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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

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
Full Text: DOI
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