Harish-Chandra's Schwartz space

In mathematical abstract harmonic analysis, Harish-Chandra's Schwartz space is a space of functions on a semisimple Lie group whose derivatives are rapidly decreasing, studied by Harish-Chandra (1966, section 9). It is an analogue of the Schwartz space on a real vector space, and is used to define the space of tempered distributions on a semisimple Lie group.

Definition

The definition of the Schwartz space uses Harish-Chandra's Ξ function and his σ function. The σ function is defined by

[math]\displaystyle{ \sigma(x)=\|X\| }[/math]

for x=k exp X with k in K and X in p for a Cartan decomposition G = K exp p of the Lie group G, where ||X|| is a K-invariant Euclidean norm on p, usually chosen to be the Killing form. (Harish-Chandra 1966).

The Schwartz space on G consists roughly of the functions all of whose derivatives are rapidly decreasing compared to Ξ. More precisely, if G is connected then the Schwartz space consists of all smooth functions f on G such that

[math]\displaystyle{ \frac{(1+\sigma)^r|Df|}{\Xi} }[/math]

is bounded, where D is a product of left-invariant and right-invariant differential operators on G (Harish-Chandra 1966).

References

• Harish-Chandra (1966), "Discrete series for semisimple Lie groups. II. Explicit determination of the characters", Acta Mathematica 116: 1–111, doi:10.1007/BF02392813, ISSN 0001-5962, http://projecteuclid.org/euclid.acta/1485889477 
• Wallach, Nolan R (1988), Real reductive groups. I, Pure and Applied Mathematics, 132, Boston, MA: Academic Press, ISBN 978-0-12-732960-4, https://archive.org/details/realreductivegro0000wall