TITLE: An integral with respect to the Haar measure on a unitary group
QUESTION [6 upvotes]: Let $A,D\in \mathbb{C}^{n \times n}$ be diagonal matrices. I need to calculate 
$$\int_{U(n)}\det{(A-HDH^\dagger)}\,\mathrm{d}H$$ 
where $dH$ is the unit invariant Haar measure on the group of unitary matrices and $H^\dagger$ is the conjugate transpose of $H$. (If $A=I$ this is very easy to solve, but I want the answer for $A\neq I$ in terms of $A$ and $D$.)

REPLY [6 votes]: Let $A={\rm diag}[a_1,\dots,a_n]$ and $B={\rm diag}[b_1,\dots,b_n]$. Let $\Delta(a)$ be the Vandermonde product in the $a_j$, and similarly $\Delta(b)$ be the Vandermonde product in the $b_k$. Suppose ${\rm min} \, a_j \ge {\rm max} \, b_k$. Let ${\rm d}U$ denote the normalised Haar measure on the unitary group $U(n)$. Then by Eq. (3.21) in Gross and Richards, "Total positivity, spherical series, and hypergeometric functions of matrix argument", J. Approx. Th. vol. 59 (1989): 224-246,
$$
\int_U \det(A - U B U^\dagger)^p {\rm d}U = c_{n,p} {\det [ (a_j - b_k)^{p+n-1}]_{j,k=1}^n \over \Delta(a) \Delta(b)},
$$
where $c_{n,p} = \prod_{j=0}^{n-1} \binom{p+n-1}{j}^{-1}$. Setting $p=1$, subject to the condition ${\rm min} \, a_j \ge {\rm max} \, b_k$, this is the sought evaluation. See Theorem 2.3 in Kieburg, Kuijlaars and Stivigny, "Singular value statistics of matrix products with truncated unitary matrices", IMRN vol. 2016 (2016) 3392-3424 for a generalisation.