Yumiko Hironaka

Fix an unramified quadratic extension $k'/k$ of $p$-adic fieds, and consider hermitian and unitary matrices with respect to $k'/k$. Denote by $a^* \in M_{nm}(k')$ the conjugate transpose of a matrix $a \in M_{mn}(k')$ and by $j_m$ the matrix whose all anti-diagonal entries are $1$ and others are $0$ in $GL_m(k')$. Set
\begin{eqnarray*}
G = U(j_m) = \left\{g \in GL_m(k') \vert gj_mg^* = j_m \right\}, \quad K = G(\mathcal{O}_{k'}),\\
X = \left\{x \in G \vert x^* = x, \; \Phi_{xj_m}(t) = \Phi_{j_m}(t) \right\} \left(= G \cap G(\overline{k})\cdot 1_m \right),
\end{eqnarray*}
where $\Phi_y(t)$ is the characteristic polynomial of matrix $y$. The group $G$ acts on $X$ by $g \cdot x = gxg^*$.
We analyse the $G$-space $X$ on the basis of {\it spherical functions} on $X$, which are $K$-invariant common eigenfunctions on $X$ with respect to Hecke algebra $\mathcal{H}(G, K)$.