Polar orbitopes Leonardo Biliotti, Alessandro Ghigi and Peter Heinzner Università di Parma - Università di Milano Bicocca - Ruhr Universität Bochum Workshop su varietà reali e complesse: geometria, topologia e analisi armonica Pisa, Scuola Normale Superiore, 3 Marzo 2013 Biliotti,L., Ghigi, A., Heinzner, P. () Polar orbitopes 1 / 26
Orbitopes Let K be a compact Lie group and let K GL(V) be a finite-dimensional representation. An orbitope is by definition the convex envelope of an orbit of K in V (Sanyal-Sottile-Sturmfels) An interesting class of orbitopes is given by the convex envelope of adjoint orbits. G be a real semisimple connected non compact Lie group g = k p Cartan decompostion K is the maximal compact subgroup of G with Lie algebra k K Gl(p) k Ad(k) p. Given x p, we consider O = K x and its convex hull Ô Biliotti,L., Ghigi, A., Heinzner, P. () Polar orbitopes 2 / 26
Polar Orbitopes Let K be a compact Lie group and let K O(V) be a finite-dimensional representation. the K-representation is called Polar, if there is a subspace a such that: Ka = V; If p V, then a (T p K p). Adjoint action is a polar action a p (maximal abelian subalgebra); By a Theorem of Dadok, up to equivalence, a Polar representation is an adjoint action. Biliotti,L., Ghigi, A., Heinzner, P. () Polar orbitopes 3 / 26
Example G = SL(2, R). K = SO(2) sl(2, R) = so(2) sym o (2, R) p is the set of 2 2 symmetric matrices of trace 0. a is the set of diagonal matrices of trace 0 A K-orbit in p is a circle or a point. Then its convex hull is a disk or a point. Biliotti,L., Ghigi, A., Heinzner, P. () Polar orbitopes 4 / 26
Motivation X = G/K be symmetric space of non compact type. ρ : G GL(V) complex irreducible representation with finite kernel K U(V) (There exists an Hermitian product, such that ρ(k) U(V)). Then we have a G-equivariant map ϕ : X P(Herm(V)), hk [hh ]. Here we identify h ρ(h). Note that hh is the radial part of h. X ρ = ϕ(x) is called the Satake-compactification of G/K. Biliotti,L., Ghigi, A., Heinzner, P. () Polar orbitopes 5 / 26
Upper half-plane H 2 = {x + iy : y > 0} = SL(2, R)/SO(2). ( [ ] a b z = az+b ) c d cz+d [ ] y y x Therefore x + iy = i and so the embedding ϕ of H 2 is 0 1 y given by ϕ(x + iy) = ( y + x2 x y y x y 1 y ) = ( y 2 + x 2 x x 1 ) SL(2, R) has 3 orbits in P(R 3 ) = {[A] : A = A t : A M 2 2 (R)}: {det < 0} {det = 0} } {{ {det > 0}. } X ρ The Satake-compactification of SL(2, R)/SO(2, R) is H 2 P(R 2 ). Biliotti,L., Ghigi, A., Heinzner, P. () Polar orbitopes 6 / 26
Bourguingnon-Li-Yau-map G = K C ρ : K C GL(V) be an holomorphic irreducible representation There exists a unique closed K C -orbit O in P(V) (unique complex K-orbit) K acts Hamiltonian on O with a momentum map Φ : O k which is a diffemorphism onto a coadjoint orbit O. Example: the momentum map of SU(2) acting on P(C 2 ) is given by ( i 1 Φ([z 1, z 2 ]) = 2( z 1 2 + z 2 2 2 ( z ) 2 2 z 1 2 ) z 2 z 1 1 ) z 1 z 2 2 ( z 1 2 z 2 2 ) ( i ) = SU(2) 2 0 0 i. 2 Biliotti,L., Ghigi, A., Heinzner, P. () Polar orbitopes 7 / 26
Bourguignon-Li-Yau map Let µ be the unique K-invariant probability measure. The Bourguignon-Li-Yau map is given by Ψ : K C /K k Ψ(gK) = Φ( gg x)dµ(x) O Theorem (Biliotti-Ghigi, Amer. J. Math. (2013)) Ψ extends to the Satake compactification X ρ and Ψ is a K-equivariant homeomorphism of X ρ onto Ô = conv(o). X ρ Ô Biliotti and Ghigi studied the integral coadjoint orbits in the sense of geometric quantization; Biliotti, Ghigi and Heinzner studied coadjoint orbitopes. Biliotti,L., Ghigi, A., Heinzner, P. () Polar orbitopes 8 / 26
Set up Let G be a semisimple, connected non compact Lie group. g = k p. Pick x p and O = K x. This orbit lies in the sphere and in general in a very complicate way. We study the face structure of Ô in the sense of convex geometry. Hence the goal is to describe the faces in terms of special submanifolds of O. Biliotti,L., Ghigi, A., Heinzner, P. () Polar orbitopes 9 / 26
Convex Geometry E (V,, ) a compact convex subset. The relative interior of E, denoted relint E, is the interior of E in its affine hull. A face F of E is a convex subset F E with the following property: if x, y E and relint[x, y] F, then [x, y] F. A face is closed The extreme points of E are the points x E such that {x} is a face. x E is an extreme point of E if and only if x cannot be written in the form x = λa + (1 λ)b with a, b E and λ (0, 1). E = conv(ext E). If F is a face of a convex set E, then ext F = F ext E. Biliotti,L., Ghigi, A., Heinzner, P. () Polar orbitopes 10 / 26
Convex Geometry Theorem If E is a compact convex set and F 1, F 2 are distinct faces of E then relint F 1 relint F 2 =. If G is a nonempty convex subset of E which is open in its affine hull, then G relint F for some face F of E. Therefore E is the disjoint union of its open faces. Biliotti,L., Ghigi, A., Heinzner, P. () Polar orbitopes 11 / 26
Convex Geometry The support function of E is the function h E : V R h E (u) = max x, u. (3) x E If u 0, the hyperplane H(E, u) := {x E : x, u = h E (u)} is called the supporting hyperplane of E for u. F u (E) := E H(E, u) is a face and it is called the exposed face of E defined by u In general not all faces of a convex subset are exposed C F := {u V : F = F u (E)} is a convex cone. If G is a compact subgroup of O(V) that preserves both E and F, then C F contains a fixed point of G. Biliotti,L., Ghigi, A., Heinzner, P. () Polar orbitopes 12 / 26
Kostant Polytope Let a be a maximal sublagebra of p and let O = K x be an adjoint orbit. We can associated to O two convex set: Ô = conv O O a = W x, where W = N K (a)/c K (a)) (Weyl group). The convex hull of P = conv(o a) = conv(w x) is called the Kostant polytope. Theorem (Kostant) Let π : p a be the orthogonal projection. Then the image of π : O a is a convex set; π(o) = π(ô) = conv(o a) = P (polytope); Biliotti,L., Ghigi, A., Heinzner, P. () Polar orbitopes 13 / 26
Theorem 1 We denote by F(Ô) the faces of Ô and by F(P) the faces of P. K acts on F(Ô) and the Weyl group W acts on F(P). Theorem Let σ F(P) and let σ. Then K σ σ is a face of F(Ô). Moreover the map σ K σ σ passes to a quotients and the resulting map F(P)/W F(Ô)/K is a bijection. Biliotti,L., Ghigi, A., Heinzner, P. () Polar orbitopes 14 / 26
Example G = SL(2, R); sl(2, R) = so(2) sym o (2, R) a is the set of diagonal matrices of trace 0 A K-orbit in p is a circle or a point. Hence its convex hull is a disk or a point. The Kostant polytope is a closed interval or a point. Biliotti,L., Ghigi, A., Heinzner, P. () Polar orbitopes 15 / 26
Exposed face Lemma ext Ô = O and if F is a face then F O = ext F. We denote by µ p : O p. Let β p. The function µ β p (x) = x, β is the height function corresponding to β. F β (Ô) = {p Ô : max Ô µβ p = µ β p (p)} ext F β (Ô) = {p O : max Oµ β p = µ β p (p)}. µ β p is a Morse-Bott function. Biliotti,L., Ghigi, A., Heinzner, P. () Polar orbitopes 16 / 26
Exposed face Proposition Let O = K x and let F β (Ô) Ô be an exposed face. Then ext F β (Ô) is both a Kβ and a (K β ) 0 -orbit (it is connected); F β (Ô) pβ = {q p : [q, β] = 0}; F β (Ô) is a Polar orbitope with respect to Gβ = K β exp(p β ). T x O = [k, x]; (dµ β p ) x ([w, x]) = [w, x], β = w, [x, β] ; Crit(µ β p ) = O p β = (K β ) o N K (a) x; Crit(µ β p ) is a finite union of K β -orbit and F β (Ô) pβ ; Biliotti,L., Ghigi, A., Heinzner, P. () Polar orbitopes 17 / 26
Critical orbits Lemma Then x is a local maximum of µ β p if and only if there exists a Weyl chamber C a such that x, β C. Proof. x is a local maximum point of µ β p if and only if the Hessian D 2 µ β p (x) is negative semidefinite. T x O = λ(x) 0 (g λ g λ ) p. [x, ξ] = λ(x) 0 λ(x)z λ; D 2 µ β p (x)(w, w) = λ(x) 0 λ(x)λ(β) z λ 2. x is a local maximum of µ β p if and only if λ(x)λ(β) 0 for every restricted root λ if and only if there exists a Weyl chamber C such that x, β C. Biliotti,L., Ghigi, A., Heinzner, P. () Polar orbitopes 18 / 26
Critical orbits Lemma Let x W x. Then x is a local maximum of µ β p if and only if w W such that w x = x and w β = β. Corollary Let β be a nonzero vector in p and let F β (Ô) be the exposed face of Ô defined by β. Then ext F β (Ô) is both a Kβ and a (K β ) 0 -orbit. Hence a local maximum of the function µ β p is a global maximum. Proof. Crit(µ β p ) = O p β = (K β ) o N K (a) x; applying the above results we get that ext F = maxµ β p is is both a K β and a (K β ) 0 -orbit. Biliotti,L., Ghigi, A., Heinzner, P. () Polar orbitopes 19 / 26
Group theoretical description of the faces Proposition Let F be a nonempty face of Ô. Then there is an abelian subalgebra s p such that F is an orbitope of (G s ) 0, i.e. F z p (s) and ext F is an orbit of (K s ) 0. If F is proper, then s {0}. Corollary If F is a nonempty face of Ô, then there exists an abelian subalgebra s p such that F is a G s = K s exp(p s ) Polar orbitope. Biliotti,L., Ghigi, A., Heinzner, P. () Polar orbitopes 20 / 26
Any face is exposed Let F be a face of Ô. Choose a subalgebra s p such that F be an orbitope of (G s ). Let a be a maximal subalgebra of p containing s. Set σ := π(ext F). Lemma π(f) = F a = σ is a proper face of P and F = K σ σ. Theorem All proper faces of Ô are exposed. Proof. σ := F a = F P is a proper face of P. Then there is a vector β a such that σ = F β (P). Since F a = F β (Ô) a we get F = Fβ(Ô). Biliotti,L., Ghigi, A., Heinzner, P. () Polar orbitopes 21 / 26
Proof of Theorem 1 Theorem The map Θ : F(P)/W = F(Ô)/K [σ] [Kσ σ] is a bijection. Proof. F is a face, then there exists s p such that F is a G s Polar orbitopes. Since two abelian maximal subalgebra are conjugate, there exists k K such that ks a and so [F] Im(Θ); σ = σ if and only if Θ([σ]) = Θ([σ ]); Biliotti,L., Ghigi, A., Heinzner, P. () Polar orbitopes 22 / 26
Faces and Parabolic subgroup g = k p; u = k ip and K x is mapped to K ix U ix; G U C, then G acts on U ix and G ix = K ix (Heinzner-Stötzel); If β p. Then G β+ := {g G : R β+ := {g G : lim exp(tβ)g exp( tβ) exists} t lim exp(tβ)g exp( tβ) = e} t G β+ = G β R β+ is a Parabolic subgroup of G. Biliotti,L., Ghigi, A., Heinzner, P. () Polar orbitopes 23 / 26
Faces and Parabolic subgroup If F is a face, we define H F := {g K : gf = F} = {g K : g ext F = ext F} Q F := {g G : g ext F = ext F} Denote by C H F F the vectors of C F that are fixed by H F. Lemma C F := {β p : F = F β (Ô)}. if β C β F, then H F = K β and F is G β = K β exp(p β ) Polar orbitope. Moreover, Q F = G β+ and ext F = G β+ x Theorem The set {ext F : F a nonempty face of Ô} coincides with the set of all closed orbits of parabolic subgroups of G. Any parabolic subgroup Q G has a unique closed orbit, which equals the set of extreme points of a unique face of F Ô. If Q = Gβ+, then F = F β (Ô). Biliotti,L., Ghigi, A., Heinzner, P. () Polar orbitopes 24 / 26
Other results Let F F(Ô), we define S = K relint F. We call S the stratum corresponding to the face F. The strata give a partition of Ô. They are smooth embedded submanifolds of p and are locally closed in Ô. For any stratum S the boundary S S is the disjoint union of strata of lower dimension. we give a description of the faces of the Polar orbitope in terms of root data. We use the formalism of x-connected subset of simple roots developed by Satake. This gives a combinatorial way to generate all the faces of the orbitope. Biliotti,L., Ghigi, A., Heinzner, P. () Polar orbitopes 25 / 26
What s next? Define a BLY map on a general symmetric space of non compact type; convexity properties of the gradient momentum map; Study the convex envelope of an elliptic orbit; Study the BLY for a general Kähler manifold. Biliotti,L., Ghigi, A., Heinzner, P. () Polar orbitopes 26 / 26