Distinction of the Steinberg representation with respect to a symmetric pair

11.06.2024 13:15 - 14:45

Jiandi Zou (U Vienna)

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
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.


H. Grobner, A. Minguez-Espallargas, A. Mellit


BZ 9, 9. OG, OMP1