Morse Groups in Symmetric Spaces Corresponding to the Symmetric Group

Let θ : g → g be an involution of a complex semisimple Lie algebra, k ⊂ g the fixed points of θ, and V = g/k the corresponding symmetric space. The adjoint form K of k naturally acts on V . The orbits and invariants of this representation were studied by Kostant and Rallis in [KR]. Let X = K\\V be the invariant theory quotient, and f : V → X be the quotient map. The space X is isomorphic to C. When g = g ⊕ g and θ acts by interchanging the factors, K |V is just the adjoint representation of G. A detailed study of the singularities of f in this case leads to the Springer representations of the Weyl group W ′ of g (see [BM], [M], [Sp], [Sl]). More precisely, if P is the nearby cycles sheaf of f along the nilcone N = f−1(0), then one obtains representations of W ′ on the stalks of P . In [Gr] we studied the singularities of f for an arbitrary symmetric space (in fact, for a more general class of objects). The main result of [Gr] is a description of the nearby cycles sheaf P in terms of the Fourier transform. In this paper, we study the microlocal geometry of f in three particular examples: V I = sln, V II = sln/son, and V III = sl2n/sp2n. Of all the classical symmetric spaces, these are the ones that have the symmetric group Σn as their small Weyl group. The questions we discuss are equally interesting for other symmetric spaces, but for the time being, they remain open outside of the three examples above. Our main results (Theorems 1.2 and 1.3) describe the Morse groups of P with two monodromy structures (see Section 1.2). Experts in singularity theory may also find technical Lemma 2.2 to be of some independent interest. In the case V = sln, we recover the computation by Evens and Mirković [EM] of the local Euler obstructions for the nilcone in sln. We also borrow from [EM] the idea of exploiting torus symmetry (see our proof of Lemma 4.1). In his Ph.D. thesis [Groj], Grojnowski studied a class of equivariant perverse sheaves on symmetric spaces which is related to the nearby cycles sheaf P we dis-