##
**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)].

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 |

### Keywords:

fixed point varieties on affine flag manifolds; simply connected semisimple algebraic group; variety of Borel subalgebras; Iwahori subalgebras; projective algebraic varieties; nilpotent orbits
PDF
BibTeX
XML
Cite

\textit{D. Kazhdan} and \textit{G. Lusztig}, Isr. J. Math. 62, No. 2--3, 129--168 (1988; Zbl 0658.22005)

Full Text:
DOI

### References:

[1] | N. Bourbaki,Groupes et algèbres de Lie, Ch. IV, V, VI, Hermann, Paris, 1968. · Zbl 0186.33001 |

[2] | P. Bala and R. W. Carter,Classes of unipotent elements in simple algebraic groups, Math. Proc. Camb. Phil. Soc.79 (1976), 401–425 and80 (1976), 1–18. · Zbl 0364.22006 |

[3] | J. Bernstein, P. Deligne, D. Kazhdan and M-F. Vigneras,Représentations des groups réductifs sur un corps local, Hermann, Paris, 1984. |

[4] | F. Bruhat and J. Tits,Groupes réductifs sur un corps local, I, Publ. Math. IHES,41 (1972), 5–251. |

[5] | R. W. Carter and G. B. Elkington,A note on the parametrization of conjugacy classes, J. Alg.20 (1972), 350–354. · Zbl 0239.20053 |

[6] | L. Corwin and R. Howe,Computing characters of tamely ramified p-adic division algebras, Pac. J. Math.73 (1977), 461–477. · Zbl 0385.22008 |

[7] | C. De Concini, G. Lusztig and C. Procesi,Homology of the zero-set of a nilpotent vector field on a flag manifold, J. Am. Math. Soc.1 (1988). · Zbl 0646.14034 |

[8] | M. Demazure and P. Gabriel,Groupes algébriques, North-Holland, Amsterdam, 1970. · Zbl 0203.23401 |

[9] | P. Deligne and G. Lusztig,Representations of reductive groups over finite fields. Ann. of Math.103 (1976), 103–161. · Zbl 0336.20029 |

[10] | Harish-Chandra,Harmonic Analysis on Reductive p-adic Groups, Lecture Notes in Mathematics162, Springer-Verlag, Berlin, 1970. · Zbl 0202.41101 |

[11] | D. Kazhdan,Proof of Springer’s hypothesis, Isr. J. Math.28 (1977), 272–286. · Zbl 0391.22006 |

[12] | V. Kac,Constructing groups associated to infinite-dimensional algebras, inInfinite Dimensional Groups with Applications, MSRI Publications, Springer-Verlag, Berlin, 1985. · Zbl 0614.22006 |

[13] | D. Kazhdan and G. Lusztig,A topological approach to Springer’s representations, Adv. Math.38 (1980), 222–228. · Zbl 0458.20035 |

[14] | B. Kostant,The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group, Am. J. Math.81 (1959), 973–1032. · Zbl 0099.25603 |

[15] | G. Lusztig,Singularities, characters, formulas and a q-analog of weight multiplicities, Astérisque101–102 (1983), 208–229. · Zbl 0561.22013 |

[16] | G. Lusztig and N. Spaltenstein,Induced unipotent classes, J. London Math. Soc.19 (1979), 41–52. · Zbl 0407.20035 |

[17] | J.-P. Serre,Corps locaux, Hermann, Paris, 1962. · Zbl 0137.02601 |

[18] | J.-P. Serre,Cohomologie galoisienne, Lecture Notes in Math.5, Springer-Verlag, Berlin, 1965. · Zbl 0136.02801 |

[19] | T. A. Springer,Regular elements of finite reflection groups, Invent. Math.25 (1974), 159–198. · Zbl 0287.20043 |

[20] | T. A. Springer,A construction of representations of Weyl groups, Invent. Math.44 (1978), 278–293. · Zbl 0376.17002 |

[21] | T. A. Springer,Generalization of Green’s polynomials, in Proc. Symp. in Pure Math., Vol. 21, Am. Math. Soc., Providence, 1971, 149–154. · Zbl 0247.20049 |

[22] | N. Spaltenstein,On the fixed point set of a unipotent element on the variety of Borel subgroups, Topology16 (1977), 203–204. · Zbl 0445.20021 |

[23] | R. Steinberg,Conjugacy classes in algebraic groups, Lecture Notes in Math.366, Springer-Verlag, Berlin, 1974. · Zbl 0281.20037 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.