Am I the last person to notice that for , the even moment

of a standard gaussian random variable (with expectation zero and variance one) is the same as the index of the Weyl group of inside the Weyl group of (in other words, the index of the groups of permutations of elements commuting with a fixed-point free involution among all permutations)?

If “Yes”, what else have I been missing in the same spirit?

I am having a hard time understanding what you mean by “in the same spirit.”

Well, it could be about other probability distributions whose moments (or some of them) have algebraic interpretations, or cases where this interpretation gives easy proofs of convergence to the normal law….

Are you looking for something like Theorem 1 in my paper “Averages over classical Lie groups, twisted by characters”

http://user.math.uzh.ch/dehaye/papers/Averages_over_classical_Lie_groups,_twisted_by_characters.pdf

?

The paper does the orthogonal case as well. (Most certainly there are other anterior references for what you asked, but this might be a starting point).