The uniform distribution on a Stiefel manifold is a matrix-variate distribution that plays an important role in multivariate statistics. There one often encounters integrals over the orthogonal group or over the Stiefel manifold with respect to an invariant measure. For example, this distribution arises in the study of the functional determinant under transformations involving orthogonal or semi-orthogonal matrices. The uniform distribution on the Stiefel manifold corresponds to the normalized Haar measure on the Stiefel manifold.
A random matrix uniformly distributed on the Stiefel manifold is invariant under the two-sided group action of the product formula_1 of orthogonal groups, i.e. formula_2 for all formula_3 and formula_4.
Uniform Distribution on a Stiefel Manifold.
Introduction.
Let formula_5 be the Stiefel manifold, i.e., the set of all orthonormal formula_6-frames in formula_7 for formula_8. This manifold can also be represented as the matrix set 
formula_9. 
The Stiefel manifold is homeomorphic to the quotient space of the orthogonal groups 
formula_10 These two can be identified, and in the case formula_11 we obtain the full orthogonal group. The Stiefel manifold inherits the left group action 
formula_12
Here, formula_13 is a compact, closed Lie subgroup of formula_14. By Haar's theorem there exists a Haar measure on formula_14 which induces an invariant measure on the quotient space formula_16.
Derivation of the Haar Measure on the Stiefel Manifold.
Let formula_17. Differentiating formula_18 yields: formula_19 Let formula_20 be the columns of formula_21. The exterior product of the superdiagonal elements defines a differential form
formula_22
of degree formula_23. This form is invariant under both left and right group actions of the orthogonal group. Integration of this form gives the Haar measure on formula_14.
Let formula_25 be an element of the Stiefel manifold with the form formula_26. We extend this to an orthogonal matrix formula_27 by choosing formula_28. The induced differential form on the Stiefel manifold is
formula_29
and of maximal degree formula_30.
This differential form is independent of the specific choice of formula_31 and remains invariant under the left and right actions of the orthogonal group.
Integration of the Haar Measure.
It can be shown that integration with respect to the invariant measure over the Stiefel manifold satisfies the recursion:
formula_32 
where formula_33 denotes the invariant measure on formula_34.
This leads to the formula 
formula_35 
where formula_36 is the multivariate gamma function.
Uniform Distribution on the Stiefel Manifold.
The uniform distribution is the unique Haar probability measure given by
formula_37 
where 
formula_38 
and the normalization constant is 
formula_39