(June 8, 2018) The Classical Groups and Domains Paul Garrett [email protected] http:=/www.math.umn.edu/egarrett/ The complex unit disk D = fz 2 C : jzj < 1g has four families of generalizations to bounded open subsets in Cn with groups acting transitively upon them. Such domains, defined more precisely below, are bounded symmetric domains. First, we recall some standard facts about the unit disk, the upper half-plane, the ambient complex projective line, and corresponding groups acting by linear fractional (M¨obius)transformations. Happily, many of the higher- dimensional bounded symmetric domains behave in a manner that is a simple extension of this simplest case. 1. The disk, upper half-plane, SL2(R), and U(1; 1) 2. Classical groups over C and over R 3. The four families of self-adjoint cones 4. The four families of classical domains 5. Harish-Chandra's and Borel's realization of domains 1. The disk, upper half-plane, SL2(R), and U(1; 1) The group a b GL ( ) = f : a; b; c; d 2 ; ad − bc 6= 0g 2 C c d C acts on the extended complex plane C [ 1 by linear fractional transformations a b az + b (z) = c d cz + d with the traditional natural convention about arithmetic with 1. But we can be more precise, in a form helpful for higher-dimensional cases: introduce homogeneous coordinates for the complex projective line P1, by defining P1 to be a set of cosets u 1 = f : not both u; v are 0g= × = 2 − f0g = × P v C C C where C× acts by scalar multiplication. That is, P1 is the set of equivalence classes of non-zero 2-by-1 vectors under the equivalence relation u u0 u u0 ∼ if and only if = c · for some non-zero c 2 × v v0 v v0 C In other words, P1 is the quotient of C2 − f0g by the action of C×. The complex line C1 is imbedded in this model of P1 by z z ! 1 The point at infinity 1 is now easily and precisely described as 1 1 = 0 1 Paul Garrett: The Classical Groups and Domains (June 8, 2018) 2 Certainly the matrix multiplication action of GL2(C) on C stabilizes non-zero vectors, and commutes 1 with scalar multiplication, so GL2(C) has a natural action on P . This action, linear in the homogeneous coordinates, becomes the usual linear fractional action in the usual complex coordinates: a b a b z az + b (az + b)=(cz + d) (z) = = = c d c d 1 cz + d 1 when cz + d 6= 0, since we can multiply through by (cz + d)−1 to normalize the lower entry back to 1, unless it is 0, in which case we've mapped to 1. 1 2 1 2 The action of GL2(C) is transitive on P , since it is transitive on C − f0g, and P is the image of C − f0g. The condition defining the open unit disk D = fz 2 C : jzj < 1g ⊂ C can be rewritten as ∗ z −1 0 z D = fz 2 : > 0g C 1 0 1 1 where T ∗ is conjugate transpose of a matrix T . The standard unitary group U(1; 1) of signature (1; 1) is −1 0 −1 0 U(1; 1) = fg 2 GL ( ): g∗ g = g 2 C 0 1 0 1 We can easily prove that U(1; 1) stabilizes the disk D, as follows. For z 2 D and a b g = 2 U(1; 1) c d compute ∗ ∗ g(z) −1 0 g(z) 1 az + b −1 0 az + b 1 1 − jg(z)j2 = = · · 1 0 1 1 (cz + d)∗ cz + d 0 1 cz + d (cz + d) ∗ ∗ 1 z −1 0 z 1 z −1 0 z = · g∗ g = · jcz + dj2 1 0 1 1 jcz + dj2 1 0 1 1 since g is in the unitary group. Then 1 1 − jg(z)j2 = (1 − jzj2) > 0 jcz + dj2 In fact, to be sure that cz + d 6= 0 in this situation, we should have really computed (by the same method) that jcz + dj2 · (1 − jg(z)j2) = 1 − jzj2 Since 1 − jzj2 > 0 and jcz + dj ≥ 0, necessarily jcz + dj > 0 and 1 − jg(z)j2 > 0. The complex upper half-plane is H = fz 2 C : Imz > 0g In the spirit of giving matrix expressions for defining inequalities, ∗ 1 z 0 −1 z Imz = · 2i 1 1 0 1 2 Paul Garrett: The Classical Groups and Domains (June 8, 2018) This suggests taking 0 −1 0 −1 G = fg 2 GL ( ): g∗ g = g 2 C 1 0 1 0 Just as in the discussion of the disk, for z 2 H and a b g = 2 G c d compute ∗ ∗ g(z) 0 −1 g(z) 1 az + b 0 −1 az + b 1 2i · Img(z) = = · · 1 1 0 1 (cz + d)∗ cz + d 1 0 cz + d (cz + d) ∗ ∗ 1 z 0 −1 z 1 z 0 −1 z = · g∗ g = · jcz + dj2 1 1 0 1 jcz + dj2 1 1 0 1 since g is in G. Then 1 Img(z) = Imz > 0 jcz + dj2 To be sure that cz + d 6= 0, compute in the same way that jcz + dj2 · Img(z) = Imz Since Imz > 0 and jcz + dj ≥ 0, necessarily jcz + dj > 0 and Img(z) > 0. Since the center t 0 Z = f : t 2 g 0 t C 1 of GL2(C) acts trivially on C ⊂ P , one might expect some simplification of matters by restricting our attention to the action of SL2(C) = fg 2 GL2(C) : det g = 1g since under the quotient map GL2(C) ! GL2(C)=Z the subgroup SL2(C) maps surjectively to the whole quotient GL2(C)=Z. The special but well-known formula −1 a b d −b = ad−bc ad−bc c d −c a ad−bc ad−bc for the inverse of a two-by-two matrix implies that 0 −1 0 −1 SL ( ) = fg 2 GL ( ): g> g = g 2 C 2 C 1 0 1 0 Thus, the subgroup in SL2(C) stabilizing the upper half-plane is 0 −1 0 −1 0 −1 0 −1 G0 = fg 2 GL ( ): g> g = and g∗ g = g 2 C 1 0 1 0 1 0 1 0 In particular, for such g we have g> = g∗, so the entries of g are real. Thus, we have shown that fg 2 SL2(C): g · H = Hg = SL2(R) = fg 2 GL2(R) : det g = 1g This is the group usually prescribed to act on the upper half-plane. Again, arbitrary complex scalar matrices act trivially on P1, so it is reasonable that we get a clearer determination of this stabilizing group by restricting attention to SL2(C). 3 Paul Garrett: The Classical Groups and Domains (June 8, 2018) The Cayley map c : D ! H is the holomorphic isomorphism given by the linear fractional transformation 1 1 i c = p · 2 SL2(C) 2 i 1 If we use −1 0 −1 0 SU(1; 1) = fg 2 SL ( ): g∗ g = g 2 C 0 1 0 1 instead of U(1; 1) as the group stabilizing the disk D, then it is easy to check that −1 c · SL2(R) · c = SU(1; 1) Thus, the disk and the upper half-plane are the same thing apart from coordinates. On the upper half-plane, the subgroup 1 b a 0 f · g ⊂ SL ( ) 0 1 0 a−1 2 R acts by affine transformations 1 b a 0 · (z) = a2z + b 0 1 0 a−1 In particular, this shows that SL2(R) is transitive on H. On the other hand, in the action of SU(1; 1) on the disk D, the subgroup µ 0 f : jµj = 1g 0 µ−1 acts by affine transformations µ 0 (z) = µ2 · z 0 µ−1 This is the subgroup of SU(1; 1) fixing the point 0 2 D. When the latter subgroup is conjugated by the Cayley element c to obtain the corresponding subgroup of SL2(R) fixing c(0) = i 2 H, consisting of elements −1 −1 ! µ 0 µ+µ µ−µ cos θ sin θ c · · c−1 = 2 2i = 0 µ−1 µ−µ−1 µ+µ−1 − sin θ cos θ − 2i 2 for θ 2 R such that µ = eiθ. The special unitary group is −1 0 −1 0 α β SU(1; 1) = fg 2 SL ( ): g∗ g = g = f : jaj2 − jbj2 = 1g 2 C 0 1 0 1 β¯ α¯ The Cayley element c conjugates SL2(R) and SU(1; 1) back and forth. Since SL2(R) acts transitively on the upper half-plane, SU(1; 1) acts transitively on the disk. Also, this can be seen directly by verifying that for jzj < 1 the matrix 1 1 z · p1 − jzj2 z¯ 1 lies in SU(1; 1) and maps 0 to z. The isotropy group in SL2(R) of the point i 2 H is the special orthogonal group cos θ − sin θ SO(2) = fg 2 SL ( ): g>g = 1 g = f : θ 2 g 2 R 2 sin θ cos θ R 4 Paul Garrett: The Classical Groups and Domains (June 8, 2018) The natural map SL2(R)=SO(2) −! H given by g · SO(2) −! g(i) is readily verified to be a diffeomorphism. The disk is an instance of Harish-Chandra's realization of quotients analogous to SL2(R)=SO(2) as bounded subsets of spaces Cn.