st.statistics - Has the Lie group preserving a probability distribution been used in Bayesian statistics?

For a (possibly signed) nondegenerate probability measure $pi$ on ${1,dots,n}$ define
$$langle pi rangle := {R in operatorname{STO}(n): pi R = pi }.$$
Here $operatorname{STO}(n)$ denotes the group of invertible stochastic matrices, i.e., the set of matrices (with entries of either sign) whose row sums are unity. Also, write $langle R rangle equiv pi$ for $R in langle pi rangle$. $langle pi rangle$ is a $(n-1)^2$-dimensional Lie group under matrix multiplication. Its Lie algebra $mathfrak{lie}(langle pi rangle)$ has a basis of the form
$$e_{(j,k)}^{langle pi rangle} := e_{(j,k)} - frac{pi_j}{pi_n}e_{(n,k)}, quad 1 le j, k le n-1$$
where $e_{(j,k)} = e_j(e_k^* - e_n^*)$ and we use typical notation for the standard basis of $mathbb{R}^n$. A block matrix decomposition shows that $mathfrak{lie}(langle pi rangle) cong mathfrak{gl}(n-1)$, and it is easy to show that
$$exp Ce_{(j,k)}^{langle pi rangle} = I + frac{e^{C(pi_j/pi_n + delta_{jk})} - 1}{pi_j/pi_n + delta_{jk}} e_{(j,k)}^{langle pi rangle}.$$
In particular, this object seems pretty tractable (although I haven't bothered to think about the Haar measure).

My question is, has it been used in
Bayesian statistics (or elsewhere)? It seems
naturally suited for such an
application.