Let \(G\) be a reductive group over a non-archimedean local field \(F\) of residual characteristic \(p\neq 2\), let \(\theta\) be an involution of \(G\) over \(F\) and let \(H\) be the connected component of the \(\theta\)-fixed subgroup of \(G\). We are interested in the problem of distinction of the Steinberg representation \(\mathrm{St}_{G}\) of \(G\) restricted to \(H\). More precisely, first we give a reasonable upper bound of the dimension of the complex vector space
\[\mathrm{Hom}_{H}(\mathrm{St}_{G}|_{H},\mathbb{C})\]
which was previously known to be finite, and secondly we calculate this dimension for special symmetric pairs \((G,H)\). For instance, the most interesting case for us is when \(G\) is a general linear group and \(H\) is an orthogonal subgroup of \(G\).

Our method follows from the previous result of Broussous--Court\`es on Prasad's conjecture. The basic idea is to realize \(\mathrm{St}_{G}\) as the \(G\)-space of complex harmonic cochains on the Bruhat--Tits building of \(G\). Thus the problem is somehow reduced to the combinatorial geometry of Bruhat--Tits building. This is a joint work with Chuijia Wang.