The index of compact simple Lie groups

Abstract

Abstract. Let M be an irreducible Riemannian symmetric space. The index i(M ) of M is the minimal codimension of a (non-trivial) totally geodesic submanifold of M . The purpose of this note is to determine the index i(M ) for all irreducible Riemannian symmetric spaces M of type (II) and (IV).

On the index of symmetric spaces

“…The second inequality follows from the fact that the reflective index of M = Sp 2r /Sp r Sp r is equal to 4r for r ≥ 3 (see [2] [20] for details). In [3] we proved that i(Sp r ) = 4r − 4. Using Proposition 4.7 we then obtain 4r − 4 = i(Sp r ) = i(Σ) ≤ i(M).…”

“…Since π(h) = {0} for at least one of the two projections, we proved (6.1). If h ∼ = sp r , then dim(h) = dim(H) ≤ dim(Sp r ) − i r (Sp r ) = (2r 2 + r) − 4(r − 1) = 2r 2 − 3r + 4, since the subgroup of Sp r with Lie algebra π(h) is a totally geodesic submanifold of Sp r and i r (Sp r ) = 4(r − 1) by [3]. Since k ′ is a subalgebra of sp r ⊕ sp r , we obviously have rk(k ′ ) ≤ rk(sp r ⊕ sp r ) = 2r and dim(k ′ ) < dim(sp r ⊕ sp r ) = 2(2r 2 + r) = 2r(2r + 1).…”

“…We can now state the main result of this paper: [26] (for rk(M) = 1), [23] (for rk(M) = 2), [2], [3], [4] (for compact simple Lie groups and many symmetric spaces of higher rank), [5] (for exceptional symmetric spaces), and finally Theorems 5.7, 6.9 and 7.3 in this paper. In Table 1 we list the index for all irreducible Riemannian symmetric spaces M of noncompact type together with examples of totally geodesic submanifolds Σ with codim(Σ) = i(M).…”

Submanifold geometry in symmetric spaces of noncompact type

“…A submanifold of a symmetric space M is called reflective if it is a connected component of the fixed point set of an involutive isometry of M ; or, equivalently, if it is a totally geodesic submanifold such that the exponentiation of one (and hence all) normal space is also a totally geodesic submanifold. Finally, let us mention that the index of symmetric spaces (that is, the smallest possible codimension of a proper totally geodesic submanifold) has been recently investigated by Berndt and Olmos [12,13,14], who proved, in particular, that the index of an irreducible symmetric space is bounded from below by the rank. Further information on totally geodesic submanifolds of symmetric spaces can be found in [8, §11.1].…”

“…For a fixed l ∈ {0, 1}, let γ ∈ Σ + be the root of minimum level in its α l -string, which consists of the roots γ and γ + α l . Fix a normal unit vector ξ l ∈ V l and define (13) φ ξ l = |α l | −1 ad(ξ l ) and φ θξ l = −|α l | −1 ad(θξ l ).…”