$(S,E)$ runs over the symplectic vector space endowed with a positive polarization, is the unique collection of continuous map satisfying (\[eq:ind2d\]) for any two dimensional space and (\[eq:indprod\]) for any product.

Unitary group of $E$
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The subgroup of ${\operatorname{Sp}}(S)$ consisting of the elements commuting with $j$ is isomorphic with the unitary group of $E$. The isomorphism is the map $\iota $ sending $h \in U(E)$ to $g \in {\operatorname{Sp}}(S)$ whose complexification acts as $$g ( x ) = \begin{cases} h (x) \text{ if } x \in E \\ \overline{h
   } (x) \text{ if } x \in \overline E
   \end{cases}$$ Denote by $U_*(E)$ the subset of $U(E)$ consisting of the $h$ such that $h -id $ is invertible. Obviously $h \in U_*(E)$ iff $\iota (h) \in
{\operatorname{Sp}}_* (S)$. Next lemma will be used to compare our trace estimates with the holomorphic Lefschetz fixed point formula.

\[lem:indexholom\] For any $h \in U_*(E)$, we have $$\operatorname{det}^{1/2} \bigl( \tfrac{1}{2} {\operatorname{id}}+  i A ( \iota ( h)
)  \bigr) =  \frac{1}{
  \det ( {\operatorname{id}}- h^{-1})} .$$

Let $(e_i)$ be an orthonormal basis of $E$ diagonalising $h$. Then the matrix of $A(g)$ in the base $(e_1, \ldots , e_n, \overline{e}_1,
\ldots , \overline{e}_n)$ is given 