Center of infinitesimal Cherednik algebras of 𝔤𝔩_{𝔫}

Tikaradze, Akaki

doi:10.1090/S1088-4165-10-00363-8

Center of infinitesimal Cherednik algebras of $\mathfrak {gl}_n$
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Represent. Theory 14 (2010), 1-8 Request permission

Abstract:

We show that the center of an infinitesimal Cherednik algebra of $\mathfrak {gl}_n$ is isomorphic to the polynomial algebra of $n$ variables. As consequences of this fact, we show that an analog of Duflo’s theorem holds and all objects in the category $\mathcal {O}$ have finite length.

References

K. Brown and K. Changtong, Symplectic reflection algebras in positive characteristic, arxiv:0709.2338 (2007).
K. A. Brown and K. R. Goodearl, Homological aspects of Noetherian PI Hopf algebras of irreducible modules and maximal dimension, J. Algebra 198 (1997), no. 1, 240–265. MR 1482982, DOI 10.1006/jabr.1997.7109
Pavel Etingof and Victor Ginzburg, Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism, Invent. Math. 147 (2002), no. 2, 243–348. MR 1881922, DOI 10.1007/s002220100171
Pavel Etingof, Wee Liang Gan, and Victor Ginzburg, Continuous Hecke algebras, Transform. Groups 10 (2005), no. 3-4, 423–447. MR 2183119, DOI 10.1007/s00031-005-0404-2
Victor Ginzburg, On primitive ideals, Selecta Math. (N.S.) 9 (2003), no. 3, 379–407. MR 2006573, DOI 10.1007/s00029-003-0338-2
Dmitri I. Panyushev, On the coadjoint representation of $\Bbb Z_2$-contractions of reductive Lie algebras, Adv. Math. 213 (2007), no. 1, 380–404. MR 2331248, DOI 10.1016/j.aim.2006.12.011
A. Khare and A. Tikaradze, Center and representations of infinitesimal Hecke algebras of $\mathfrak {sl}_2$, arxiv:0807.4776 (2008).
Mustapha Raïs, La représentation coadjointe du groupe affine, Ann. Inst. Fourier (Grenoble) 28 (1978), no. 1, xi, 207–237 (French, with English summary). MR 500922
M. Rais, Les invariants polynômes de la représentation coadjointe de groupes inhomogénes, arxiv:0903.5146 (2009).
S. P. Smith, A class of algebras similar to the enveloping algebra of $\textrm {sl}(2)$, Trans. Amer. Math. Soc. 322 (1990), no. 1, 285–314. MR 972706, DOI 10.1090/S0002-9947-1990-0972706-5
A. Tikaradze, Infinitesimal Cherednik algebras of $\mathfrak {gl}_2$, arxiv:0810.2001 (2008).