MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

## Who can tell me the properties for the delta function in Sato’s theory?

In Sato's theory, the following formal delta function is defined:

$\delta(\lambda,z)=\frac{1}{\lambda}\sum_{n=-\infty}^\infty(\frac{z}{\lambda})^n=\frac{1}{z}\frac{1}{1-\lambda/z}+\frac{1}{\lambda}\frac{1}{1-z/\lambda}$

Given a function $f(z)=\sum a_iz^i$,

$f(\lambda)\delta(\lambda,z)=f(z)\delta(\lambda,z)$.

I want to know the properties as many as possible. Or useful references are welcome to be provided. Thanks!

-

The formal delta function obeys the usual properties that the Dirac delta function does, but relative to the pairing defined by the residue. For instance, $$\mathrm{Res}_z f(z)\delta(z,w) = f(w)$$ for any formal distribution $f(z)$.

This and more can be found in Kac's Vertex algebras for beginners, particularly Proposition 2.1 in §2.1.

-
thanks for the answer, which is very helpful. I want to know whether the following relation is right, $\delta(-\lambda,z+\mu-\lambda)=???\delta(-\mu,z)$ Thanks again. – Jack Cheng May 6 2010 at 12:29
I think so. Although formally $\delta(x,y)$ is not a "function" of $x-y$, for all practical purposes it is. – José Figueroa-O'Farrill May 6 2010 at 13:25

In addition to Kac's excellent book I thought I'd mention Vertex algebras and algebraic curves by Ed Frenkel and me, which goes a little more into formal delta functions, D-modules and the like --- see section 1.1 and chapter 19 (2nd edition) in particular.

-
 thanks. It's very helpful. – Jack Cheng May 7 2010 at 1:42
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8288970589637756, "perplexity": 554.5525168750225}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368709379061/warc/CC-MAIN-20130516130259-00051-ip-10-60-113-184.ec2.internal.warc.gz"}