III The The Shilov Shilov boundary boundary III
V Positive Positive Hermitian Hermitian Jordan Triple System Jordan Triple System V {x,y,z} C -linear linear in x in x and and z, z, conjugate {x,y,z} C - conjugate linear linear in y in y symmetric in (x,z) in (x,z) symmetric a quintic quintic identity a identity τ (x,y) = tr (z → {x,y,z}) positive τ (x,y) = tr (z → {x,y,z}) positive definite definite hermitian hermitian form form c tripotent {c,c,c} = 2c c tripotent {c,c,c} = 2c lxl spectral spectral norm norm of x of x lxl D unit ball ball for for the the spectral spectral norm norm D unit G = Hol Hol(D) (D) 0 0 , K , K stabilizer stabilizer of 0 in G. of 0 in G. G = Example. . V = Mat (p,q, V = Mat (p,q, C ) (p ≤ ≤ q, n=p+q) q, n=p+q) Example C ) (p {x,y,z} = xy xy*z+ *z+zy zy*x, *x, lxl lxl = = llxll llxll op {x,y,z} = op D = {x in V, lxl lxl<1}, G = PU(p,q), K = P(U(p) <1}, G = PU(p,q), K = P(U(p)xU xU(q)) (q)) D = {x in V, k(z,w)= det det(1 (1 p -zw*) zw*) - k(z,w)= p - -n n
III.1 Tripotents III.1 Tripotents and and Peirce Peirce frames frames Let V be be a PHJTS, a PHJTS, with with triple triple product product {.,.,.}. Assume, for {.,.,.}. Assume, for convenience convenience, , Let V that V that V is is simple (i.e. simple (i.e. cannot cannot be be written written as a as a sum sum of of two two PHJTS). PHJTS). Recall that that a tripotent a tripotent is is an an element element c c which which satisfies satisfies {c,c,c} = 2c. {c,c,c} = 2c. Recall There is is a (partial) a (partial) order order on on tripotents tripotents : if c : if c and and d are d are two two tripotents tripotents, , There then say say c c < d if < d if there there exists exists a tripotent f ≠ 0, orthogonal to c a tripotent f ≠ 0, orthogonal to c and and such such then that d=c+f. d=c+f. that A tripotent c is is primitive primitive (or minimal) if c (or minimal) if c can can not not be be written written as a as a sum sum of of A tripotent c two (non (non zero zero) ) tripotents tripotents. . Any Any tripotent tripotent can can be be written written as a as a sum sum of of two primitive orthogonal tripotents tripotents. . primitive orthogonal Any two two minimal minimal tripotents tripotents are are conjugate conjugate under under an an automorphism automorphism of V. of V. Any A Peirce frame is is a maximal set of a maximal set of orthognal orthognal primitive primitive tripotents tripotents. . Any Any A Peirce frame two Peirce Peirce frames frames are are conjugate conjugate under under an an automorphism automorphism of V. In of V. In two particular particular, , the the number number of of elements elements is is the the same same for all for all frames frames (call (call it it the the rank of V). of V). rank
Let c be be a tripotent. a tripotent. Then Then TFAE TFAE Let c (i) c= c 1 +…+c c r , where where (c (c 1 ,…, c c r ) is is a Peirce a Peirce frame frame (i) c= c 1 +…+ r , 1 ,…, r ) (ii ii) c ) c is is a maximal tripotent a maximal tripotent ( (iii iii) V = V ) V = V 2 (c) + V 1 (c) (i.e. V 0 (c) = 0) ( 2 (c) + V 1 (c) (i.e. V 0 (c) = 0) (iv iv) ) R R c c 1 + R R c c 2 +… + R R c c r is a maximal flat a maximal flat space space in V. in V. ( 1 + 2 +… + r is
III.2 The The Shilov Shilov boundary boundary III.2 Let D be be a a domain domain in in some some complex complex vector vector space space. . Let D The Shilov boundary S of D S of D is is the the smallest smallest closed closed subset subset of of the the The Shilov boundary boundary of D, for of D, for which which the the maximum maximum principle principle for for the the modulus modulus boundary of holomorphic of holomorphic function function applies applies. . The The Shilov Shilov boundary boundary may may be be much smaller smaller than than the the topological topological boundary boundary. . much Example 1 1. Let D . Let D be be the the product product of of two two copies of copies of the the complex complex unit unit Example disc. Then Then the the Shilov Shilov boundary boundary of D is of D is the the product product of of two two copies copies disc. of the the unit unit circle circle, as , as can can be be seen seen by by applying applying twice twice the the maximum maximum of principle w.r.t. principle w.r.t. each each variable. variable. Example 2 Example 2. Let D . Let D be be the the unit unit ball ball in Mat in Mat (p,q) (p,q) with with p ≤ q, p ≤ q, then then � x x is is in in the the topological topological boundary boundary of D of D iff iff 1 1 is is an an eigenvalue eigenvalue of of xx xx* * � x x is is in in the the Shilov Shilov boundary boundary iff iff xx*= xx *= Id Id p � � p
Example 3. 3. The The Siegel disc Siegel disc and and the the Lagrangian Lagrangian manifold manifold Example Let V = Symm Symm(r, (r, C ) be be the the PHJTS, PHJTS, with with product product Let V = C ) {x,y,z} = xy xy*z+ *z+zy zy*x . *x . {x,y,z} = D = {x in V ; 1- -xx*>>0} xx*>>0} D = {x in V ; 1 The group G group G is is Sp Sp(2r, (2r, R )( mod mod{±1}), {±1}), and and K K is is isomorphic isomorphic to U(r) to U(r) The R )( t . acting on V by (u,X) → → uXu acting on V by (u,X) uXu t . D is is called called the the Siegel disc . Its Its Shilov Shilov boundary boundary is is D Siegel disc . S = { σ σ in V ; in V ; σ σ σ σ * = 1}. S = { * = 1}. S is is isomorphic isomorphic to to the the Lagrangian Lagrangian manifold ( manifold (also also to U(r)/O(r)). to U(r)/O(r)). S . (E, ω) ω) a real Recall. (E, Recall a real symplectic symplectic vector vector space space , of dimension 2r. , of dimension 2r. A Lagrangian Lagrangian L L is is a maximal a maximal totally totally isotropic isotropic vector vector subspace subspace of E of E A (hence hence of dimension r). of dimension r). The The Lagrangian manifold is is the the set of all set of all ( Lagrangian manifold lagrangians. . It It sits sits in in the the Grassmanian Grassmanian of of r r- -subspaces subspaces in E. in E. lagrangians
The Shilov Shilov boundary boundary S of S of the the open unit open unit ball ball D in V D in V can can be be The described in described in the the following following equivalent equivalent ways ways : : S is is the the set of maximal set of maximal tripotents tripotents S i) i) S is is the the set of set of extremal extremal points of points of the the closed closed open open ball ball (as a (as a S ii) ii) convex set) set) convex S is is the the set of points in set of points in the the closed closed unit unit ball ball which which are are at at S iii) iii) maximal distance of the the origin origin for for the the distance distance associated associated to to maximal distance of form τ τ . Hermitian form . Hermitian
III.3 Action of G on S and and S S x S III.3 Action of G on S x S The action of a action of a holomorphic holomorphic diffeomorphism diffeomorphism of D of D always always extend extend to to The some neigh’d neigh’d of of the the closure closure of D. of D. Hence Hence the the action of G action of G extends extends some to the to the closure closure of D. In of D. In particular particular, G , G acts acts on S. on S. Proposition 1 S Proposition 1 S is is a a connected connected compact manifold. G compact manifold. G acts acts transitively on S, on S, and and S S is is the the unique unique closed closed G G- -orbit orbit in in the the transitively boundary of D. K (a maximal compact of D. K (a maximal compact subgroup subgroup of G) of G) acts acts boundary already transitively transitively on S. on S. The The stabilizer stabilizer of a point in S of a point in S is is a a already (maximal) parabolic (maximal) parabolic subgroup subgroup of G. of G.
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