Check section 9.1.2 of these notes There I compute the volumes of real Grassmannians. A similar computation works in the complex case.

Update Using the description $\mathrm{Gr}\;(k, N)\cong U(N/U(k)\times U(N-k)$ and a bi-invariant metric on $U(N)$, this induces bi-invaraint metrics on $U(k),U(N-k)\subset U(n)$ and an invariant metric on $\mathrm{Gr}(k,N)$. The volume of $\mathrm{Gr}(k,N)$ with respect to this metric is

$$ {\rm vol} \mathrm{Gr}(k, N)= \frac{ {\rm vol}\; U(N)}{{\rm vol}\; U(k)\cdot {\rm vol}\; U(N-k)}. $$

The volume of a compact Lie group $G$ with respect to a bi-invariant metric $g$ weas computed by I.G. Macdonald,

The volume of a compact Lie group, Invent. Math. 56(1980), no. 2, 93–95.

For the Lie group $U(n)$ this takes the form

$$ {\rm vol}\; U(n)=\frac{1}{(2P_n)^2(2\pi)^n}\times {\rm vol}\; T^n\times \prod_{k=1}^n {\rm vol}\;S^{2k-1}, $$

where ${\rm vol}\; T^n$ denotes the volume of the maximal torus of $U(n)$ equipped with the induced bi-invariant metric, and $P_n$ is the product of the lengths of the positive roots of $U(n)$.

Check section 9.1.2 of these notes There I compute the volumes of real Grassmannians. A similar computation works in the complex case.

Update Using the description $\mathrm{Gr}\;(k, N)\cong U(N/U(k)\times U(N-k)$ and a bi-invariant metric on $U(N)$, this induces bi-invaraint metrics on $U(k),U(N-k)\subset U(n)$ and an invariant metric on $\mathrm{Gr}(k,N)$. The volume of $\mathrm{Gr}(k,N)$ with respect to this metric is

$$ {\rm vol} \mathrm{Gr}(k, N)= \frac{ {\rm vol}\; U(N)}{{\rm vol}\; U(k)\cdot {\rm vol}\; U(N-k)}. $$

The volume of a compact Lie group $G$ with respect to a bi-invariant metric $g$ was computed by I.G. Macdonald,

The volume of a compact Lie group, Invent. Math. 56(1980), no. 2, 93–95.

For the Lie group $U(n)$ this takes the form

$$ {\rm vol}\; U(n)=\frac{1}{(2P_n)^2(2\pi)^n}\times {\rm vol}\; T^n\times \prod_{k=1}^n {\rm vol}\;S^{2k-1}, $$

where ${\rm vol}\; T^n$ denotes the volume of the maximal torus of $U(n)$ equipped with the induced bi-invariant metric, and $P_n$ is the product of the lengths of the positive roots of $U(n)$.

Check section 9.1.2 of these notes There I compute the volumes of real Grassmannians. A similar computation works in the complex case.

Check section 9.1.2 of these notes There I compute the volumes of real Grassmannians. A similar computation works in the complex case.

Update Using the description $\mathrm{Gr}\;(k, N)\cong U(N/U(k)\times U(N-k)$ and a bi-invariant metric on $U(N)$, this induces bi-invaraint metrics on $U(k),U(N-k)\subset U(n)$ and an invariant metric on $\mathrm{Gr}(k,N)$. The volume of $\mathrm{Gr}(k,N)$ with respect to this metric is

$$ {\rm vol} \mathrm{Gr}(k, N)= \frac{ {\rm vol}\; U(N)}{{\rm vol}\; U(k)\cdot {\rm vol}\; U(N-k)}. $$

The volume of a compact Lie group $G$ with respect to a bi-invariant metric $g$ weas computed by I.G. Macdonald,

The volume of a compact Lie group, Invent. Math. 56(1980), no. 2, 93–95.

For the Lie group $U(n)$ this takes the form

$$ {\rm vol}\; U(n)=\frac{1}{(2P_n)^2(2\pi)^n}\times {\rm vol}\; T^n\times \prod_{k=1}^n {\rm vol}\;S^{2k-1}, $$

where ${\rm vol}\; T^n$ denotes the volume of the maximal torus of $U(n)$ equipped with the induced bi-invariant metric, and $P_n$ is the product of the lengths of the positive roots of $U(n)$.