Abstract
In this article, by studying the Bernstein degrees and Goldie rank polynomials, we establish a comparison between the irreducible representations of G = GLn(ℂ) possessing the minimal Gelfand–Kirillov dimension and those induced from finite-dimensional representations of the maximal parabolic subgroup of G of type (n − 1,1). We give the transition matrix between the two bases for the corresponding coherent families.
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Alesker, S.: The α-cosine transform and intertwining integrals on real Grassmannians, In: Geometric Aspects of Functional Analysis. Lecture Notes in Math., Vol. 2050, Springer, Heidelberg, 2012, 1–21
Barbasch, D.: Orbital integrals of nilpotent orbits, In: The Mathematical Legacy of Harish-Chandra (Baltimore, MD, 1998). Proc. Sympos. Pure Math., Vol. 68, Amer. Math. Soc., Providence, RI, 2000, 97–110
Barbasch, D., Vogan, D.: Primitive ideals and orbital integrals in complex classical groups. Math. Ann., 259(2), 153–199 (1982)
Barbasch, D., Vogan, D.: Unipotent representations of complex semisimple groups. Ann. of Math. (2), 121(1), 41–110 (1985)
Beilinson, A., Bernstein, J.: Localisation de \(\mathfrak{g}\)-modules. C. R. Acad. Sci. Paris Sér. I Math., 292(1), 15–18 (1981)
Björner, A., Brenti, F.: Combinatorics of Coxeter groups, Graduate Texts in Mathematics, Vol. 231, Springer, New York, 2005
Brenti, F.: Kazhdan–Lusztig and R-polynomials, Young’s lattice, and Dyck partitions. Pacific J. Math., 207(2), 257–286 (2002)
Brundan, J.: Moeglin’s theorem and Goldie rank polynomials in Cartan type A. Compos. Math., 147(6), 1741–1771 (2011)
Brylinski, J.-L., Kashiwara, M.: Kazhdan–Lusztig conjecture and holonomic systems. Invent. Math., 64(3), 387–410 (1981)
Deodhar, V.: On some geometric aspects of Bruhat orderings. II. The parabolic analogue of Kazhdan–Lusztig polynomials. J. Algebra, 111(2), 483–506 (1987)
Douglass, J.: An inversion formula for relative Kazhdan–Lusztig polynomials. Comm. Algebra, 18(2), 371–387 (1990)
Gourevitch, D.: Composition series for degenerate principal series of GL(n). C. R. Math. Acad. Sci. Soc. R. Can., 39(1), 1–12 (2017)
Howe, R., Lee, S. T.: Degenerate principal series representations of GLn(ℂ)and GLn(ℝ). J. Funct. Anal., 166(2), 244–309 (1999)
Humphreys, J.: Representations of Semisimple Lie Algebras in the BGG Category \({\cal O}\), Graduate Studies in Math., Vol. 94, AMS, Providence, RI, 2008
Joseph, A.: On the annihilators of the simple subquotients of the principal series. Ann. Sci. école Norm. Sup., 10(4), 419–439 (1977)
Joseph, A.: Kostant’s problem, Goldie rank and the Gel’fand–Kirillov conjecture. Invent. Math., 56(3), 191–213 (1980)
Joseph, A.: Goldie rank in the enveloping algebra of a semisimple Lie algebra. I. J. Algebra, 65(2), 269–283 (1980)
Kazhdan, D., Lusztig, G.: Representations of Coxeter groups and Hecke algebras. Invent. Math., 53(2), 165–184 (1979)
Sagan, B. E.: The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions, Graduate Texts in Mathematics, Vol. 203, Springer, New York, 2001
Speh, B., Vogan, D.: Reducibility of generalized principal series representations. Acta Math., 145(3–4), 227–299 (1980)
Vogan, D.: Gelfand–Kirillov dimension for Harish-Chandra modules. Invent. Math., 48(1), 75–98 (1978)
Vogan, D.: Irreducible characters of semisimple Lie groups. I. Duke Math. J., 46(1), 61–108 (1979)
Vogan, D.: The algebraic structure of the representation of semisimple Lie groups I. Ann. of Math., 109, 1–60 (1979)
Vogan, D.: Representations of Real Reductive Lie Groups. Progress in Math., Vol. 15, Birkhauser, Boston, Mass., 1981
Vogan, D.: Irreducible characters of semisimple Lie groups. IV. Character-multiplicity duality. Duke Math. J., 49(4), 943–1073 (1982)
Xue, H.: Homogeneous distributions on finite dimensional vector spaces. J. Lie Theory, 28(1), 33–41 (2018)
Zhelobenko, D. P.: Harmonic Analysis on Semisimple Complex Lie Groups (Russian). Izdat. Nauka, Moscow, 240 pp. (1974)
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Bin Yong Sun is an editorial board member for Acta Mathematica Sinica English Series and was not involved in the editorial review or the decision to publish this article. All authors declare that there are no competing interests.
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Z. Bai was supported in part by the National Natural Science Foundation of China (Grant No. 12171344) and the National Key R& D Program of China (Grant Nos. 2018YFA0701700 and 2018YFA0701701); Y. Chen was supported in part by the National Natural Science Foundation of China (Grant No. 12301035) and the Natural Science Foundation of Jiangsu Province (Grant No. BK20221057); D. Liu was supported by National Key R&D Program of China (Grant No. 2022YFA1005300) and the National Natural Science Foundation of China (Grant No. 12171421); B. Sun was supported by National Key R&D Program of China (Grant Nos. 2022YFA1005300 and 2020YFA0712600) and New Cornerstone Investigator Program
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Bai, Z.Q., Chen, Y.Y., Liu, D.W. et al. Irreducible Representations of GLn(ℂ) of Minimal Gelfand–Kirillov Dimension. Acta. Math. Sin.-English Ser. 40, 639–657 (2024). https://doi.org/10.1007/s10114-024-3207-x
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DOI: https://doi.org/10.1007/s10114-024-3207-x