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Fixed point varieties on affine flag manifolds. (English) Zbl 0658.22005
Let G be a simply connected semisimple algebraic group over \({\mathbb{C}}\) with Lie algebra \({\mathfrak G}\) and let \({\mathcal B}\) denote the variety of Borel subalgebras of \({\mathfrak G}\). For any nilpotent element \(N_ 0\in {\mathfrak G}\) let \({\mathcal B}_{N_ 0}\) denote the closed subvariety of \({\mathcal B}\) of Borel subalgebras containing \(N_ 0\). The geometry \({\mathcal B}_{N_ 0}\) has been studied by Springer, Steinberg, Spaltenstein and others, and has interesting applications to representation theory.
In this article the authors study the affine analogue of this situation. Let F be the field of formal power series \(F={\mathbb{C}}((\epsilon))\) and let \({\mathfrak G}_ F={\mathfrak G}\otimes_{{\mathbb{C}}}F\). Let \(\hat {\mathcal B}\) be the set of all Iwahori subalgebras of \({\mathfrak G}_ F\). It is known that \(\hat {\mathcal B}\) is an increasing union of ordinary projective algebraic varieties over \({\mathbb{C}}\). For any \(N\in {\mathfrak G}\) let \(\hat {\mathcal B}_ N\) denote the space of Iwahori subalgebras containing \(N\). The authors restrict their attention to the case where \(N\) is a nil-element, that is, \(ad(N)^ r\to 0\) in End(\({\mathfrak G}_ F)\) for \(r\to \infty\). This condition implies that \(\hat {\mathcal B}_ N\) is nonempty. The authors show that \(\hat {\mathcal B}_ N\) is infinite dimensional unless N is regular semisimple in which case \(\hat {\mathcal B}_ N\) is a locally finite union of ordinary irreducible projective algebraic varieties of the same dimension over \({\mathbb{C}}\). Moreover, there is a free abelian group \(\Lambda_ N\) of finite rank which acts on \(\hat {\mathcal B}_ N\) without fixed points and \(\hat {\mathcal B}_ N/\Lambda_ N\) is an algebraic variety. If N is elliptic, that is, the centralizer of N is an anisotropic torus, then \(\Lambda_ N=1\) and \(\hat {\mathcal B}_ N\) is an algebraic variety with finitely many components.
Let \(A={\mathbb{C}}[[ \epsilon ]]\) denote the ring of integers of F and let \(\hat G=G(F)\). The set X of all \(\hat G\)-conjugates of \({\mathfrak G}_ A={\mathfrak G}\otimes A\subset {\mathfrak G}_ F\) is, like \(\hat {\mathcal B}\), an increasing union of projective algebraic variaties over \({\mathbb{C}}\). There is a \(\hat G\)-equivariant map \(p: \hat {\mathcal B}\to X\) which maps \(\hat {\mathcal B}_ N\) onto the set of \(X_ N\) of subalgebras in \(X\) which contain \(N\). The authors show that the dimensions of \(\hat {\mathcal B}_ N\) and \(X_ N\) are equal which implies that N is \(\hat G\)-conjugate to an element of \({\mathfrak G}_ A\) whose image in \({\mathfrak G}_ A/\epsilon {\mathfrak G}_ A\) is regular nilpotent. The authors conjecture a formula for dim(\({\mathcal B}_ N)\) which they verify for the case where N is elliptic.
The authors define a map \(\sigma\) from the nilpotent orbits in \({\mathfrak G}\) to the Weyl group W as follows. First they show that the \(\hat G- \)conjugacy classes of Cartan subalgebras of \({\mathfrak G}_ F\) are parameterized by conjugacy classes in the Weyl group. If \(N_ 0\) is a nilpotent element of \({\mathfrak G}\) and \(Y\in {\mathfrak G}_ A\) then \(N=N_ 0+\epsilon Y\) is a nil-element which is in fact regular semisimple for ‘almost all’ choices of Y. There is a unique Cartan subalgebra which contains N and this subalgebra is associated to a conjugacy class \(\sigma\) (N) in W. The authors show that this conjugacy class is independent of Y and dim(\(\hat {\mathcal B}_ N)=\dim ({\mathcal B}_{N_ 0})\). They also show that \(\sigma\) takes ‘distinguished’ nilpotent orbits in the sense of P. Bala and R. W. Carter [Math. Proc. Camb. Philos. Soc. 79, 401-425 (1976; Zbl 0364.22006) and 80, 1-18 (1976; Zbl 0364.22007)] to Weyl group elements without eigenvalue 1 and that this map restricts on a certain subset of nilpotent orbits to a map defined by R. W. Carter and G. B. Elkington [J. Algebra 20, 350-354 (1972; Zbl 0239.20053)].
Reviewer: D.M.Snow

MSC:
22E60 Lie algebras of Lie groups
20G20 Linear algebraic groups over the reals, the complexes, the quaternions
14L30 Group actions on varieties or schemes (quotients)
14M17 Homogeneous spaces and generalizations
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