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Help for literature on entrywise invariant kernels

I am looking for literature on entrywise invariant kernels. The specific example I have in mind is $K:R^{d}\times R^{d}\to R$ and locally compact groups acting on vector space $R^{d}$. … More precisely I am looking to find properties of the eigenvectors and eigenvalues of kernels such that for any elements of the group $g,h\in R^{d\times d}$ I have $K(gx,hy)=K(x,y)$. Any suggestion? …
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Green function of the triangular kernel?

What is the green function of the triangular kernel $K$: $$ K(x,y)=1-|x-y| $$ where $x,y\in R$ such that $|x-y|<1$?
Fabio's user avatar
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