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Let $K$ be a CM-field with totally real subfield $F$. Let $(V_1, h_1)$ and $(V_2, h_2)$ be two $n$-dimensional $K$-vector spaces with nondegenerate Hermitian forms, where $n\geq 3$.

Question 1 Does every $F$-group isomorphism $PU(V_1, h_1)\cong PU(V_1, h_1)$ arise from an isometry of $(V_1, h_1)\cong (V_2, \lambda h_2)$ for some $\lambda\in F^{\times}$ ?

Edit. Question 2 Suppose there exists an $F$-group isomorphism $PU(V_1, h_1)\cong PU(V_1, h_1)$, can we deduce $(V_1, h_1)\cong (V_2, \lambda h_2)$ for some $\lambda\in F^{\times}$, or any other relations between $h_1$ and $h_2$?

A related statement for orthogonal group is given in the comment to the question $p$-adic orthogonal groups in four variables .

We are interested in the case that $K$ is a cyclotomic field and $h$ has signature $(1,n-1)$ for an embedding $F\to \mathbb{R}$ and is definite for all the other embeddings.

Let $K$ be a CM-field with totally real subfield $F$. Let $(V_1, h_1)$ and $(V_2, h_2)$ be two $n$-dimensional $K$-vector spaces with nondegenerate Hermitian forms, where $n\geq 3$.

Question 1 Does every $F$-group isomorphism $PU(V_1, h_1)\cong PU(V_2, h_2)$ arise from an isometry of $(V_1, h_1)\cong (V_2, \lambda h_2)$ for some $\lambda\in F^{\times}$ ?

Edit. Question 2 Suppose there exists an $F$-group isomorphism $PU(V_1, h_1)\cong PU(V_2, h_2)$, can we deduce $(V_1, h_1)\cong (V_2, \lambda h_2)$ for some $\lambda\in F^{\times}$, or any other relations between $h_1$ and $h_2$?

A related statement for orthogonal group is given in the comment to the question $p$-adic orthogonal groups in four variables .

We are interested in the case that $K$ is a cyclotomic field and $h$ has signature $(1,n-1)$ for an embedding $F\to \mathbb{R}$ and is definite for all the other embeddings.

Let $K$ be a CM-field with totally real subfield $F$. Let $(V_1, h_1)$ and $(V_2, h_2)$ be two $n$-dimensional $K$-vector spaces with nondegenerate Hermitian forms, where $n\geq 3$.

Question. Does every $F$-group isomorphism $PU(V_1, h_1)\cong PU(V_1, h_1)$ arise from an isometry of $(V_1, h_1)\cong (V_2, \lambda h_2)$ for some $\lambda\in F^{\times}$ ?

A related statement for orthogonal group is given in the comment to the question $p$-adic orthogonal groups in four variables .

We are interested in the case that $K$ is a cyclotomic field and $h$ has signature $(1,n-1)$ an embedding $F\to \mathbb{R}$ and is definite for all the other embeddings.

Let $K$ be a CM-field with totally real subfield $F$. Let $(V_1, h_1)$ and $(V_2, h_2)$ be two $n$-dimensional $K$-vector spaces with nondegenerate Hermitian forms, where $n\geq 3$.

Question 1 Does every $F$-group isomorphism $PU(V_1, h_1)\cong PU(V_1, h_1)$ arise from an isometry of $(V_1, h_1)\cong (V_2, \lambda h_2)$ for some $\lambda\in F^{\times}$ ?

Edit. Question 2 Suppose there exists an $F$-group isomorphism $PU(V_1, h_1)\cong PU(V_1, h_1)$, can we deduce $(V_1, h_1)\cong (V_2, \lambda h_2)$ for some $\lambda\in F^{\times}$, or any other relations between $h_1$ and $h_2$?

A related statement for orthogonal group is given in the comment to the question $p$-adic orthogonal groups in four variables .

We are interested in the case that $K$ is a cyclotomic field and $h$ has signature $(1,n-1)$ for an embedding $F\to \mathbb{R}$ and is definite for all the other embeddings.

Let $K$ be a CM-field with totally real subfield $F$. Let $(V_1, h_1)$ and $(V_2, h_2)$ be two $n$-dimensional $K$-vector spaces with nondegenerate Hermitian forms, where $n\geq 3$.

Question. Does every $F$-group isomorphism $PU(V_1, h_1)\cong PU(V_1, h_1)$ arise from an isometry of $(V_1, h_1)\cong (V_2, \lambda h_2)$ for some $\lambda\in F^{\times}$ ?

A related statement for orthogonal group is given in the comment to the question $p$-adic orthogonal groups in four variables .

We are interested in the case that $K$ is a cyclotomic field and $h$ has signature $(1,n-1)$ an embedding $F\to \mathbb{C}$ and is definite for all the other embeddings.

Let $K$ be a CM-field with totally real subfield $F$. Let $(V_1, h_1)$ and $(V_2, h_2)$ be two $n$-dimensional $K$-vector spaces with nondegenerate Hermitian forms, where $n\geq 3$.

Question. Does every $F$-group isomorphism $PU(V_1, h_1)\cong PU(V_1, h_1)$ arise from an isometry of $(V_1, h_1)\cong (V_2, \lambda h_2)$ for some $\lambda\in F^{\times}$ ?

A related statement for orthogonal group is given in the comment to the question $p$-adic orthogonal groups in four variables .

We are interested in the case that $K$ is a cyclotomic field and $h$ has signature $(1,n-1)$ an embedding $F\to \mathbb{R}$ and is definite for all the other embeddings.