The Classical Groups and Domains

September 22, 200 The Classical Groups and Domains Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ The complex unit disk D { C : < has four families of generaliations to bounded open subsets in C n with groups acting transitively upon them. Such domains, defined more precisely below, are bounded symmetriomains. 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 Möbius transformations. Happily, many of the higherdimensional bounded symmetriomains behave in a manner that is a simple extension of this simplest case.. The disk, upper half-plane, SL 2 R, and U, 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 realiation of domains The group. The disk, upper half-plane, SL 2 R, and U, GL 2 C { : a, b, c, d C, ad bc 0 acts on the extended complex plane C by linear fractional transformations a + b c + d with the traditional natural convention about arithmetic with. But we can be more precise, in a form helpful for higher-dimensional cases: introduce homogeneous coordinates for the complex projective line P, by defining P to be a set of cosets P u { : not both u, v are 0/C C 2 {0 /C v where C acts by scalar multiplication. That is, P is the set of equivalence classes of non-ero 2-by- vectors under the equivalence relation u v u v if and only if u c v u v for some non-ero c C In other words, P is the quotient of C 2 {0 by the action of C. The complex line C is imbedded in this model of P by The point at infinity is now easily and precisely described as 0

Paul Garrett: The Classical Groups and Domains September 22, 200 Certainly the matrix multiplication action of GL 2 C on C 2 stabilies non-ero vectors, and commutes with scalar multiplication, so GL 2 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 c + d a + b/c + d when c + d 0, since we can multiply through by c + d to normalie the lower entry back to, unless it is 0, in which case we ve mapped to. The action of GL 2 C is transitive on P, since it is transitive on C 2 {0, and P is the image of C 2 {0. The condition defining the open unit disk D { C : < C can be rewritten as D { C : 0 0 > 0 where T is conjugate transpose of a matrix T. The standard unitary group U, of signature, is U, {g GL 2 C : g 0 g 0 0 0 We can easily prove that U, stabilies the disk D, as follows. For D and compute g 2 g c + d 2 0 0 since g is in the unitary group. Then g g U, g 0 g 0 c + d a + b c + d c + d 2 0 0 0 0 a + b c + d c + d g 2 c + d 2 2 > 0 In fact, to be sure that c + d 0 in this situation, we should have really computed by the same method that c + d 2 g 2 2 Since 2 > 0 and c + d 0, necessarily c + d > 0 and g 2 > 0. The complex upper half-plane is H { C : Im > 0 In the spirit of giving matrix expressions for defining inequalities, Im 2i 0 0 2

This suggests taking Paul Garrett: The Classical Groups and Domains September 22, 200 G {g GL 2 C : g 0 0 Just as in the discussion of the disk, for H and g G compute 2i Img g since g is in G. Then c + d 2 0 0 g g 0 g 0 Im g c + d To be sure that c + d 0, compute in the same way that g a + b c + d c + d 2 c + d 2 Im > 0 c + d 2 Img Im Since Im > 0 and c + d 0, necessarily c + d > 0 and Img > 0. Since the center t 0 Z { : t C 0 t 0 0 0 0 0 0 a + b c + d c + d of GL 2 C acts trivially on C P, one might expect some simplification of matters by restricting our attention to the action of SL 2 C {g GL 2 C : det g since under the quotient map GL 2 C GL 2 C/Z the subgroup SL 2 C maps surjectively to the whole quotient GL 2 C/Z. The special but well-known formula for the inverse of a two-by-two matrix implies that d ad bc c ad bc SL 2 C {g GL 2 C : g 0 0 Thus, the subgroup in SL 2 C stabiliing the upper half-plane is G {g GL 2 C : g 0 g 0 0 0 b ad bc a ad bc g 0 0 and g 0 g 0 0 0 In particular, for such g we have g g, so the entries of g are real. Thus, we have shown that {g SL 2 C : g H H SL 2 R {g GL 2 R : det g This is the group usually prescribed to act on the upper half-plane. Again, arbitrary complex scalar matrices act trivially on P, so it is reasonable that we get a clearer determination of this stabiliing group by restricting attention to SL 2 C. 3

Paul Garrett: The Classical Groups and Domains September 22, 200 The Cayley map c : D H is the holomorphic isomorphism given by the linear fractional transformation If we use c 2 i SL i 2 C SU, {g SL 2 C : g 0 g 0 0 0 instead of U, as the group stabiliing the disk D, then it is easy to check that c SL 2 R c SU, Thus, the disk and the upper half-plane are the same thing apart from coordinates. On the upper half-plane, the subgroup b a 0 { 0 0 a SL 2 R acts by affine transformations b a 0 0 0 a a 2 + b In particular, this shows that SL 2 R is transitive on H. On the other hand, in the action of SU, on the disk D, the subgroup µ 0 { 0 µ : µ acts by affine transformations µ 0 0 µ µ 2 This is the subgroup of SU, fixing the point 0 D. When the latter subgroup is conjugated by the Cayley element c to obtain the corresponding subgroup of SL 2 R fixing c0 i H, consisting of elements for θ R such that µ e iθ. The special unitary group is µ 0 c 0 µ c µ+µ 2 µ µ 2i SU, {g SL 2 C : g 0 g 0 µ µ 2i µ+µ 2 0 { 0 cos θ sin θ sin θ cos θ α β : a β 2 b 2 ᾱ The Cayley element c conjugates SL 2 R and SU, back and forth. Since SL 2 R acts transitively on the upper half-plane, SU, acts transitively on the disk. Also, this can be seen directly by verifying that for < the matrix lies in SU, and maps 0 to. 2 The isotropy group in SL 2 R of the point i H is the special orthogonal group SO2 {g SL 2 R : g cos θ sin θ g 2 { : θ R sin θ cos θ 4

The natural map given by Paul Garrett: The Classical Groups and Domains September 22, 200 is readily verified to be a diffeomorphism. SL 2 R/SO2 H g SO2 gi The disk is an instance of Harish-Chandra s realiation of quotients analogous to SL 2 R/SO2 as bounded subsets of spaces C n. The ambient complex projective space P in which both the bounded disk and unbounded upper half-plane models of SL 2 R/SO2 are realied is an instance of the compact dual, whose general construction is due to Borel. The Cayley map from ounded to an unbounded model was studied in general circumstances by Piatetski-Shapiro for the classical domains, and later by Koranyi and Wolf instrinsically. 2. Classical groups over C and over R We list the classical complex groups and the classical real groups. Further, we tell the compact real groups, and indicate the maximal compact subgroups of the real groups. There are very few different classical groups over C, because C is algebraically closed and of characteristic ero. They are GL n C invertible n-by-n matrices with complex entries general linear groups Sp n C {g GL 2n C : g 0n n n 0 n On, C {g GL n C : g g n g 0n n n 0 n symplectic groups orthogonal groups Sometimes what we ve denoted Sp n C is written Sp 2n C. Also, for some purposes the orthogonal groups with n odd are distinguished from those with n even. Closely related to GL n C are and likewise related to orthogonal groups are It is not hard to show that special linear groups SL n C {g GL n C : det g special orthogonal groups SOn, C On, C SL n C Sp n C SL 2n C A special case of the latter is Sp C SL 2 C. E. Cartan s labels for these families are type A n SL n+ C type B n O2n +, C type C n Sp n C type D n O2n, C Over R there is a greater variety of classical groups, since R has the proper algebraic extension C, and also is the center of the non-commutative division algebra H, the Hamiltonian quaternions H {a + bi + cj + dk : a, b, c, d R where i 2 j 2 k 2 ij ji k jk kj i ki ik j 5

Paul Garrett: The Classical Groups and Domains September 22, 200 Each family of complex group breaks up into several different types of real groups. We will not worry about determinant-one conditions. Type A: The groups over R affiliated with the complex general linear groups GL n C include are the obvious general linear groups GL n R GL n C GL n H over the real numbers, complex numbers, and Hamiltonian quaternions. To this family also belong the unitary groups Up, q {g GL p+q C : g p 0 p 0 g 0 q 0 q of signature p, q. As usual, often Un, 0 and U0, n are denoted Un. Types B,D: The groups over R affiliated with the complex orthogonal groups On, C include On, C itself, and real orthogonal groups Op, q {g SL p+q R : g p 0 p 0 g 0 q 0 q of signature p, q. As usual, On, 0 and O0, n are denoted On. There are also the quaternion skewhermitian groups O 2n {g GL n H : g i n g i n where g denotes the conjugate-transpose with respect to the quaternion conjugation entry-wise a + bi + cj + dk a bi cj dk The i occurring in the definition of the group is the quaternion i, and is not in the center of the division algebra of quaternions. Types C: The groups over R affiliated with the complex symplectic groups Sp n C include Sp n C itself, and real symplectic groups Sp n R {g GL 2n R : g 0n n 0n g n n 0 n n 0 n and also the quaternion hermitian groups Sp p, q {g GL p+q H : g p 0 0 q with signature p, q. As usual, Sp n, 0 and Sp 0, n are denoted Sp n. p 0 g 0 q The fact that the signatures p, q are the unique invariants for groups Op, q, Up, q, and Sp p, q and that the other groups have only dimension as invariant is the Inertia Theorem. There are minor variations of the above families, obtained in two ways. The first is by adding a determinantone condition: SL n C {g GL n C : det g SL n R {g GL n R : det g SUp, q Up, q SL p+q C SL n H GL n H SL 2n C SOp, q Op, q SL p+q R SOn, C On, C SL n C SO 2n O 2n SL n nh A bit of thought is necessary to make sense of SL n H, since H is not commutative, and thus determinants of matrices with entries in H do not behave as simply as for entries from commutative rings. The second 6

Paul Garrett: The Classical Groups and Domains September 22, 200 variations, for groups preserving a quadratic, hermitian, or alternating form, is to allow similitudes, meaning that the form is possibly changed by a scalar. For example, GUp, q {g GL p+q C : g p 0 p 0 g µg for some µg R 0 q 0 q GOp, q {g GL p+q R : g p 0 p 0 g µg for some µg R 0 q 0 q GSp n R {g GL 2n R : g 0n n 0n g µg n for some µg R n 0 n n 0 n The list of the classical compact groups among the groups over R is short: Un On Sp n Each of the classical groups over R has a maximal compact subgroup, unique up to conjugation by a theorem of E. Cartan, which can be proven case-by-case for these groups. We list the isomorphism classes of the maximal compact subgroups in the format G K, where G runs over groups over R and K is a maximal compact subgroup. GL n C Un GL n R On GL n H Sp n Up, q Up Uq On, C On Op, q Op Oq O 2n Un Sp n C Sp n Sp n R Un Sp p, q Sp p Sp q In a few of these cases it is non-trivial to understand the copy of the compact group. For example, the copy of Un inside Sp n R is a b Un { : a + ib Un Sp b a n R Similarly, if relying upon an inertia theorem we use skew form 0n n n 0 n to define O 2n, the analogously defined copy of Un is a maximal compact in O 2n. 3. The four families of self-adjoint cones A cone in a real vectorspace V is a non-empty open subset C closed under scalar multiplication by positive real numbers. A cone is convex when it is convex as a set, meaning as usual that for x, y C and t [0, ] then tx + ty is also in C. Now suppose V has a positive-definite inner product,. Let C be a convex cone. The adjoint cone C of C is C {λ V : v, λ > 0 : for all v C A convex cone C is self-adjoint when C C. A cone is homogeneous when there is a group G of R-linear automorphisms of the ambient vectorspace V stabiliing C and acting transitively on C. In that case, 7

Paul Garrett: The Classical Groups and Domains September 22, 200 letting K be the isotropy group of asepoint in C, we have a homeomorphism C G/K. In the following descriptions we use the orthogonal similitude group GOQ of a quadratic form on a real vector space V, where µg R depends on g. GOQ {g GLV : Qgv µg Qv for all v V There are four families of convex, self-adjoint, homogeneous cones, described as quotients G/K. positive-definite symmetric n-by-n real matrices GL n R/On positive-definite hermitian n-by-n complex matrices GL n C/Un positive-definite quaternion-hermitian n-by-n matrices GL n H/Sp n interior of light cone, {y 0, y,..., y n : y0 2 y 2 +... + yn 2 > 0, y 0 > 0 GO, n + /{ GOn where H + denotes the connected component of the identity in a topological group H, and GOn {k GOn : k GOn, + The actions of the indicated groups are as follows. Proofs of transitivity are given after the descriptions of the actions. Elements g of G GL n R act on positive definite symmetric real n-by-n matrices S by gs gsg The orthogonal group On is the isotropy group of the n-by-n identity matrix n. Elements g of G GL n C act on positive definite hermitian complex n-by-n matrices S by gs gsg where g is conjugate-transpose. The unitary group Un is the isotropy group of n. Elements g of G GL n H act on positive definite quaternion-hermitian n-by-n matrices S by gs gsg where g is quaternion-conjugate-transpose. The symplectic group Sp n is the isotropy group of n. Elements g of G GOn, + act on vectors y y 0, y,..., y n by the natural linear action gy g y The imbedded orthogonal similitude group { GOn is the isotropy group of, 0,..., 0. Note that it is necessary to restrict ourselves to the connected component of the identity to assure that we stay in the chosen half of the interior of the light cone. Proof of transitivity and stabiliation of the cone of the indicated group has a common pattern for the first three types of cones, using spectral theorems for symmetric and hermitian operators. We do just the real case. First, for a symmetric real n-by-n matrix S positive definite or not, by the spectral theorem there is a matrix g GL n R so that S gsg is diagonal. Since S is positive definite, it must be that all the diagonal entries of S are positive. Then there is a diagonal matrix h in GL n R so that hs h n. Thus, every positive definite symmetric matrix lies in the GL n R-orbit of n, so we have the transitivity. To prove transitivity of the action of On, + on the light cone, first note that the imbedded copy of GOn is transitive on vectors y 0, y,..., y n with y,..., y n non-ero. Thus, every GOn -orbit has a representative either of the form y 0,, 0,..., 0 or y 0, 0, 0,..., 0. In the former case, we may as well think in terms of SO,, which contains matrices cosh t sinh t M t sinh t cosh t 8

Paul Garrett: The Classical Groups and Domains September 22, 200 y0 y with t R. Given y 0 there is t R so that of M t 0 with y 0 0 > 0. Thus, every orbit contains a vector y 0, 0,..., 0 with y 0 > 0. Then dilation by a suitable positive real number which is a similitude yields, 0,..., 0. Thus, every orbit contains the latter vector, which proves the transitivity. 4. The four families of classical domains Let n denote the n-by-n identity matrix, and for hermitian matrices M, N write M > N to indicate that M N is positive definite. The four families of bounded classical domains, each diffeomorphic to a quotient G/K for a classical real group G and maximal compact K, are as follows. I p,q {p-by-q complex matrix : q > 0 Up, q/up Uq II n {n-by-n complex matrix : T and n > 0 O 2n/Un III n {n-by-n complex matrix : T and n > 0 Sp n R/Un IV n { C n : 2 +... + n 2 < 2 + +... + n 2 < On, 2/On O2 The fourth family of domains is less amenable than the first three, since its description is relatively ungainly. We prove that Up, q stabilies I p,q and acts transitively upon it by generalied linear fractional transformations a + bc + d where a is p-by-p, b is p-by-q, c is q-by-p, and d is q-by-q. Then, letting 0 < A mean that A is positive-definite hermitian, 0 < q p 0 p 0 q 0 q q q 0 q q a + b p 0 c + d 0 q a + b c + d c + d a + b a + b c + d For the right-hand side to be positive definite, it must be that c + d is non-singular, which also verifies that the apparent definition of the action by linear fractional transformations is plausible. Then, in similar fashion, q g g p 0 g g q 0 q q c + d p 0 q 0 q c + d a + b p 0 c + d 0 q q a + b c + d c + d c + d c + d q c + d The latter quantity is positive definite, so the image g lies back in the specified domain. The unbounded models of these domains are I p q {, u : i u u>0, q-by-q, up q-by-q for p q, hermitian upper half-space II n {n-by-n quaternionic : i i > 0 quaternion upper half-space III n {symmetric complex n-by-n : i > 0 Siegel upper half-space IV n {, u C C n : i u 2 > 0 Note that the unbounded models I n,n, II n, and III n all are visibly tube domains in the sense that they are describable as domain { x + iy : x V, y C 9

Paul Garrett: The Classical Groups and Domains September 22, 200 where C is a self-adjoint homogeneous convex open cone in a real vector space V. The domains IV n are also tube domains, related to light cones. The domains I p,q with p q, including the case of the complex unit ball in C n which is I n,, are not tube domains. These coordinates are more convenient to verify that, for example, Sp n R stabilies and acts transitively on the Siegel upper half-space H n by generalied linear fractional transformations a + bc + d using n-by-n blocks for matrices in Sp n R. To verify that c + d is genuinely invertible, use the fact that matrices in Spn, R are real, and compute 0n n 0n n n n 0 n n n n 0 n a + b c + d 0n n n 0 n a + b c + d c + d a + b [c + d a + b] Thus, the latter expression is i times a positive-definite symmetric real matrix. Thus, for any non-ero -by-n complex matrix v, 0 v c + d a + b [c + d a + b] v In particular, c + dv 0. Thus, c + d is invertible, and the expression a + bc + d makes sense. Next, we verify that a + bc + d is symmetric. To this end, similarly compute 0 0n n n n 0 n n a + b c + d 0n n n 0 n a + b c + d n 0n n n 0 n n c + d a + b [c + d a + b] c + d [a + bc + d a + bc + d ]c + d c + d [g g ]c + d Since c + d is invertible, it must be that g is symmetric. Next, we verify that g does lie in the Siegel upper half-space H n. In the equality from above n c + d a + b [c + d a + b] left multiply by c + d and right multiply by c + d to obtain c + d c + d a + bc + d [a + bc + d ] g g Thus, since g is actually symmetric, ig g is positive-definite symmetric real. The bounded and unbounded models of domains G/K are both open subsets of a larger compact complex manifold called the compact dual, on which the complexified G acts by linear fractional transformations. The bounded model is mapped to the unbounded model by a special element in the complexified group, the Cayley element c. In the case of the domains I p q, the picture is relatively easy to understand. Let and let Ď be the quotient Ď Ω { complex p-by-q v : rank v q Ω/GL q C where GL q C acts on the right by matrix multiplication. The Ω coordinates on Ď are homogeneous coordinates. This is a Grassmannian variety. The group G C GL p+q C acts on the left on Ω by matrix 0

Paul Garrett: The Classical Groups and Domains September 22, 200 multiplication, and certainly preserves the maximal rank. Thus, the bounded model of the I p,q domain consists of points with homogeneous coordinates q and q > 0 The right action of GL q C preserves the latter relation, so this set is well-defined via homogeneous coordinates. The unbounded model has a similar definition in these terms, being the set of points with being q-by-q, u being p q-by-q, u q so that i u u>0 Again, the latter relation is stable under the right action of GL q C, so this set is well-defined in homogeneous coordinates. Let i 2 q 0 2 q c 0 p q 0 i 2 q 0 2 q Of course if p q n then this is simpler, and completely analogous to the SL 2 R situation: c 2 n i n i n n u q In the case of the domains II n the direct relation between the bounded and unbounded realiations is computationally more irksome, since the bounded model makes no mention of quaternions, while the unbounded model is most naturally described in such terms. The domain IV n is even less convivial in these ad hoc terms. 5. Harish-Chandra s and Borel s realiation of domains The general construction of this section clarifies the phenomena connected with the four families of classical domains. We will not prove anything, but only describe intrinsically what happens. We will verify that the construction duplicates the bounded and unbounded models for Sp n R. Let G be an almost simple semi-simple real Lie group, K a maximal compact subgroup of G, and suppose that the center of K contains a circle group Z that is, a group isomorphic to R/Z. It turns out that the action of Z on the complexified Lie algebra g C of G decomposes g C into 3 pieces g C p + k C p where k is the Lie algebra of K, and p ± are χ ± -eigenspaces for a non-trivial character χ of Z. It turns out that G exp p + K C expp where K C is the complexification of K. Thus, letting B K C exp p we have G/K GB/B G C /B

Paul Garrett: The Classical Groups and Domains September 22, 200 In fact, given g G, there is a unique ζg p + so that The function g exp ζg K C exp p g ζg p + is the Harish-Chandra imbedding of G/K in the complex vector space p +. Given g, h G, the action ζh g ζh of g on the point ζh is defined by g exp ζh expg ζh K C exp p We can show that the above description does really work in the case of G Sp n R, and produces the bounded model given earlier, namely In the bounded model, III n { symmetric n-by-n complex matrices : n > 0 k 0 K { 0 k : k Un k 0 K C { 0 k : k GL n C Then p + 0 { : 0 0 p 0 0 { : ζ 0 Then given try to solve for, ζ, and h GL n C such that h 0 0 0 0 h ζ This is a b One finds that h d, and bd h + h ζ h h ζ h 0 with linear fractional action and also ζ d c. This evidently requires that d is invertible. That is, bd d 0 0 0 0 d d c To compute the action of on a point, we do a similar computation: a a + b 0 c c + d which has the decomposition a a + b a + bc + d c c + d 0 c + d 0 0 c + d and thus assuming c + d is invertible the action is a + bc + d 0 c + d c 2