ) A ψ (w 0, ν) triv 1 1 sgn 1 1 ν 1+ν L(ν) is unitary for 0 ν 1. H(A1) χ 10

11 Spherical Unitary dual of a Hecke algebra of type B2 Here w 0 = s e1 e 2 s e2 s e1 e 2 s e2. For ν = (a, b), A ψ (w 0, ν) factors as: A ψ (s e1 e 2, ( b, a))a ψ (s e2, ( b, a))a ψ (s e1 e 2, (a, b))a ψ (s e2, (a, b)) The factors are computed using the formula: A ψ (s α, γ) = Id+ α,γ ψ(s α) 1+ α,γ. ψ Ŵ rel ψ(s e1 e 2 ) ψ(s e2 ) the operator A ψ (w 0, ν) (a b) 1+(a b) 1 1 (a+b) 1+(a+b) a 1+a 1 1 b 1+b ( ) ( ) 1+a tr. 2 2 a 3 b b 2 +ab+ab det (1+a)(1+b)[1+(a b)][1+(a+b)] 1 b 1 (a b) 1 (a+b) 1+b 1+(a b) 1+(a+b) 1 a 1+a 11

12 Spherical unitary dual of a Hecke algebra of type B2 a = b H(B2) a = 0 12

13 The spherical unitary dual of a Hecke algebra H: affine graded Hecke algebra with root system g: complex Lie algebra associated to The spherical unitary dual of H is a union of sets (complementary series) parameterized by nilpotent orbits in g. Fix an orbit O and its Lie triple {e, h, f}. The centralizer in g of the Lie triple is a reductive Lie subalgebra, denoted Z g (O). The complementary series attached to O, denoted by CS(O), is the set of all parameters ν such that L(ν) is unitary O is the maximal nilpotent orbit with the property ν = 1 2 h + χ, with χ Z g(o). 13

14 Complementary series attached to nilpotent orbits Fix a Hecke algebra H and an orbit O. Barbasch and Ciubotaru reduce the problem of finding the complementary series CS(O) of H to the problem of detecting the zero-complementary series CS(0) of the centralizer of O. With a few exceptions (one orbit for F4 and E 7, and 6 orbits for E8), they prove that ν = 1 2 h + χ CS H(O) χ CS H(Z(O)) (0). The zero-complementary series CS(0) = {ν : X(ν) is unitary and irreducible} is known explicitly (for all root systems). It is a union of alcoves in the complement of the reduciblity hyperplanes: α, ν = 1, α +. 14

15 Spherical unitary dual of a Hecke algebra of type B2 Complementary series are attached to nilpotent orbits in so(5, C). ν = 1 2 h + χ CS H(O) χ CS H(Z(O)) (0). The zero-complementary series for the centralizers are known. O 1 2 h χ Z(O) 1 2 h + χ CS H(O) 5 (1, 2) (0, 0) (1, 2) 311 (0, 1) (a, 0) (a, 1) a = 0 ( ( , ) (a, a) + a, a) 0 a < (0, 0) (a, b) (a, b) 0 a b < 1 a < 1 15

16 Spherical unitary dual of a Hecke algebra of type B2 a = b H(B2) (2,1) CS(5) a = 0 (1,0) CS(3, 1, 1) ( 1 2, 1 2 ) CS(2, 2, 1) (1, 0) 0 ( 1 2, 1 2 ) (1, 0) CS(1, 1, 1, 1, 1) 16

17 PART 2 Unitary dual of (double cover of) real split groups 17

18 Real split groups G K G (maximal compact) A n SL(n + 1, R) SO(n + 1) B n SO(n + 1, n) 0 SO(n + 1) SO(n) C n Sp(2n, R) U(n) D n SO(n, n) 0 SO(n) SO(n) G 2 G 2 SU(2) SU(2)/{±I} F 4 F 4 Sp(1) Sp(3)/{±I} E 6 E 6 Sp(4)/{±I} E 7 E 7 SU(8)/{±I} E 8 E 8 Spin(16)/{I, w} 18

19 Fine K-types For each root α we choose a Lie algebra homomorphism φ α : sl(2, R) g 0 whose image ( is a subalgebra ) of g 0 isomorphic to sl(2, R). Then 0 1 Z α := φ α is a generator for an so(2)-subalgebra of k A K-type µ K is level k if γ k for every root α and every eigenvalue γ of dµ(z α ). K-types of level are fine. are called pseudospherical ; the ones of level Every δ M is contained in a fine K-type µ δ (maybe not unique). Every M-type in the W-orbit of δ appears in µ δ with multiplicity 1. 19

20 Parameters Minimal Principal Series P = M AN minimal parabolic subgroup (δ, V δ ) M ν Â a real and weakly dominant Principal Series X(δ, ν) = Ind G P=MAN(δ ν triv) µ K, mult(µ, X(δ, ν) K ) = mult(δ, µ M ). G P = MAN M Z 2 A R >0 δ ν ( ) ( ) a b a 0 triv SL 2 (R) ±I ν 0 0 a 1 0 a 1 sgn 8 < triv bk Z, n M = : sgn 8 if n 2Z < L if n 2Z + 1, X(δ, ν) n SO(2)= 2n if δ = triv : L 2n + 1 if δ = sgn n 20

21 The Langlands quotient L(δ, ν) For simplicity, assume, that δ is contained in a unique fine K-type µ δ. L(δ, ν)=unique irreducible quotient of X(δ, ν) containing µ δ If w 0 W is the longest element, there is an intertwining operator (normalized on µ δ ) such that A(w 0, δ, ν) has no poles A(w 0, δ, ν): X(δ, ν) X(w 0 δ, w 0 ν) The (closure of the) image of A(w 0, δ, ν)) is L(δ, ν). Hence L(δ, ν) = X(δ,ν) Ker(A(w 0,δ,ν)). 21

22 Unitarity of the Langlands quotient L(δ, ν) L(δ, ν) is Hermitian if and only if w 0 δ δ and w 0 ν = ν. In this case, the (normalized) operator A(w 0, δ, ν) := µ δ (w 0 )A(w 0, δ, ν): X(δ, ν) X(δ, ν) induces a non-degenerate invariant Hermitian form on L(δ, ν). For every µ K, there is an operator A µ (w 0, δ, ν) on Hom M (µ, δ). (A µδ (w 0, δ, ν) = 1.) L(δ, ν) is unitary if and only if the operator A µ (w 0, δ, ν) is positive semidefinite, µ K We should compute the signature of infinitely many operators. 22

23 The operator A µ (w 0, δ, ν) on the K-type µ Fix a K-type µ. The operator A µ (w 0, δ, ν) can be written as a product of operators corresponding to simple reflections. If w 0 = s α1 s α2 s α1 is a reduced decomposition of w 0 in W, then A µ (w 0, δ, ν) = n i=1 A µ (s αi, δ i 1, ν i 1 ) with ν i = s αn+1 i s αn 1 s αn ν, and δ i = s αn+1 i s αn 1 s αn δ. Note that the full operator A µ (w 0, δ, ν) acts on Hom M (µ, δ), but the factors might not. [The reflections move δ around in its W-orbit.] 23

24 The α-factor A µ (s α, ρ, γ) An α-factor of the full intertwining operator is a map A µ (s α, ρ, γ): Hom M (µ, ρ) Hom M (µ, s α ρ). Here ρ and s α ρ are two M-types in the W-orbit of δ. They can both be realized inside µ δ (our fixed fine K-type containing δ). Let Z α be a generator for the so(2)-subalgebra attached to α. Then Z 2 α acts on Hom M (µ, ρ); decompose Hom M (µ, ρ) in Z 2 α-eigenspaces: Hom M (µ, ρ) = l E α,ρ ( l 2 ) [Here l Z+ 1 2 if δ is genuine and α is metaplectic; l Z otherwise.] The operator A µ (s α, ρ, γ) maps E α,ρ ( l 2 ) E α,s αρ ( l 2 ), l, acting by: T c l (α, γ)µ δ (σ α ) T µ(σ 1 α ). 24

25 The scalars c l (α, γ) Let l 1 2 Z, l 0. Set ξ = γ, ˇα. Then c l(α, γ) is equal to 1 if l = 0, 1 or 1 2 (for the normalization) ( 1) l/2(1 ξ)(3 ξ) (2m 1 ξ) (1 + ξ)(3 + ξ) (l 1 + ξ) ( 1) (l 1)/2(2 ξ)(4 ξ) (l 1 ξ) (2 + ξ)(4 + ξ) (l 1 + ξ) if l 2N if l 2N + 1 ( 1) (l+1/2)/2(1 2 ξ)(5 2 ξ) (l 1 ξ) ( ξ)(3 2 + ξ) (l 1 + ξ) if l N ( 1) (l 1)/2(3 2 ξ)(7 2 ξ) (l 1 ξ) ( ξ)(7 2 + ξ) (l 1 + ξ) if l N 25

26 PART 3 Petite K-types 26

27 The idea of petite K-types The unitarity of L(δ, ν) is hard to detect: It depends on the signature of infinitely many operators (one for each K-type) The computations are hard if the K-type is large. To make the problem easier: 1. Select a small set of petite K-types on which the computations are easy. 2. Compute operators only for petite K-types, hoping that the calculation will rule out large non-unitarity regions. This approach will provide necessary conditions for unitarity: L(δ, ν) unitary A µ (w 0, δ, ν) pos. semidefinite, µ petite 27

28 Main feature of petite K-types The operators {A µ (δ, ν): µ petite} resemble operators for affine graded Hecke algebras. unitarizability of Langlands quotients for real split groups relate unitarizability of Langlands quotients for Hecke algebras 28

29 Petite K-types For each root α and each M-type ρ, there is an action of Z 2 α on the space Hom M (µ, ρ), defined by T T dµ(z α ) 2. If µ δ is a fine K-type containing δ, we look at the action of Zα 2 on the space Hom M (µ, µ δ ) = Hom M (µ, δ i ). δ i W-orbit of δ Definition. A K-type is called petite for δ if the eigenvalues of Z 2 α on Hom M (µ, µ δ ) are of the form k 2 with k 2, α. This is a restriction on the eigenvalues of Z 2 α on the isotypic component in µ of every M-type in the W-orbit of δ. Corollary. K-types of level 2 are petite for every δ. 29

30 Main theorem G: real split group associated to a root system δ: an M-type for G, with fine K-type µ δ W δ : stabilizer of δ in W; δ : good roots for δ δ if δ is genuine H: affine graded Hecke algebra for ˇ δ otherwise. Theorem. For each K-type µ, there is a representation ψ µ of the stabilizer W δ on the space Hom M (µ, δ). When µ is petite, the correspondence µ K = ψ µ Ŵ δ gives rise to a matching of intertwining operators: A G µ (w 0, δ, ν) = A H ψ µ (w 0, ν). 30

31 The spherical case: δ = δ 0 is trivial If δ = δ 0 is trivial, every reflection stabilizes δ. So W δ 0 = W δ0 = H = Hecke algebra with root system ˇ δ 0 non-genuine take ˇ δ0 Theorem. If µ is petite for δ 0 and ψ µ is the representation of W on Hom M (µ, δ 0 ), then A G µ (w 0, δ 0, ν) A H ψ µ (w 0, ν) 31

32 The spherical case: δ = δ 0 is trivial Decompose A µ (w 0, δ, ν) in α-factors A µ (s α, ρ, λ): Hom M (µ, ρ) Hom M (µ, s α ρ) (with ρ an M-type in the W-orbit of δ). If δ = δ 0, then ρ = s α ρ = δ 0, so every factor A µ (s α, ρ, λ) End(Hom M (µ, δ 0 )) Hom M (µ, δ 0 ) carries a repr. ψ µ of W, and an action of dµ(z 2 α). Decompose Hom M (µ, δ 0 ) in eigenspaces of dµ(z 2 α): Hom M (µ, δ 0 ) = m N E α ( 4m 2 ). A µ (s α, ρ, λ) acts on acts on E α ( 4m 2 ) by T c(α, γ, 2m) µ δ (σ α )T(µ(σ α ) 1 ). }{{}}{{} a scalar ψ µ (s α )T 32

33 Petite Spherical K-types If the K-type µ is petite for δ 0, then (V µ ) M = E α (0) E α ( 4). E α (0) 1 ψ µ(sα) E α (0) E α ( 4) E α ( 16) 1 γ, ˇα 1+ γ, ˇα ψ µ(sα) E α ( 4) E α ( 16) E... α ( 36)... E α ( 36) The spaces E α (0) and E α ( 4) coincide with the (+1)- and ( 1)-eigenspace of ψ µ (s α ), respectively. (+1)-eig. of ψµ(sα) ( 1)-eig. of ψµ(sα) 1 (+1)-eig. of ψµ(sα) Hence A µ (s α, δ 0, λ) = I+ γ, ˇα ψ µ(s α ) 1+ γ, ˇα. = 1 γ, ˇα 1+ γ, ˇα ( 1)-eig. of ψµ(sα) this is an operator for a Hecke algebra with root system ˇ 33

34 The pseudospherical case If δ is pseudospherical, every reflection stabilizes δ. So W δ = W δ = H = Hecke algebra with root system δ genuine take δ If µ is petite for δ and ψ µ is the representation of W on Hom M (µ, δ), then A G µ (w 0, δ, ν) A H ψ µ (w 0, ν) 34

35 Similar to the spherical case: The stabilizer of δ is W. The pseudospherical case Every α-factor is an endomorphism of Hom M (µ, δ). Hom M (µ, δ) carries a repr. ψ µ of W and an action of dµ(z α ) 2. Decompose Hom M (µ, δ) in Zα-eigenspaces: 2 Hom M (µ, δ) = E α ( l 2 ). (Here l 2N l N/2 if α is metaplectic, and l 2N otherwise.) A µ (s α, ρ, λ) acts on acts on E α ( l 2 ) by T c(α, γ, l) µ δ (σ α )T(µ(σ α ) 1 ). }{{}}{{} a scalar ψ µ (s α )T The scalar depends on whether α is metaplectic or not. 35

36 Petite K-types for δ pseudospherical (α metaplectic) In this case, (V µ ) M = E α ( 1/4) E α ( 9/4). E α ( 1/4) 1 ψ µ(sα) E α (0) E α ( 9/4) E α ( 25/4) 1/2 γ, ˇα 1/2+ γ, ˇα ψ µ(sα) E α ( 4) E α ( 16) E α ( 49/4) E α ( 49/4) The spaces E α ( 1/4) and E α ( 9/4) coincide with the (+1)- and ( 1)-eigenspace of ψ µ (s α ), respectively. (+1)-eig. of ψµ(sα) 1 (+1)-eig. of ψµ(sα) ( 1)-eig. of ψµ(sα) 1/2 γ, ˇα 1/2+ γ, ˇα ( 1)-eig. of ψµ(sα) = 1 γ, α 1+ γ, α Hence A µ (s α, δ 0, λ) = I+ γ, α ψ µ(s α ) 1+ γ, α. = operator for H( ) 36

37 Petite K-types for δ pseudospherical (α not metaplectic) In this case, (V µ ) M = E α (0) E α ( 4). E α (0) 1 ψ µ(sα) E α (0) E α ( 4) 1 γ, ˇα 1+ γ, ˇα ψ µ(sα) E α ( 4) E α ( 16) E α ( 36) E α ( 16) E α ( 36) The spaces E α (0) and E α ( 4) coincide with the (+1)- and ( 1)-eigenspace of ψ µ (s α ), respectively. (+1)-eig. of ψµ(sα) ( 1)-eig. of ψµ(sα) 1 (+1)-eig. of ψµ(sα) 1 γ, ˇα 1+ γ, ˇα ( 1)-eig. of ψµ(sα) Hence A µ (s α, δ 0, λ) = I+ γ, α ψ µ(s α ) 1+ γ, α. = operator for H( ) = 1 γ, α 1+ γ, α 37

38 The non-spherical non-pseudospherical case In this case, not every reflection stabilizes δ. W δ W and δ H = Hecke algebra with root system ˇ if δ genuine otherwise Theorem. If µ is petite for δ and ψ µ is the representation of W δ on Hom M (µ, δ), then A G µ (w 0, δ, ν) A H ψ µ (w 0, ν) These operators decompose as a product of α-factors, mimicking a minimal decomposition of w 0 in W and in W δ respectively. Hence A G µ (w 0, δ, ν) has more factors. 38

39 Choose a minimal decompositions of w 0 in W δ : w 0 = s β1 s β2 s βr. Every root s β which is simple in both W δ and W gives rise to a single factor of A G µ (w 0, δ, ν) of A G µ (w 0, δ, ν) = I+ γ, β ψ µ (s β ) 1+ γ, β I+ γ, ˇβ ψ µ (s β ) 1+ γ, ˇβ if δ is genuine otherwise. Every root s β which is simple in W δ but not simple in W gives rise to I+ γ, β ψ a product µ (s β ) 1+ γ, β if δ is genuine of factors = I+ γ, ˇβ ψ µ (s β ) 1+ γ, ˇβ otherwise. This product reflects a minimal factorization of s β in W. 39

40 PART 4 What petite K-type do for us: Embedding of unitary duals 40

41 A matching of petite K-types with relevant W δ -types If µ is petite, the correspondence µ K ψ µ = Hom M (µ, δ) Ŵ δ gives rise to a matching of operators: A G µ (w 0, δ, ν) A H ψ µ (w 0, ν). Theorem. If set {ψ µ : µ K petite} includes all the relevant W δ -types, then we obtain an embedding of unitary duals: δ-complementary series of G spherical unitary dual of H {ν : L G (δ, ν) unitary} {ν : L H (ν) unitary} 41

42 An embedding of unitary duals Suppose that for each relevant W δ -type ψ, there is a petite K-type µ such that ψ µ = ψ. ( ). L G (δ, ν) is unitary =============> L H ( ν) is unitary A G µ (w 0, δ, ν) A G µ (w 0, δ, ν) A H ψ (w 0, ν) A H ψ (w 0, ν) pos. semidef. pos. semidef. ( ) pos. semidef. ( ) pos. semidef. µ K µ petite ψ relevant ψ Ŵ ( ) Relevant W-types detect unitarity for H. [Barbasch-Ciubotaru] 42

43 Spherical unitary duals for split groups Theorem [Barbasch] If δ = δ 0, the matching holds for all real split groups G. Let H be the affine graded Hecke algebra with root system dual to the one of G. Then spherical unitary dual of G This gives us: spherical unitary dual of H this is an equality for classical groups the full spherical unitary dual of real split classical groups non-unitarity certificates for spherical representations of real split exceptional groups. 43

44 An example: the spherical unitary dual of Sp(4) Let H be an affine graded algebra of type B2. The spherical unitary dual of Sp(4) embeds into the one of H(B2): a = b H(B2) a = 0 Actually, they are equal. 44

45 Pseudospherical unitary duals for split groups Theorem [Adams, Barbasch, Paul, Trapa, Vogan] Let δ be pseudospherical. The matching holds for the double cover of every real split group of classical type. Let H be the affine graded Hecke algebra with the same root system as G. Then pseudosph. unitary dual of G spherical unitary dual of H this is an equality for Mp(2n) This gives us: the full pseudospherical unitary dual of Mp(2n) non-unitarity certificates for the other classical groups. 45

46 An example: the pseudospherical unitary dual of Mp(4) Let H be an affine graded algebra Hecke of type C2. The pseudospherical unitary dual of Mp(4) coincides with the spherical unitary dual of H (which, in turn, coincides with the spherical unitary dual of of SO(3, 2)). ν 1 = ν 2 SO(3,2) ν 2 = 0 46

47 Genuine complementary series of Mp(2n) Genuine M-types of Mp(2n) are parameterized by pairs of non-negative integers (p, q) with p + q = n. If δ = δ p,q, then δ = C p C q. Theorem [Paul, P., Salamanca] The matching holds for every genuine M-type δ p,q of Mp(2n). Let H be the affine graded Hecke algebra for the root system C p C q. Then δ p,q -compl. series of Mp(2n) spherical unitary dual of H an equality for n 4 (all p, q), and for some special families of parameters (all n) [Barbasch]: spherical unitary dual of H = spherical unitary dual of SO(p + 1, p) SO(q + 1, q). 47

48 CS(Mp(6), δ 2,1 ) = CS(SO(3, 2) 0, δ 0 ) CS(SO(2, 1) 0, δ 0 ) ν 2 1/2 3/2 ν 1 CS(SO(3,2) 0, δ 0 ) CS(SO(2,1) 0, δ 0 ) CS(Mp(6), δ 2,1 ) equals the product 48

49 Complementary series of SO(n + 1, n) 0 M-types of SO(n + 1, n) 0 are parameterized by pairs of non-negative integers (p, q) with p + q = n. If δ = δ p,q, then δ = B p B q. Theorem [Paul, P., Salamanca] The matching holds for every M-type δ p,q of SO(n + 1, n) 0. Let H be affine graded Hecke algebra for the root system C p C q. Then CS(SO(n + 1, n) 0, δ p,q ) spherical unitary dual of H an equality for n 4 (all p, q), and for some special families of parameters (all n) [Barbasch]: spherical unitary dual of H = spherical unitary dual of SO(p + 1, p) SO(q + 1, q). 49

50 CS(SO(4, 3) 0, δ 2,1 ) = CS(SO(3, 2) 0, δ 0 ) CS(SO(2, 1) 0, δ 0 ) 50

51 The double cover of split E6 joint with D.Barbasch repr. of M note dim δ matching δ 1 trivial 1 E6 δ 8 pseudosph. 8 E6 NO ( ) δ 27 non-genuine 27 1 D 5 δ 36 non-genuine 36 1 A 5 A 1 ( ) One relevant W(E6)-type is missing. 51

52 The double cover of split F4 joint with D.Barbasch repr. of M note dim δ matching δ 1 trivial 1 F4 δ 2 pseudosph. 2 F4 δ 3 non-genuine 3 1 C4 No ( ) δ 6 genuine 3 2 B4 No ( ) δ 12 non-genuine 12 1 B3 A 1 ( ) One relevant W(C4)-type is missing. ( ) Two relevant W(B4)-types are missing. 52