Weyl Group Representations and Unitarity of Spherical Representations. Alessandra Pantano, University of California, Irvine Windsor, October 23, 2008 ν1 = ν2 β S S ν α β Sβν Sαν ν SO(3,2) 0 α SαSβSαSβν = SβSαSβν SβSαSβSαν S S SαSβSαν β αν 0 1 2 ν = 0 [type B2] 2 1 0: Introduction Spherical unitary dual of real split semisimple Lie groups spherical equivalence classes of unitary-dual = irreducible unitary = ? of G spherical repr.s of G aim of the talk Show how to compute this set using the Weyl group 2 0: Introduction Plan of the talk ² Preliminary notions: root system of a split real Lie group ² De¯ne the unitary dual ² Examples (¯nite and compact groups) ² Spherical unitary dual of non-compact groups ² Petite K-types ² Real and p-adic groups: a comparison of unitary duals ² The example of Sp(4) ² Conclusions 3 1: Preliminary Notions Lie groups Lie Groups A Lie group G is a group with a smooth manifold structure, such that the product and the inversion are smooth maps Examples: ² the symmetryc group Sn=fbijections on f1; 2; : : : ; ngg à finite ² the unit circle S1 = fz 2 C: kzk = 1g à compact ² SL(2; R)=fA 2 M(2; R): det A = 1g à non-compact 4 1: Preliminary Notions Root systems Root Systems Let V ' Rn and let h; i be an inner product on V . If v 2 V -f0g, let hv;wi σv : w 7! w ¡ 2 hv;vi v be the reflection through the plane perpendicular to v. A root system for V is a ¯nite subset R of V such that ² R spans V , and 0 2= R ² if ® 2 R, then §® are the only multiples of ® in R h®;¯i ² if ®; ¯ 2 R, then 2 h®;®i 2 Z ² if ®; ¯ 2 R, then σ®(¯) 2 R A1xA1 A2 B2 C2 G2 5 1: Preliminary Notions Root systems Simple roots Let V be an n-dim.l vector space and let R be a root system for V . The roots ®1; ®2 : : : ®n 2 R are called simple if ² they are a basis of V PP ² every root in R can written as i ai®i, with all ai ¸ 0 or all ai · 0. −γ 1−γ 2 −γ 2 −2γ 1−γ 2 − γ 1 γ1 γ2 γ1+γ 2 2γ 1+γ 2 type B2 Each choice of simple roots determines a set of positive roots . 6 1: Preliminary Notions Root systems The Weyl group Let R be a root system for V . The Weyl group of the root system is the ¯nite group of orthogonal transformations on V generated by the reflections through the simple roots. 3 P 2 [type A2] V =fv 2 R j j vj = 0g ' R . Simple roots: ® = e1 ¡ e2, ¯ = e2 ¡ e3. 3 st nd ² s®=se1¡e2 acts v 2 R by switching the 1 and 2 coordinate 3 nd rd ² s¯=se2¡e3 acts v 2 R by switching the 2 and 3 coordinate. W =hs®; s¯i= the symmetric group S3 (permutations of 1; 2; 3). 7 1: Preliminary Notions Root systems The root system of a real split Lie group Let G be a Lie group and let g be its Lie algebra. For all x 2 G, there is an inner automorphism Int(x): G ! G; g 7! xgx¡1. The di®erential of Int(x) is a linear transformation on g, denoted by Ad(x): We extend Ad(x) to the complexi¯ed Lie algebra gC, by linearity. The map Ad: G ! GL(gC); x 7! Ad(x) is a representation of G, called the adjoint representation. Assume that G is a real split group of rank n. Then G contains a n subgroup A ' (R¸0) such that the operators fAd(g)gg2A are simultaneously diagonalizable. Decompose gC in simultaneous eigenspaces for Ad(A). The nonzero eigenfunctions form a root system. The Weyl group is NK (A) : CK (A) 8 1: Preliminary Notions Real Split Lie groups Every \abstract" root system ¢ appears as the root system of a real split semisimple Lie group G. ¢ G K ½ G (maximal compact) An SL(n + 1; R) SO(n + 1) Bn SO(n + 1; n)0 SO(n + 1) £ SO(n) Cn Sp(2n; R) U(n) Dn SO(n; n)0 SO(n) £ SO(n) G2 G2 SU(2) £ SU(2)=f§Ig F4 F4 Sp(1) £ Sp(3)=f§Ig E6 E6 Sp(4)=f§Ig E7 E7 SU(8)=f§Ig E8 E8 Spin(16)=fI; wg 9 2: De¯nitions Unitary dual of a Lie group Unitary Representations Let G be a Lie group, H be a complex Hilbert space. A representation of G on H is a group homomorphism ¼ : G !B(H) = fbounded linear operators on Hg such that the map ¼ : G £ H ! H; (g; v) 7! ¼(g)v is continuous. ¼ is called unitary if ¼(g) is a unitary operator on H; 8 g 2 G Examples: ² H = C, ¼(g)v = v à trivial representation ² H = L2(G; ¹), ¼(g)f = f(¢ g) à right regular representation 10 2: De¯nitions Unitary dual of a Lie group Irreducibility and Equivalence of Representations A representation (¼; H) of G is irreducible if f0g and H are the only closed G-stable subspaces. [W is G-stable if ¼(G)W ½ W ] Two representations (¼1; H1), (¼2; H2) of G are equivalent if there is a bounded linear operator T : H1 !H2 (with bounded inverse) such that T ± ¼1(g) = ¼2(g) ± T; 8 g 2 G For ¯nite-dimensional representations, de¯ne the character by ¼ : G 7! C; g 7! trace(¼(g)). If ¼1, ¼2 are ¯nite-dimensional, then ¼1 w ¼2 , ¼1 = ¼2 . 11 2: De¯nitions Unitary dual of a Lie group The Unitary Dual Let G be a Lie group. Equivalence classes of irreducible Unitary Dual of G = = ? unitary representations of G 12 3: Examples Unitary dual of the symmetric group The symmetric group on n letters ¡ ¡ Sn = fpermutations on the set f1; 2; : : : ; ngg ¡ ¡ e.g. S4 is the group of symmetries of the cube (permutations of the diagonals). ¡¡ ¡¡ Every permutation can be written as a product of cycles: σ = (15)(24) 2 S5 is the bijection: 1 $ 5; 2 $ 4; 3 ª 3: Two permutations are conjugate , same cycle structure. e.g. the partitions (1324)(56) and (1523)(46) are conjugate in S6. Conjugacy classes in Sn Ã! partitions of n. 13 3: Examples Unitary dual of the symmetric group Irreducible representations of the symmetric group S3 Elements of S : (123); (132); (12); (13); (23); Id = (1)(2)(3): 3 | {z } | {z } conjugate conjugate 3 2,1 1,1,1 Conjugacy classes in S3: For every ¯nite group G, the number of equivalence classes of irred. representations equals the number of conjugacy classes. ) S3 has 3 irreducible inequivalent representations. 14 3: Examples Unitary dual of the symmetric group ² The Trivial representation: H1 = C and ¼1(σ)v = v 8 σ 2 S3; v 2 C: ² The Sign representation: H2 = C and 8 <+v if σ is even ¼2(σ)v = :¡v if σ is odd. [σ is even if it is a product of an even number of 2-cycles.] ² The Permutation representation: H = C3 and 3 σ ¢ (v1; v2; v3) = (vσ(1); vσ(2); vσ(3)) 8 σ 2 S3; 8 v 2 C . ½ ¾ 3 ? 3 P Not irreducible! U=h1; 1; 1i and W =U = v 2 C : vi = 0 i=1 are (closed and) G-stable. Let ¼3 be the restriction of σ to W . ¼1, ¼2 and ¼3 are all the irreducible repr.s of S3, up to equivalence. 15 3: Examples Unitary dual of the symmetric group What about unitarity? If G is a ¯nite group, any representation (¼; V ) of G is unitarizable. ² Start from any inner product (¢; ¢) on V ² Construct a new inner product by averaging over the group: 1 X hv; wi ´ (¼(g)v; ¼(g)w) 8 v; w 2 V: #G g2G ² h; i is invariant under the action of G, so ¼ unitary. P R True for compact groups. Replace by ¢ ¢ ¢dg: g2G g2G 16 3: Examples Unitary dual of ¯nite groups The unitary dual of ¯nite groups Let G be a ¯nite group. ² The number of equivalence classes of irreducible representations equals the number of conjugacy classes. ² Every irreducible representation is ¯nite-dimensional. ² Every irreducible representation is unitary. ² Two irreducible representations are equivalent if and only if they have the same character. ² The characters can be computed explicitly. This gives a complete classi¯cation. What we are still missing is an explicit model of the representations ::: 17 3: Examples Unitary dual of compact groups The unitary dual of compact groups Let G be a compact group. ² G has in¯nitely many irreducible inequivalent repr.s. ² Every irreducible representation is ¯nite-dimensional. ² Every irreducible representation is unitary. The irreducible repr.s of compact connected semisimple groups are known. They are classi¯ed by highest weight. We have formulas for the character and the dimension of each representations. What we are missing is an explicit construction. 18 3: Examples Unitary dual of non-compact groups What about the non-compact group SL(2; R)? SL(2; R) has many interesting unitary irreducible representations, but only one is ¯nite-dimensional: the trivial representation! non-spherical spherical principal series principal series Unitary dual of SL(2; R) ¡ + Bargmann; 1947 Piº Piº º > 0 º ¸ 0 holomorphic discrete series + + + + D1 D2 D3 Dn pps s s s s s s ppp pppp ppppppppp sC à trivial ppp ppp ppp sp p s Ks pps s s s s s s s 2 (0; 1) D+ D¡ ¡ ¡ ¡ ¡ 0 0 D1 D2 D3 Dn limits of complementary anti-holomorphic discrete series discrete series series 19 3: Examples Unitary dual of non-compact groups The unitary dual of non-compact groups G: real reductive group, e.g.