The non-degeneracy of the bilinear form of m-Quasi-Invariants
- 作者：A.M. Garsia and N. Wallach
- 刊名：Advances in Applied Mathematics
- 出版年：2006
- 期刊代码：61_01968858
- 类别：mt
- 出版时间：September 2006
- 卷：37
- 期：3
- 页码：309-359
- 文件大小：340 K
摘要
We give here a new proof of the non-degeneracy of the fundamental bilinear form for me="mml1">method=retrieve&_udi=B6W9D-4K5HW64-1&_mathId=mml1&_user=10&_cdi=6680&_rdoc=5&_handle=V-WA-A-W-E-MsSAYZA-UUW-U-AAZCBUYAUV-AAZWYYYEUV-VBBDVADY-E-U&_acct=C000050221&_version=1&_userid=10&md5=ac1b5686e0b26a74ada5cbdae0fc39fe" title="Click to view the MathML source">Sn-m-Quasi-Invariants and for m-Quasi-Invariants of classical Weyl groups. We also indicate how our approach can be extended to other Coxeter groups. This bilinear form plays a crucial role in the original proof [P. Etingof, V. Ginzburg, On m-quasi-invariants of a Coxeter group, arXiv: math.QA/0106175 v1, June 2001] that m-Quasi-Invariants are a free module over the invariants as well as in all subsequent proofs [Y. Berest, P. Etingof, V. Ginsburg, Cherednik algebras and differential operators on quasi-invariants, math.QA/0111005; A. Garsia, N. Wallach, Some new applications of orbit harmonics, Sém. Lothar. Combin. 50 (2005), Article B50j]. However, in previous literature this non-degeneracy was stated and used without proof with reference to some deep results of Opdam [E.M. Opdam, Some applications of shift operators, Invent. Math. 98 (1989) 1–18] on shift-differential operators. This result hinges on the validity of a deceptively simple identity on Dunkl operators which, at least in the me="mml2">method=retrieve&_udi=B6W9D-4K5HW64-1&_mathId=mml2&_user=10&_cdi=6680&_rdoc=5&_handle=V-WA-A-W-E-MsSAYZA-UUW-U-AAZCBUYAUV-AAZWYYYEUV-VBBDVADY-E-U&_acct=C000050221&_version=1&_userid=10&md5=dac01759922a64dbce755d814c6a5903" title="Click to view the MathML source">Sn case, begs for an elementary painless proof. An elementary but by all means not painless proof of this identity can be found in a paper of Dunkl and Hanlon [C. Dunkl, P. Hanlon, Integrals of polynomials associated with tableaux and the Garsia–Haiman conjecture, Math. Z. 228 (1998) 537–567. 71]. Our proof here is not elementary but hopefully it should be painless and informative.