Let $\mathcal{O}\_\lambda$ be the set of hermitian $n+1 \times n+1$ matrices with Eigenvalues $\lambda = (\lambda\_1, \dots, \lambda\_{n+1})$.
and $\mathcal{O}^\mu$ the set of hermitian $n \times n$ matrices with Eigenvalues $\mu = (\mu\_1, \dots, \mu\_n)$.


Let $\pi \colon \mathcal{O}\_\lambda \to \mathbb{C}^{n\times n}, \ 
\begin{pmatrix}
a\_0 & \alpha^T \\
\overline{\alpha} & A
\end{pmatrix} \mapsto A$ be the projection on the lower right "corner".


How could I prove that $\pi^{-1}(\mathcal{O}^\mu)$ is a submanifold of $\mathcal{O}\_\lambda$ and that $(d\_x\pi)^{-1}T\_{\pi(x)}\mathcal{O}^\mu=T\_x (\pi^{-1}(\mathcal{O}^\mu))$?


My idea was to show that the map, assigning each hermitian matrix the corresponding eigenvalues in non-decreasing order, is a differentiable map and is a submersion. But I'm not quite sure if thats true or how to prove this.


Edit: Thanks to Robert Bryant we now know, that the condition for the tangentspaces doesn't have to be fulfilled, if $\lambda\_j < \lambda\_{j+1}$ and $\mu\_j = \lambda\_j$ or $\mu\_j = \lambda\_{j+1}$ for some $j$. But is it fufilled, if we look only at the generic cases, where for $\lambda\_j < \lambda\_{j+1}$ we have $\lambda\_j < \mu\_j < \lambda\_{j+1}$?



subset of hermitian matrices given by eigenvalues form a submanifold
The set $\pi^{-1}(\mathcal{O}^\mu)$ is always a smooth submanifold of $\mathcal{O}\_\lambda$ (though it may well be empty). When it is not empty, it is a single orbit of $\mathrm{U}(n)\subset \mathrm{U}(n{+}1)$, where $\mathrm{U}(n)$ is the subgroup of $\mathrm{U}(n{+}1)$ consisting of those unitary $(n{+}1)$-by-$(n{+}1)$ matrices whose top row is $(1,0,\ldots,0)$.


To see this, note that any element $B\in\pi^{-1}(\mathcal{O}^\mu)$ can be written in the form
$$
B = \begin{pmatrix}a\_0& v^T\bar P^T\\ Pv & PM\bar P^T\end{pmatrix}
= \begin{pmatrix}1& 0\\ 0 & P\end{pmatrix} \begin{pmatrix}a\_0& v^T\\ v & M\end{pmatrix} \begin{pmatrix}1& 0\\ 0 & \bar P^T\end{pmatrix}
$$
where $P\ \bar P^T = I\_n$ and where $M = \mathrm{diag}(\mu\_1,\ldots,\mu\_n)$ and $v$ lies in $\mathbb{R}^n$ and has nonnegative real entries. Writing
$$
v = \begin{pmatrix} \sqrt{a\_1}\\ \vdots \\ \sqrt{a\_n}\end{pmatrix},
$$
where $a\_i\ge 0$, we can now compute the characteristic polynomial of $B$ and set it equal to $(\lambda\_0-t)(\lambda\_1-t)\cdots(\lambda\_n-t)$ to arrive at the equation
$$
(a\_0{-}t) - \frac{a\_1}{\mu\_1{-}t} - \cdots - \frac{a\_n}{\mu\_n{-}t}
= \frac{(\lambda\_0-t)(\lambda\_1-t)\cdots(\lambda\_n-t)}{(\mu\_1-t)\cdots(\mu\_n-t)}.
$$
Note that it follows from this that, if $\mu\_i$ occurs with multiplicity $r\_i$ in the list $(\mu\_1,\ldots,\mu\_n)$ then it must occur with multiplicity at least $r\_i-1$ in the list $(\lambda\_0,\ldots,\lambda\_n)$. For example, suppose that $\mu\_1 = \mu\_2 = \cdots \mu\_{r\_1}$ with $\mu\_i\not=\mu\_1$ for $i>r\_1$ and number the $\lambda\_i$ so that $\lambda\_k = \mu\_1$ for $0\le k\le r\_1-2$. Then we find that the above equation implies
$$
a\_1 + a\_2 + \cdots + a\_{r\_1} = 
-\frac{(\lambda\_{r\_1-1}-\mu\_1)(\lambda\_{r\_1}-\mu\_1)\cdots(\lambda\_n-\mu\_1)}{(\mu\_{r\_1+1}-\mu\_1)\cdots(\mu\_n-\mu\_1)}.
\tag 1
$$
Of course, the right hand side of this equation must be nonnegative, or there is no solution. In particular, $v$ is a sum of eigenvectors of $M$ and the norm of the eigenvector component of $v$ with eigenvalue $\mu\_i$ is uniquely determined by the above equation. Of course, we also have 
$$
a\_0 = \lambda\_0 + \cdots + \lambda\_n - \mu\_1 - \cdots - \mu\_n\ .
$$
Thus, $B$ lies in a $\mathrm{U}(n)$-orbit of a single element of $\pi^{-1}(\mathcal{O}^\mu)$, as claimed. Since $\pi^{-1}(\mathcal{O}^\mu)$ is a single $\mathrm{U}(n)$-orbit, it is a smooth submanifold of $\mathcal{O}\_\lambda$ (when it is non-empty). 


Moreover, the map $\pi:\pi^{-1}(\mathcal{O}^\mu)\to \mathcal{O}^\mu$ is a smooth submersion when $\pi^{-1}(\mathcal{O}^\mu)$ is not empty (i.e., when the multiplicity conditions and inequalities mentioned above are satisfied), and the fibers are products of spheres $S^{2r\_i-1}$ for each eigenvalue $\mu\_i$ of multiplicity $r\_i\ge 1$ for which the corresponding expression on the right hand side of the above equation $(1)$ is positive.


As for the characterization of the tangent space that the OP wanted, that does not always hold. It suffices to consider $n=1$ and the case that $(\lambda\_0,\lambda\_1) = (1,2)$, when the manifold $\pi^{-1}(\mathcal{O}^\mu)$ is empty unless $1\le \mu\_1\le 2$, and is a point when $\mu\_1 = 1$ or $\mu\_1 = 2$, but is a circle when $1 < \mu\_1 < 2$. The manifold $\mathcal{O}\_\lambda$ is a $2$-sphere in this case and the map $\pi:\mathcal{O}\_\lambda\to \mathbb{R}$ is the projection onto the interval $[1,2]$, so the desired tangent characterization fails when $\mu\_1 = 1$ or $\mu\_1 = 2$.


