## 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

### Citations:

Zbl 0364.22006; Zbl 0364.22007; Zbl 0239.20053
Full Text:

### References:

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