Let \({\mathcal O}\) be the ring of integers of a \(p\)-adic field \(K\). Consider the \(\text{GL}_l\)-action on \(\text{Alt}(l)\) given by \(h\cdot y= hy^th\), where \(\text{Alt}(l)= \{y\in M(l)\mid^t y= -y\}\).
It is known that \[ \{y\in \text{Alt}(l)\mid\text{of maximal rank}\}= \bigcup_{\lambda\in \Lambda^+_m} \text{GL}_l({\mathcal O})\cdot(\pi^\lambda)_l, \] where \(m= [{l\over 2}]\), \(\lambda^+_m= \{\lambda\in \mathbb{Z}^m\mid\lambda_1\geq\cdots\geq \lambda_m\geq 0\}\), and \((\pi^\lambda)_l\in \text{Alt}(l)\) is the standard element attached to \(\lambda\) with respect to a prime element \(\pi\) of \(K\).
The author shows the following identity for any \(\mathbb{C}\)-valued continuous function \(F\) on \(\text{Alt}(l;{\mathcal O})\): \[ \int_{M_{2n,l}({\mathcal O})}F(^t xJ_nx)\,dx= \sum_{\lambda\in \Lambda^+_m} A_{\lambda,l}\cdot \int_{\text{GL}_l({\mathcal O})\cdot(\pi^\lambda)_l} F(y)\,dy, \] with explicitly given constants \(A_{\lambda,l}\).
As an application, the author gives explicit expressions of \(p\)-adic local zeta functions over \(K\) of certain prehomogeneous vector spaces in which \(\text{Sp}(n)\) appear.