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On Lusztig’s asymptotic Hecke algebra for 𝑆𝐿₂
Proceedings of the American Mathematical Society ( IF 1 ) Pub Date : 2020-10-20 , DOI: 10.1090/proc/15259 Stefan Dawydiak
Proceedings of the American Mathematical Society ( IF 1 ) Pub Date : 2020-10-20 , DOI: 10.1090/proc/15259 Stefan Dawydiak
Abstract:Let 
be a split connected reductive algebraic group, let 
be the corresponding affine Hecke algebra, and let 
be the corresponding asymptotic Hecke algebra in the sense of Lusztig. When 
, and the parameter 
is specialized to a prime power, Braverman and Kazhdan showed recently that for generic values of 
, 
has codimension two as a subalgebra of 
, and described a basis for the quotient in spectral terms. In this note we write these functions explicitly in terms of the basis 
of 
, and further invert the canonical isomorphism between the completions of 
and 
, obtaining explicit formulas for each basis element 
in terms of the basis 
of 
. We conjecture some properties of this expansion for more general groups. We conclude by using our formulas to prove that 
acts on the Schwartz space of the basic affine space of 
, and produce some formulas for this action.
中文翻译：
关于Lusztig的𝑆𝐿2的渐近Hecke代数
摘要：在Lusztig的意义上，让它成为分裂连接的还原代数群，让其为对应的仿射Hecke代数，并让其为对应的渐近Hecke代数。当，并且参数专用于质数时，Braverman和Kazhdan最近证明，对于的泛型值，余维2为的子代数，并用频谱术语描述了商的基础。在这份说明中，我们在基础方面明确写入这些功能的，并进一步转化的完井之间的同构规范和，获得每个基础元素明确的公式在基础方面的 。我们为更一般的群体推测了这种扩展的某些性质。我们通过使用公式来证明对的基本仿射空间的Schwartz空间起作用，并为此得出一些公式。
更新日期：2020-10-20
be a split connected reductive algebraic group, let be the corresponding affine Hecke algebra, and let be the corresponding asymptotic Hecke algebra in the sense of Lusztig. When , and the parameter is specialized to a prime power, Braverman and Kazhdan showed recently that for generic values of , has codimension two as a subalgebra of , and described a basis for the quotient in spectral terms. In this note we write these functions explicitly in terms of the basis of , and further invert the canonical isomorphism between the completions of and , obtaining explicit formulas for each basis element in terms of the basis of . We conjecture some properties of this expansion for more general groups. We conclude by using our formulas to prove that acts on the Schwartz space of the basic affine space of , and produce some formulas for this action. 中文翻译：
关于Lusztig的𝑆𝐿2的渐近Hecke代数
摘要：在Lusztig的意义上
，让它成为分裂连接的还原代数群，让其为对应的仿射Hecke代数，并让其为对应的渐近Hecke代数。当，并且参数专用于质数时，Braverman和Kazhdan最近证明，对于的泛型值，余维2为的子代数，并用频谱术语描述了商的基础。在这份说明中，我们在基础方面明确写入这些功能的，并进一步转化的完井之间的同构规范和，获得每个基础元素明确的公式在基础方面的 。我们为更一般的群体推测了这种扩展的某些性质。我们通过使用公式来证明对的基本仿射空间的Schwartz空间起作用，并为此得出一些公式。 
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