Components of Springer fibers associated to closed orbits for the symmetric pairs \((Sp(2n),GL(n))\) and \((O(n),O(p)\times O(q))\). I (Q408505)

The authors describe (irreducible) components of Springer fibers associated to closed \(K\)-orbits in the flag variety of a complex classical group \(G\) for the pairs NEWLINE\[NEWLINE(G,K) = (\text{Sp}(2n),GL(n)) \;\text{ and } \;(G,K) = (O(n),O(p)\times O(q)),\tag{*}NEWLINE\]NEWLINE where \(p + q = n\), arising from the real groups \(Sp(2n,\mathbb{R})\) and \(O(p,q)\).NEWLINENEWLINELet \(G\) be a reductive complex algebraic group and \(K\) the fixed point group of an involution \(\Theta\) of \(G\). Let \(\mathfrak g\) be the Lie algebra of \(G\) and \(\mathfrak g = \mathfrak k + \mathfrak p\) its decomposition into the \(+1\)-eigenspace \(\mathfrak k\) and the \(-1\)-eigenspace \(\mathfrak p\) defined by the differential \(\theta\) of \(\Theta\). Suppose \(\mathfrak{b}\) is an element of the variety \(\mathcal B\) of all Borel subalgebras of \(\mathfrak{g}\) (the flag variety of \(G\)) which contains a Cartan subalgebra \(\mathfrak h\) of \(\mathfrak g\) and has a Levi decomposition \(\mathfrak b = \mathfrak h + \mathfrak n^-\). Then, if \(\mathcal Q = K\cdot\mathfrak b\) is a closed orbit in \(\mathcal B\) and \(T^\star\mathcal B\) is the cotangent bundle of \(\mathcal B\), the conormal bundle in \(T^\star\mathcal B\) to this orbit is denoted by \(T^\star_{\mathcal Q}\mathcal B\). The moment mapping \(\mu\) for the natural action of \(G\) on \(T^\star\mathcal B\) is given by \(\mu(g,\xi) = \text{Ad}(g)\xi\) \((g\in G; \xi\in\mathfrak n^-)\) and its restriction to \(T^\star_{\mathcal Q}\mathcal B\) is denoted by \(\gamma_{\mathcal Q}\). An element \(f\) of \(\mathfrak n^-\cap\mathfrak p\) is said to be generic in \(\mathfrak n^-\cap\mathfrak p\) if \(K\cdot f\) is dense in \(\gamma_{\mathcal Q}(T^\star_{\mathcal Q}\mathcal B)\) and then \(\gamma_{\mathcal Q}^{-1}(f)\) is said to be the Springer-fiber over \(f\) associated to \(\mathcal Q\). Let the subscript \(e\) denote the identity component of a given group and \(A_K(f)\) the component group \(Z_K(f)/Z_K(f)_e\), where \(Z_K(f)\) is the centralizer of \(f\) in \(K\).NEWLINENEWLINEThe authors give an algorithm to construct a generic element \(f\) in \(\mathfrak n^-\cap\mathfrak p\) for the pairs \((G,K)\) in (*) and define a sequence \(((G_0,K_0),(G_1,K_1),\dotsc,(G_m,K_m))\) of group pairs, where \(G_0 = G\), \(K_0 = K\), \(G_i\subset G_{i-1}\), \(K_i = K\cap\,G_i\) for \(1\leqslant i\leqslant m\), as well as sequences \((Q_0,Q_1,\dotsc,Q_m)\), \((L_0,L_1,\dotsc,L_m)\), where \(Q_i\) is a parabolic subgroup of \(K_i\) and \(L_i\) is the Levi factor of \(Q_i\). Then, they prove the main theorem of this paper.NEWLINENEWLINE\(C_f = L_{m,e}\dotsm L_{1,e}L_{0,e}\cdot\mathfrak b \, (= Q_{m,e}\dotsm Q_{1,e}Q_{0,e}\cdot\mathfrak b)\) is a component of the Springer fiber \(\mu^{-1}(f)\) that is contained in \(\gamma_{\mathcal Q}^{-1}(f)\); furthermore, all the components of \(\mu^{-1}(f)\) contained in \(\gamma_{\mathcal Q}^{-1}(f)\) are \(A_K(f)\) translates of \(C_f\) and \(\gamma_{\mathcal Q}^{-1}(f) = L_m\dotsm L_1L_0\cdot\mathfrak b\).NEWLINENEWLINEThe paper ends with the application of this description of \(\gamma_{\mathcal Q}^{-1}(f)\) to derive an algorithm for computing associated cycles of Harish-Chandra modules of discrete series representations of the real groups \(\text{Sp}(2n,\mathbb{R})\) and \(SO_e(p,q)\) with \(p\,q\) even.