Workshop on the occasion of the 70th birthday of. Francesco Mercuri. Universität zu Köln. Classical Symmetric Spaces. Gudlaugur Thorbergsson

Transcription

1 Classical I Universität zu Köln Workshop on the occasion of the 70th birthday of Francesco Mercuri Parma, September 19-20, 2016

2 I We say that a symmetric space M = G/K is classical if the groups in the pair (G, K ) are classical matrix groups. We will be interested in those that are irreducible and of compact type. There are ten series of such spaces, which can be divided into Type I consisting of seven series and Type II which consists of the three series of classical compact simple Lie groups.

3 Here is a list of the spaces of Type I. I AI AII AIII BDI DIII CI CII SU(n)/SO(n) SU(2n)/Sp(n) SU(p + q)/s(u(p) U(q)) SO(p + q)/so(p) SO(q)) SO(2n)/U(n) Sp(n)/U(n) Sp(p + q)/sp(p) Sp(q)) The symmetric spaces of Type II are the compact Lie groups SU(n), SO(n), and Sp(n).

4 I Our goal in this talk will be to introduce these ten series in a systematic way. There will be a consiting of three series and there will be a I consisting of seven series, but this division in classes will not coincide with the usual divsion into types that we presented above.

5 I consists of the Grassmann manifolds G k (F n ) of k-planes in F n where F is the field of real numbers R, the complex numbers C, and the quaternions H. We thus get three series of Grassmannians each depending on two natural numbers (k, n). The seven series in I will consist of certain submanifolds of the Grassmannians.

6 I (σ, ɛ)-sesquilinear We will need (σ, ɛ)-sesquilinear to define the seven series of submanifolds in the Grassmannians constituting I. Again, F will denote R, C, or H and F n will be considered to be a right vector space. Let σ : F F be an antiautomorphism, i.e., σ(αβ) = σ(β)σ(α). The only examples we will consider are the identity and the conjugation. We let ɛ stand for 1 or 1.

7 I Definition. A (σ, ɛ)-sesquilinear form f is a map f : F n F n F that is additive in both arguments and satisfies f (xα, yβ) = σ(α)f (x, y)β and f (x, y) = ɛσ(f (y, x)) for all x and y in F n and all α and β in F. We will always assume f to be nondegenerate.

8 A subspace W of F n is called totally isotropic if f W W = 0. I The Witt index r of f is by definition the maximal dimension of a totally isotropic subspace of F n. Let N i (F n, f ) denote the space of i-dimensional totally isotropic subspaces of (F n, f ) for i r. The automorphism group Aut(F n, f ) acts transitively on N i (F n, f ) by a theorem of Witt.

9 I Let G denote the subgroup of Aut(F n, f ) consisting of the automorphism with determinant one. Theorem. Assume the Witt index r of (F n, f ) is positive. Then G is a noncompact semi-simple Lie group that acts transitively on N i (F n, f ) for all i r. If K is a maximal compact subgroup of G, then K also acts transitively on N i (F n, f ).

10 I I Theorem. Let N i (F n, f ) be endowed witht a K -invariant Riemannian metric. Then N i (F n, f ) is a symmetric space if and only if i is equal to the Witt index r and n = 2i. Remark. We will explain below that the theorem above gives us up to equivalence seven different series of symmetric spaces. These will be our class II spaces. Remark. Two of the class II spaces are disconnected and three are locally a product of a real line and an irreducible symmetric space of compact type. After choosing a component or splitting off a one dimensional factor, we get the seven examples of irreducible symmetric spaces of compact type which are not Grassmannians.

11 I The seven cases of (σ, ɛ)-sesquilinear f on F n R σ = identity ɛ = +1 symmetric ɛ = 1 symplectic C σ = identity ɛ = +1 symmetric ɛ = 1 symplectic σ = conjugation ɛ = +1 Hermitian ɛ = 1 skew-hermitian H σ = conjugation ɛ = +1 Hermitian ɛ = 1 skew-hermitian If f is Hermitian over C, then i times f is skew-hermitian. These two cases are genuinely different over H. The eight cases therefore reduce to seven.

12 I It will be convenient to group the seven types of (σ, ɛ)-sesquilinear according to their normal. We get the following three groups. 1 The symmetric over R and the Hermitian over C and H. 2 The symplectic over R and C 3 The symmetric over C and the skew-hermitian over H.

13 I 1. Let f be a form on F n that is symmetric if F = R and Hermitian if F = C or H. Then there is a basis of F n such that f can be written as f (x, y) = k x i y i i=1 n i=k+1 x i y i. It turns out that the Witt index of f is equal to min{k, n k}. Hence N k (F n, f ) is a symmetric space if and only if n = 2k.

14 I In fact, N k (R 2k, f ) = O(k), N k (C 2k, f ) = U(k), N k (H 2k, f ) = Sp(k). Note that N k (R 2k, f ) is disconnected and that N k (C 2k, f ) has a one-dimensional Euclidean factor. The correspondence between N k (R 2k, f ) and O(k) is that a totally isotropic subspace in (R 2k, f ) is the graph of an orthogonal map in O(k), and vice versa. An analogous correspondence holds in the other two cases.

15 I 2. Let f be a symplectic form on F n. Then n = 2k and F is either R or C. There is a basis of F n such that f can be written as f (x, y) = k (x i y k+i x k+i y i ). i=1 The Witt index of f is equal to k which is half the dimension n = 2k. Hence N k (F n, f ) is a symmetric space. The totally isotropic subspaces are usually called Lagrangian subspaces and N k (F n, f ) is called Lagrangian Grassmannian.

16 We have N k (R 2k, f ) = U(k)/O(k), N k (C 2k, f ) = Sp(k)/U(k). I Note that N k (R 2k, f ) has a one-dimensional Euclidean factor. To see that N k (R 2k, f ) is U(k)/O(k), one identifies R 2k with C k = R k + ir k and proves that the Langrangian subspaces of R 2k are precisely the k-dimensional real subspaces W with the property that the orthonormal bases of W are also unitary bases of C k. One identifies N k (C 2k, f ) and Sp(k)/U(k) similarly.

17 I 3a. Let f be a symmetric form on C n. There is a basis of C n such that f can be written as f (x, y) = n x i y i. The Witt index r of f is equal to [ n 2]. The spaces N r (C n, f ) are called orthogonal Grassmannians. It follows that N k (C n, f ) is a symmetric space if and only if n = 2k. i=1 We will identify N k (C 2k, f ) with O(2k)/U(n) thereby showing that N k (C n, f ) is disconnected. We denote the two components by N + k (C2k, f ) and N k (C2k, f ).

18 I We prove that a k-dimensional subspace S of C 2k is contained in N k (C 2k, f ) if and only if a unitary basis z 1 = u 1 + iv 1,..., z k = u k + iv k with u i and v i in R k has the property that 2u1,..., 2u n, 2v 1,..., 2v n is an orthonormal basis of R 2k. Now we clearly can identify N k (C 2k, f ) and O(2k)/U(k). It depends on the orientation of the basis 2u1,..., 2u n, 2v 1,..., 2v n in which component of N k (C 2k, f ) the subspace S lies.

19 I 3b. Let f be a skew Hermitian form on H n. Then there is a basis of H n such that f can be written as f (x, y) = n x i jy i i=1 where j is the third element in the standard basis of H. The Witt index r of f is equal to [ n 2]. The space Nr (H n, f ) is called a quaternionic orthogonal Grassmannian. It follows that N k (H n, f ) is a symmetric space if and only if n = 2k. We will identify N k (H 2k, f ) with U(2k)/Sp(k). Note that N k (H 2k, f ) has a one-dimensional Euclidean factor.

20 I We prove that a k-dimensional subspace S of H 2k is contained in N + k (H2k, f ) if and only if a quaternionic unitary basis z 1 = u 1 + jv 1,... z k = u k + jv k with u i and v i in C k has the property that 2u1,..., 2u n, 2v 1,..., 2v n is a unitary basis of R 2k. Now we clearly can identify N k (H 2k, f ) and U(2k)/Sp(k).

21 I Summarizing we have the following ten series. : The three series of Grassmann manifolds. I: 1 The three series of compact classical groups. 2 The two series of Lagrangian Grassmannians. 3 The two series of orthogonal Grassmannians and the quaternionic orthogonal Grassmannians if the Witt index of f is half the dimension of F n.

22 I Classical Let a f be a (σ, ɛ)-sesquilinear form on F n with positive Witt index r. Let G denote the subgroup of Aut(F n, f ) consisting of the automorphism with determinant one. As we observed, G is a noncompact semi-simple Lie group. Let K be a maximal compact subgroup of G. Then (G, K ) is a symmetric pair of noncompact type and N i (F n, f ) = G/P for all 1 i r where P is a maximal parabolic subgroup of G.

23 I More generally, one can consider so-called which are by definition quotients G/P where G is semisimple and P is some parabolic subgroup of G. All Grassmannians G k (F n ) and all spaces of type N i (F n, f ) are, also when they are not symmetric. We will now discuss the remaining examples of of classical type.

24 I of type A n or Projective Geometry. If G = SL(n, F), then the corresponding are the flag manifolds F(d 1, d 2,... d k, F n+1 ) consisting of nested sequences of subspaces in F n+1 of dimensions d 1 < d 2 <... < d k. The Coxeter group of the symmetric space SL(n; F)/SU(n; F) is of type A n.

25 I of type B n (= C n ) or Polar Geometry. If G is the semi-simple group of a (σ, ɛ)-sesquilinear form f on F N and N r (F N, f ) is connected where r is the Witt index, then the are the flag manifolds F(d 1, d 2,... d k, F N, f ) consisting of flags of totally isotropic subspaces. The Coxeter group of the symmetric space G/K is of type B r (=C r ).

26 I of type D n or Oriflamme Geometry. Here we assume that N n (F 2n, f ) is disconnected where n is the Witt index of f. We denote the two components of N n (F n, f ) by N + n (F n, f ) and N n (F n, f ). We do not allow totally isotropic spaces of dimension n 1 in the flags. If no more than one totally isotropic subspaces in a flag has dimension n, then the situation is as in type B n. If there are two maximal isotropic subspaces (i.e. of dimension n), then one must be in N + n (F n, f ) and the other in N n (F n, f ) and their intersection must be (n 1)-dimensional.

27 This corresponds to the diagram D n I The Coxeter group of the symmetric space G/K is of type D n.

28 I The oriflamme