The space of invariant functions on a finite Lie algebra

Lehrer, G.

doi:10.1090/S0002-9947-96-01492-4

The space of invariant functions on a finite Lie algebra
HTML articles powered by AMS MathViewer

by G. I. Lehrer PDF

Trans. Amer. Math. Soc. 348 (1996), 31-50 Request permission

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 $G$.

References

Armand Borel, Linear algebraic groups, W. A. Benjamin, Inc., New York-Amsterdam, 1969. Notes taken by Hyman Bass. MR 0251042
Armand Borel and Jacques Tits, Groupes réductifs, Inst. Hautes Études Sci. Publ. Math. 27 (1965), 55–150 (French). MR 207712, DOI 10.1007/BF02684375
Charles W. Curtis, Truncation and duality in the character ring of a finite group of Lie type, J. Algebra 62 (1980), no. 2, 320–332. MR 563231, DOI 10.1016/0021-8693(80)90185-4
C. W. Curtis, G. I. Lehrer, and J. Tits, Spherical buildings and the character of the Steinberg representation, Invent. Math. 58 (1980), no. 3, 201–210. MR 571572, DOI 10.1007/BF01390251
P. Deligne, Cohomologie étale, Lecture Notes in Mathematics, vol. 569, Springer-Verlag, Berlin, 1977 (French). Séminaire de géométrie algébrique du Bois-Marie SGA $4\frac {1}{2}$. MR 463174, DOI 10.1007/BFb0091526
P. Deligne and G. Lusztig, Representations of reductive groups over finite fields, Ann. of Math. (2) 103 (1976), no. 1, 103–161. MR 393266, DOI 10.2307/1971021
François Digne and Jean Michel, Representations of finite groups of Lie type, London Mathematical Society Student Texts, vol. 21, Cambridge University Press, Cambridge, 1991. MR 1118841, DOI 10.1017/CBO9781139172417
R. B. Howlett and G. I. Lehrer, On Harish-Chandra induction and restriction for modules of Levi subgroups, J. Algebra 165 (1994), no. 1, 172–183. MR 1272585, DOI 10.1006/jabr.1994.1104
N. Kawanaka, Fourier transforms of nilpotently supported invariant functions on a simple Lie algebra over a finite field, Invent. Math. 69 (1982), no. 3, 411–435. MR 679766, DOI 10.1007/BF01389363
N. Kawanaka, Generalized Gel′fand-Graev representations and Ennola duality, Algebraic groups and related topics (Kyoto/Nagoya, 1983) Adv. Stud. Pure Math., vol. 6, North-Holland, Amsterdam, 1985, pp. 175–206. MR 803335, DOI 10.2969/aspm/00610175
G. I. Lehrer, Rational tori, semisimple orbits and the topology of hyperplane complements, Comment. Math. Helv. 67 (1992), no. 2, 226–251. MR 1161283, DOI 10.1007/BF02566498

Characters and the Jordan decomposition in reductive groups over finite fields

George Lusztig, A unipotent support for irreducible representations, Adv. Math. 94 (1992), no. 2, 139–179. MR 1174392, DOI 10.1016/0001-8708(92)90035-J
George Lusztig, Fourier transforms on a semisimple Lie algebra over $\textbf {F}_q$, Algebraic groups Utrecht 1986, Lecture Notes in Math., vol. 1271, Springer, Berlin, 1987, pp. 177–188. MR 911139, DOI 10.1007/BFb0079237
J. G. M. Mars and T. A. Springer, Character sheaves, Astérisque 173-174 (1989), 9, 111–198. Orbites unipotentes et représentations, III. MR 1021511
Toshiaki Shoji, Geometry of orbits and Springer correspondence, Astérisque 168 (1988), 9, 61–140. Orbites unipotentes et représentations, I. MR 1021493
T. A. Springer, The Steinberg function of a finite Lie algebra, Invent. Math. 58 (1980), no. 3, 211–215. MR 571573, DOI 10.1007/BF01390252
T. A. Springer, Trigonometric sums, Green functions of finite groups and representations of Weyl groups, Invent. Math. 36 (1976), 173–207. MR 442103, DOI 10.1007/BF01390009
T. A. Springer, The unipotent variety of a semi-simple group, Algebraic Geometry (Internat. Colloq., Tata Inst. Fund. Res., Bombay, 1968) Oxford Univ. Press, London, 1969, pp. 373–391. MR 0263830
Bhama Srinivasan, On the Steinberg character of a finite simple group of Lie type, J. Austral. Math. Soc. 12 (1971), 1–14. MR 0291317, DOI 10.1017/S1446788700008247