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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The space of invariant functions on a finite Lie algebra

Author: G. I. Lehrer
Journal: Trans. Amer. Math. Soc. 348 (1996), 31-50
MSC (1991): Primary 20G40, 20G05; Secondary 22E60, 11T24
MathSciNet review: 1322953
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Abstract: We show that the operations of Fourier transform and duality on the space of adjoint-invariant functions on a finite Lie algebra commute with each other. This result is applied to give formulae for the Fourier transform of a ``Brauer function''---i.e. one whose value at $X$ depends only on the semisimple part $X_s$ of $X$ and for the dual of the convolution of any function with the Steinberg function. Geometric applications include the evaluation of the characters of the Springer representations of Weyl groups and the study of the equivariant cohomology of local systems on $G/T$, where $T$ is a maximal torus of the underlying reductive group $G$.

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  • [B] A. Borel, Linear algebraic groups, Benjamin, N.Y. (1969), MR 40:4273.
  • [BT] A. Borel and J. Tits, Groupes réductifs, Publ. Math IHES 27 (1965), 55--152, MR 34:7527.
  • [C] C.W. Curtis, Truncation and duality in the character ring of a finite group of Lie type, J. of Algebra 62 (1980), 320--332, MR 81e:20011.
  • [CLT] C.W. Curtis, G.I. Lehrer and J. Tits, Spherical buildings and the character of the Steinberg representation, Inventiones Math. 58 (1980), 201--210, MR 81f:20060.
  • [D] P. Deligne, Cohomologie Étale (SGA 4$\frac{1}{2}$), Springer L.N.M., Springer-Verlag Berlin 509 (1977), MR 57:3132.
  • [DL] P. Deligne and G. Lusztig, Representations of reductive groups over finite fields, Ann. Math. 103 (1976), 103--161, MR 52:14076.
  • [DM] F. Digne and J. Michel, Representations of finite groups of Lie type, Cambridge U.P., Cambridge (1991), MR 92g:20063.
  • [HL] R.B. Howlett and G.I. Lehrer, On Harish-Chandra induction and restriction for modules of Levi subgroups, J. of Algebra 165 (1994), 172--183, MR 95d:20025.
  • [K] N. Kawanaka, Fourier transforms of nilpotently supported invariant functions on a simple Lie algebra over a finite field, Inventiones Math. 69 (1982), 411--435, MR 84c:20053.
  • [K2] N. Kawanaka, Generalized Gelfand-Graev representations and Ennola duality, Adv. St. Pure Math. 6 (1985), 175--206, MR 87e:20075.
  • [L1] G.I. Lehrer, Rational tori, semisimple orbits and the topology of hyperplane complements, Commentarii Math. Helv. 67 (1992), 226--251, MR 93e:20065.
  • [L2] G.I. Lehrer, Characters and the Jordan decomposition in reductive groups over finite fields, U. Warwick preprint series (1980).
  • [Lu1] G. Lusztig, A unipotent support for irreducible representations, Advances in Math. 94 (1992), 139--179, MR 94a:20073.
  • [Lu2] G. Lusztig, Fourier transforms on a semisimple Lie algebra over $\mathbb{F} _q$, in ``Algebraic groups Utrecht 1986'', Springer, MR 89b:17015.
  • [MS] J.G.M. Mars and T.A. Springer, Character sheaves, Soc. Math. France Astérisque 173-174 (1989), 111--198, MR 91a:20044.
  • [Sh] T. Shoji, Geometry of orbits and Springer correspondence, Soc. Math. France Astérisque
    168 (1988), 61--140, MR 91b:20057.
  • [Sp1] T.A. Springer, The Steinberg function of a finite Lie algebra, Inventiones Math. 58 (1980), 211--215, MR 81g:20090.
  • [Sp2] T.A. Springer, Trigonometric sums, Green functions of finite groups and representations of Weyl Groups, Inventiones Math. 36 (1976), 173--207, MR 56:491.
  • [Sp2] T.A. Springer, The unipotent variety of a semisimple group, Proc. Coll. Alg. Geom., Tata Institute (1969), 373--391, MR 41:8429.
  • [Sr] B. Srinivasan, On the Steinberg character of a finite simple group of Lie type, J. Aust. Math. Soc. 12 (1977), 1--14, MR 45:411.

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Additional Information

G. I. Lehrer
Affiliation: address School of Mathematics and Statistics, University of Sydney, Sydney N.S.W. 2006, Australia

Received by editor(s): February 15, 1994
Article copyright: © Copyright 1996 American Mathematical Society

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